Reduction of the optimal tracking problem in the presence of noise

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Abstract

In this paper, the decomposition method based on the theory of fast and slow integral manifolds is used to analyze the optimal tracking problem. We consider a singularly perturbed optimal tracking problem with a given reference trajectory in the case of incomplete information about the state vector in the presence of random external perturbations.

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Введение

Известно, что линейно-квадратичная задача слежения ставится следующим образом (см., например, [1]). Рассматривается управляемая система вида

X ˙ =A(t)X+B(t)u+F(t), X( t 0 )= X 0 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGabmiwayaacaGaaGypamrr1ngBPrwtHrhAYaqeguuD JXwAKbstHrhAGq1DVbacfaGae8hGWhKaaGikaiaadshacaaIPaGaam iwaiabgUcaRiab=fi8cjaaiIcacaWG0bGaaGykaiaadwhacqGHRaWk caWGgbGaaGikaiaadshacaaIPaGaaGilaiaaiccacaaIGaGaamiwai aaiIcacaWG0bWaaSbaaSqaaiaaicdaaeqaaOGaaGykaiaai2dacaWG ybWaaSbaaSqaaiaaicdaaeqaaOGaaGilaaaa@5CC0@ (1)

Y=(t)X. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamywaiaai2datuuDJXwAK1uy0HMmaeHbfv3ySLgz G0uy0HgiuD3BaGqbaiab=jqidjaaiIcacaWG0bGaaGykaiaadIfaca aIUaaaaa@4829@ (2)

Здесь X MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaaaa@38E0@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ вектор состояния системы, вектор Y MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamywaaaa@38E1@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ вектор наблюдаемых параметров, u MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyDaaaa@38FD@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ вектор управляющих параметров, F MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOraaaa@38CE@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ вектор внешних возмущений. Эталонное движение задается в явном виде ξ=ξ(t) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOVdGNaaGypaiabe67a4jaaiIcacaWG0bGaaGyk aaaa@3EAE@ , а функционал качества имеет вид

J= t 0 t 1 (t)X(t))ξ(t) T (t) (t)X(t))ξ(t) + u T (t)(t)u(t) dt MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOsaiaai2dadaWdXbqabSqaaiaadshadaWgaaqa aiaaicdaaeqaaaqaaiaadshadaWgaaqaaiaaigdaaeqaaaqdcqGHRi I8aOWaamWaaeaadaqadaqaamrr1ngBPrwtHrhAYaqeguuDJXwAKbst HrhAGq1DVbacfaGae8NaHmKaaGikaiaadshacaaIPaGaamiwaiaaiI cacaWG0bGaaGykaiaaiMcacqGHsislcqaH+oaEcaaIOaGaamiDaiaa iMcaaiaawIcacaGLPaaadaahaaWcbeqaaiaadsfaaaGccqWFAesuca aIOaGaamiDaiaaiMcadaqadaqaaiab=jqidjaaiIcacaWG0bGaaGyk aiaadIfacaaIOaGaamiDaiaaiMcacaaIPaGaeyOeI0IaeqOVdGNaaG ikaiaadshacaaIPaaacaGLOaGaayzkaaGaey4kaSIaamyDamaaCaaa leqabaGaamivaaaakiaaiIcacaWG0bGaaGykaiab=1risjaaiIcaca WG0bGaaGykaiaadwhacaaIOaGaamiDaiaaiMcaaiaawUfacaGLDbaa caWGKbGaamiDaaaa@795D@ (3)

или

J= t 0 t 1 Y(t)ξ(t) T (t) Y(t)ξ(t) + u T (t)(t)u(t) dt. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOsaiaai2dadaWdXbqabSqaaiaadshadaWgaaqa aiaaicdaaeqaaaqaaiaadshadaWgaaqaaiaaigdaaeqaaaqdcqGHRi I8aOWaamWaaeaadaqadaqaaiaadMfacaaIOaGaamiDaiaaiMcacqGH sislcqaH+oaEcaaIOaGaamiDaiaaiMcaaiaawIcacaGLPaaadaahaa WcbeqaaiaadsfaaaWefv3ySLgznfgDOjdaryqr1ngBPrginfgDObcv 39gaiuaakiab=PrirjaaiIcacaWG0bGaaGykamaabmaabaGaamywai aaiIcacaWG0bGaaGykaiabgkHiTiabe67a4jaaiIcacaWG0bGaaGyk aaGaayjkaiaawMcaaiabgUcaRiaadwhadaahaaWcbeqaaiaadsfaaa GccaaIOaGaamiDaiaaiMcacqWFDeIucaaIOaGaamiDaiaaiMcacaWG 1bGaaGikaiaadshacaaIPaaacaGLBbGaayzxaaGaamizaiaadshaca aIUaaaaa@721F@ (4)

В дальнейшем будем предполагать, что все матричные функции, входящие в (1) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ (3) непрерывно дифференцируемы при t[ t 0 , t 1 ] MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiDaiabgIGiolaaiUfacaWG0bWaaSbaaSqaaiaa icdaaeqaaOGaaGilaiaadshadaWgaaWcbaGaaGymaaqabaGccaaIDb aaaa@40D5@ , тогда решение данной задачи дается следующей формулой для оптимального управления 

u opt = 1 B T (Px+χ). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyDamaaBaaaleaacaWGVbGaamiCaiaadshaaeqa aOGaaGypaiabgkHiTmrr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq 1DVbacfaGae8xhHi1aaWbaaSqabeaacqGHsislcaaIXaaaaOGae8xG Wl0aaWbaaSqabeaacaWGubaaaOGaaGikaiaadcfacaWG4bGaey4kaS Iaeq4XdmMaaGykaiaai6caaaa@5430@

Здесь P MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuaaaa@38D8@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ решение матричного дифференциального уравнения Риккати 

P ˙ +PA+ A T PPSP+M=0, P( t 1 )=0, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGabmiuayaacaGaey4kaSIaamiuamrr1ngBPrwtHrhA YaqeguuDJXwAKbstHrhAGq1DVbacfaGae8hGWhKaey4kaSIae8hGWh 0aaWbaaSqabeaacaWGubaaaOGaamiuaiabgkHiTiaadcfacqWFsc=u caWGqbGaey4kaSIae8hJW3KaaGypaiaaicdacaaISaGaaGiiaiaaic cacaWGqbGaaGikaiaadshadaWgaaWcbaGaaGymaaqabaGccaaIPaGa aGypaiaaicdacaaISaaaaa@5D75@

S=B 1 B T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFsc=ucaaI9aGae8xGWlKae8xhHi1aaWbaaSqabeaacqGHsi slcaaIXaaaaOGae8xGWl0aaWbaaSqabeaacaWGubaaaaaa@4D31@ , M= T MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqb a8aacqWFmcFtcaaI9aGae8NaHm0aaWbaaSqabeaacaWGubaaaOGae8 NgHeLae8NaHmeaaa@48B2@ , а χ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdmgaaa@39BA@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ решение линейной дифференциальной системы

χ ˙ = (ASP) T χ+ T ξPF=0, χ( t 1 )=0. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGafq4XdmMbaiaacaaI9aGaeyOeI0IaaGikamrr1ngB PrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbacfaGae8hGWhKaeyOeI0 Iae8NKWpLaamiuaiaaiMcadaahaaWcbeqaaiaadsfaaaGccqaHhpWy cqGHRaWkcqWFceYqdaahaaWcbeqaaiaadsfaaaGccqWFAesucqaH+o aEcqGHsislcaWGqbGaamOraiaai2dacaaIWaGaaGilaiaaiccacaaI GaGaeq4XdmMaaGikaiaadshadaWgaaWcbaGaaGymaaqabaGccaaIPa GaaGypaiaaicdacaaIUaaaaa@62A0@

Рассмотрим управляемую систему вида

ε x ¨ A(t)xH(t) x ˙ =B(t)u+f(t). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTduMabmiEayaadaGaeyOeI0IaamyqaiaaiIca caWG0bGaaGykaiaadIhacqGHsislcaWGibGaaGikaiaadshacaaIPa GabmiEayaacaGaaGypaiaadkeacaaIOaGaamiDaiaaiMcacaWG1bGa ey4kaSIaamOzaiaaiIcacaWG0bGaaGykaiaai6caaaa@4EA6@

y=C(t)x, x( t 0 )= x 10 , x ˙ ( t 0 )= x 20 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyEaiaai2dacaWGdbGaaGikaiaadshacaaIPaGa amiEaiaaiYcacaaIGaGaaGiiaiaadIhacaaIOaGaamiDamaaBaaale aacaaIWaaabeaakiaaiMcacaaI9aGaamiEamaaBaaaleaacaaIXaGa aGimaaqabaGccaaISaGaaGiiaiaaiccaceWG4bGbaiaacaaIOaGaam iDamaaBaaaleaacaaIWaaabeaakiaaiMcacaaI9aGaamiEamaaBaaa leaacaaIYaGaaGimaaqabaGccaaIUaaaaa@5235@

Здесь x n , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiEaiabgIGioprr1ngBPrwtHrhAYaqeguuDJXwA KbstHrhAGq1DVbacfaGae8xhHi1aaWbaaSqabeaacaWGUbaaaOGaaG ilaaaa@471C@ эталонное движение задается функцией ξ=ξ(t) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOVdGNaaGypaiabe67a4jaaiIcacaWG0bGaaGyk aaaa@3EAE@ , а функционал качества имеет вид

null (5)

Полагая x ˙ = x 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGabmiEayaacaGaaGypaiaadIhadaWgaaWcbaGaaGym aaqabaaaaa@3BB4@ , приходим к задаче вида (1) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ (3) при

X= x x 1 , A= 0 I ε 1 A ε 1 H , B= 0 ε 1 B , = C 0 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiwaiaai2dadaqadaqaauaabeqaceaaaeaacaWG 4baabaGaamiEamaaBaaaleaacaaIXaaabeaaaaaakiaawIcacaGLPa aacaaISaGaaGiiaiaaiccatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy 0HgiuD3BaGqbaiab=bi8bjaai2dadaqadaqaauaabeqaciaaaeaaca aIWaaabaGaamysaaqaaiabew7aLnaaCaaaleqabaGaeyOeI0IaaGym aaaakiaadgeaaeaacqaH1oqzdaahaaWcbeqaaiabgkHiTiaaigdaaa GccaWGibaaaaGaayjkaiaawMcaaiaaiYcacaaIGaGaaGiiaiab=fi8 cjaai2dadaqadaqaauaabeqaceaaaeaacaaIWaaabaGaeqyTdu2aaW baaSqabeaacqGHsislcaaIXaaaaOGaamOqaaaaaiaawIcacaGLPaaa caaISaGaaGiiaiaaiccacqWFceYqcaaI9aWaaeWaaeaafaqabeqaca aabaGaam4qaaqaaiaaicdaaaaacaGLOaGaayzkaaGaaGilaiaaicca caaIGaaaaa@6D16@

null

Представим P MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuaaaa@38D8@ и χ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdmgaaa@39BA@ в следующем виде:

P= P 1 ε P 2 ε P 2 T ε P 3 , χ= χ 1 ε χ 2 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuaiaai2dadaqadaqaauaabeqaciaaaeaacaWG qbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeqyTduMaamiuamaaBaaale aacaaIYaaabeaaaOqaaiabew7aLjaadcfadaqhaaWcbaGaaGOmaaqa aiaadsfaaaaakeaacqaH1oqzcaWGqbWaaSbaaSqaaiaaiodaaeqaaa aaaOGaayjkaiaawMcaaiaaiYcacaaIGaGaaGiiaiabeE8aJjaai2da daqadaqaauaabeqaceaaaeaacqaHhpWydaWgaaWcbaGaaGymaaqaba aakeaacqaH1oqzcqaHhpWydaWgaaWcbaGaaGOmaaqabaaaaaGccaGL OaGaayzkaaGaaGOlaaaa@55F1@

Тогда для матриц P 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIXaaabeaaaaa@39BF@ , P 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIYaaabeaaaaa@39C0@ , P 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIZaaabeaaaaa@39C1@ получается нелинейная система матричных уравнений вида

P ˙ 1 = P 2 A A T P 2 T + P 2 S P 2 T M 1 = F 1 ( P 1 , P 2 ,t,ε), ε P ˙ 2 = P 1 P 2 H A T P 3 + P 2 S P 3 =f( P 1 , P 2 , P 3 ,t,ε), ε P ˙ 3 = P 3 H A 4 T P 3 + P 3 S P 3 ε( P 2 T + P 2 )= F 3 ( P 2 , P 3 ,t,ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaqbaeGabmWaaaqaaiqadcfagaGaamaaBaaaleaacaaI Xaaabeaakiaai2dacqGHsislcaWGqbWaaSbaaSqaaiaaikdaaeqaaO GaamyqaiabgkHiTiaadgeadaahaaWcbeqaaiaadsfaaaGccaWGqbWa a0baaSqaaiaaikdaaeaacaWGubaaaOGaey4kaSIaamiuamaaBaaale aacaaIYaaabeaakiaadofacaWGqbWaa0baaSqaaiaaikdaaeaacaWG ubaaaOGaeyOeI0IaamytamaaBaaaleaacaaIXaaabeaakiaai2daca WGgbWaaSbaaSqaaiaaigdaaeqaaOGaaGikaiaadcfadaWgaaWcbaGa aGymaaqabaGccaaISaGaamiuamaaBaaaleaacaaIYaaabeaakiaaiY cacaWG0bGaaGilaiabew7aLjaaiMcacaaISaaabaaabaaabaGaeqyT duMabmiuayaacaWaaSbaaSqaaiaaikdaaeqaaOGaaGypaiabgkHiTi aadcfadaWgaaWcbaGaaGymaaqabaGccqGHsislcaWGqbWaaSbaaSqa aiaaikdaaeqaaOGaamisaiabgkHiTiaadgeadaahaaWcbeqaaiaads faaaGccaWGqbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamiuamaa BaaaleaacaaIYaaabeaakiaadofacaWGqbWaaSbaaSqaaiaaiodaae qaaOGaaGypaiaadAgacaaIOaGaamiuamaaBaaaleaacaaIXaaabeaa kiaaiYcacaWGqbWaaSbaaSqaaiaaikdaaeqaaOGaaGilaiaadcfada WgaaWcbaGaaG4maaqabaGccaaISaGaamiDaiaaiYcacqaH1oqzcaaI PaGaaGilaaqaaaqaaaqaaiabew7aLjqadcfagaGaamaaBaaaleaaca aIZaaabeaakiaai2dacqGHsislcaWGqbWaaSbaaSqaaiaaiodaaeqa aOGaamisaiabgkHiTiaadgeadaqhaaWcbaGaaGinaaqaaiaadsfaaa GccaWGqbWaaSbaaSqaaiaaiodaaeqaaOGaey4kaSIaamiuamaaBaaa leaacaaIZaaabeaakiaadofacaWGqbWaaSbaaSqaaiaaiodaaeqaaO GaeyOeI0IaeqyTduMaaGikaiaadcfadaqhaaWcbaGaaGOmaaqaaiaa dsfaaaGccqGHRaWkcaWGqbWaaSbaaSqaaiaaikdaaeqaaOGaaGykai aai2dacaWGgbWaaSbaaSqaaiaaiodaaeqaaOGaaGikaiaadcfadaWg aaWcbaGaaGOmaaqabaGccaaISaGaamiuamaaBaaaleaacaaIZaaabe aakiaaiYcacaWG0bGaaGilaiabew7aLjaaiMcaaeaaaeaaaaaaaa@A400@ (6)

с граничными условиями

P 1 ( t 1 )=0, P 2 ( t 1 )=0, P 3 ( t 1 )=0, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIXaaabeaakiaaiIcacaWG 0bWaaSbaaSqaaiaaigdaaeqaaOGaaGykaiaai2dacaaIWaGaaGilai aayIW7caaMi8UaaGjcVlaadcfadaWgaaWcbaGaaGOmaaqabaGccaaI OaGaamiDamaaBaaaleaacaaIXaaabeaakiaaiMcacaaI9aGaaGimai aaiYcacaaMi8UaaGjcVlaayIW7caWGqbWaaSbaaSqaaiaaiodaaeqa aOGaaGikaiaadshadaWgaaWcbaGaaGymaaqabaGccaaIPaGaaGypai aaicdacaaISaaaaa@5750@

а система уравнений для χ 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaaaa@3AA1@ и χ 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaSbaaSqaaiaaikdaaeqaaaaa@3AA2@ имеет вид

χ ˙ 1 = (AS P 2 T ) T χ 2 + C 1 T Qξ P 2 f, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGafq4XdmMbaiaadaWgaaWcbaGaaGymaaqabaGccaaI 9aGaeyOeI0IaaGikaiaadgeacqGHsislcaWGtbGaamiuamaaDaaale aacaaIYaaabaGaamivaaaakiaaiMcadaahaaWcbeqaaiaadsfaaaGc cqaHhpWydaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcaWGdbWaa0baaS qaaiaaigdaaeaacaWGubaaaOGaamyuaiabe67a4jabgkHiTiaadcfa daWgaaWcbaGaaGOmaaqabaGccaWGMbGaaGilaaaa@5115@ (7)

ε χ ˙ 2 = χ 1 (HS P 3 ) T χ 2 P 3 f MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTduMafq4XdmMbaiaadaWgaaWcbaGaaGOmaaqa baGccaaI9aGaeyOeI0Iaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaOGaey OeI0IaaGikaiaadIeacqGHsislcaWGtbGaamiuamaaBaaaleaacaaI ZaaabeaakiaaiMcadaahaaWcbeqaaiaadsfaaaGccqaHhpWydaWgaa WcbaGaaGOmaaqabaGccqGHsislcaWGqbWaaSbaaSqaaiaaiodaaeqa aOGaamOzaaaa@4EBD@ (8)

с граничными условиями

χ 1 ( t 1 )=0, χ 2 ( t 1 )=0. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaOGaaGikaiaa dshadaWgaaWcbaGaaGymaaqabaGccaaIPaGaaGypaiaaicdacaaISa GaaGjcVlaayIW7caaMi8Uaeq4Xdm2aaSbaaSqaaiaaikdaaeqaaOGa aGikaiaadshadaWgaaWcbaGaaGymaaqabaGccaaIPaGaaGypaiaaic dacaaIUaaaaa@4D15@

Для анализа задач управления с сингулярными возмущениями обычно применяется метод пограничных функций Васильевой (см. обзоры [2-4]). В настоящей статье будет применен метод декомпозиции [5]. Суть метода декомпозиции состоит в следующем. При некоторых естественных предположениях о гладкости и нормальной гиперболичности сингулярно возмущенная система

X ˙ =F(X,Y,t,ε),ε Y ˙ =G(X,Y,t,ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGabmiwayaacaGaaGypaiaadAeacaaIOaGaamiwaiaa iYcacaWGzbGaaGilaiaadshacaaISaGaeqyTduMaaGykaiaaiYcacq aH1oqzceWGzbGbaiaacaaI9aGaam4raiaaiIcacaWGybGaaGilaiaa dMfacaaISaGaamiDaiaaiYcacqaH1oqzcaaIPaaaaa@4F16@

преобразованием

Y=Z+L(X,t,ε), X+V+εΠ(V,Z,t,ε) (Π(V,0,t,ε)0) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamywaiaai2dacaWGAbGaey4kaSIaamitaiaaiIca caWGybGaaGilaiaadshacaaISaGaeqyTduMaaGykaiaaiYcacaaIGa GaaGiiaiaadIfacqGHRaWkcaWGwbGaey4kaSIaeqyTduMaeuiOdaLa aGikaiaadAfacaaISaGaamOwaiaaiYcacaWG0bGaaGilaiabew7aLj aaiMcacaaIGaGaaGiiaiaaiIcacqqHGoaucaaIOaGaamOvaiaaiYca caaIWaGaaGilaiaadshacaaISaGaeqyTduMaaGykaiabggMi6kaaic dacaaIPaaaaa@618A@

приводится к виду

V ˙ =F(V,L(V,t,ε), ε Z ˙ =W((V,Z,t,ε) (W(V,0,t,ε)0), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGabmOvayaacaGaaGypaiaadAeacaaIOaGaamOvaiaa iYcacaWGmbGaaGikaiaadAfacaaISaGaamiDaiaaiYcacqaH1oqzca aIPaGaaGilaiaaiccacaaIGaGaeqyTduMabmOwayaacaGaaGypaiaa dEfacaaIOaGaaGikaiaadAfacaaISaGaamOwaiaaiYcacaWG0bGaaG ilaiabew7aLjaaiMcacaaIGaGaaGiiaiaaiIcacaWGxbGaaGikaiaa dAfacaaISaGaaGimaiaaiYcacaWG0bGaaGilaiabew7aLjaaiMcacq GHHjIUcaaIWaGaaGykaiaaiYcaaaa@6132@

в котором первое уравнение не зависит от быстрой переменной, а решениями второго уравнения являются так называемые правые пограничные функции, для которых справедливы оценки типа Z(t,ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaarqqr1ngBPrgifHhDYfgaiuaapaGae8xjIaLaamOwaiaa iIcacaWG0bGaaGilaiabew7aLjaaiMcacqWFLicucqGHKjYOaaa@4626@ Cexp c(t t 1 )/ε MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyizImQaam4qaiGacwgacaGG4bGaaiiCamaabmaa baGaam4yaiaaiIcacaWG0bGaeyOeI0IaamiDamaaBaaaleaacaaIXa aabeaakiaaiMcacaaIVaGaeqyTdugacaGLOaGaayzkaaaaaa@4761@ , t 0 t t 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiDamaaBaaaleaacaaIWaaabeaakiabgsMiJkaa dshacqGHKjYOcaWG0bWaaSbaaSqaaiaaigdaaeqaaaaa@402F@ при не зависящих от малого параметра числах c MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4yaaaa@38EB@ и C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4qaaaa@38CB@ ( 0<c MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGimaiaaiYdacaWGJbaaaa@3A6B@ , 1C MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGymaiabgsMiJkaadoeaaaa@3B3B@ ). При этом L MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamitaaaa@38D4@ соответствует медленному интегральному многообразию исходной системы, а εΠ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTduMaeuiOdafaaa@3B28@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ быстрому многообразию некоторой вспомогательной системы. При этом переменная V MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvaaaa@38DE@ соответствует регулярной составляющей решения исходной системы, а переменная Z MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwaaaa@38E2@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFuaca aa@3C73@ погранслойной составляющей. Важно отметить, что если Z=O(ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwaiaai2dacaWGpbGaaGikaiabew7aLjaaiMca aaa@3D89@ при t= t 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiDaiaai2dacaWG0bWaaSbaaSqaaiaaigdaaeqa aaaa@3BA3@ , то и функция Z MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwaaaa@38E2@ содержит в качестве множителя малый параметр. Матричная функция L MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamitaaaa@38D4@ удовлетворяет так называемому уравнению инвариантности

ε L t +ε L X F(X,L,t,ε)=G(X,L,t,ε). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTdu2aaSaaaeaacqGHciITcaWGmbaabaGaeyOa IyRaamiDaaaacqGHRaWkcqaH1oqzdaWcaaqaaiabgkGi2kaadYeaae aacqGHciITcaWGybaaaiaadAeacaaIOaGaamiwaiaaiYcacaWGmbGa aGilaiaadshacaaISaGaeqyTduMaaGykaiaai2dacaWGhbGaaGikai aadIfacaaISaGaamitaiaaiYcacaWG0bGaaGilaiabew7aLjaaiMca caaIUaaaaa@5823@

Следует отметить, что применение в реальных системах управления управляющих воздействий с использованием пограничных функций далеко не всегда целесообразно, так как предполагает резкое изменение напряжения в цепях управления на очень коротком промежутке времени. С другой стороны, отказ от использования таких функций может незначительно сказываться на погрешности функционала качества. Ниже будет показано, что субоптимальное управление, не содержащее правых пограничных функций, приводит к погрешности порядка O( ε 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4taiaaiIcacqaH1oqzdaahaaWcbeqaaiaaikda aaGccaaIPaaaaa@3CD6@ в функционале (5), что вполне приемлемо с прикладной точки зрения.

1. Оценка погрешности функционала

Для оценки погрешности при построении субоптимального управления функционал (3) можно представить в следующем виде:

J= 0 t f C(t)x(t)ξ(t) T Q(t) C(t)x(t))ξ(t) + u T (t)R(t)u(t) dt= MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOsaiaai2dadaWdXbqabSqaaiaaicdaaeaacaWG 0bWaaSbaaeaacaWGMbaabeaaa0Gaey4kIipakmaadmaabaWaaeWaae aacaWGdbGaaGikaiaadshacaaIPaGaamiEaiaaiIcacaWG0bGaaGyk aiabgkHiTiabe67a4jaaiIcacaWG0bGaaGykaaGaayjkaiaawMcaam aaCaaaleqabaGaamivaaaakiaadgfacaaIOaGaamiDaiaaiMcadaqa daqaaiaadoeacaaIOaGaamiDaiaaiMcacaWG4bGaaGikaiaadshaca aIPaGaaGykaiabgkHiTiabe67a4jaaiIcacaWG0bGaaGykaaGaayjk aiaawMcaaiabgUcaRiaadwhadaahaaWcbeqaaiaadsfaaaGccaaIOa GaamiDaiaaiMcacaWGsbGaaGikaiaadshacaaIPaGaamyDaiaaiIca caWG0bGaaGykaaGaay5waiaaw2faaiaadsgacaWG0bGaaGypaaaa@6E65@

= 0 t f u+ R 1 B T (t)P(t)x(t)+ R 1 B T (t)χ(t) T R(t) u+ R 1 B T (t)P(t)x(t)+ R 1 B T (t)χ(t) dt+ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGypamaapehabeWcbaGaaGimaaqaaiaadshadaWg aaqaaiaadAgaaeqaaaqdcqGHRiI8aOWaamWaaeaadaqadaqaaiaadw hacqGHRaWkcaWGsbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaamOq amaaCaaaleqabaGaamivaaaakiaaiIcacaWG0bGaaGykaiaadcfaca aIOaGaamiDaiaaiMcacaWG4bGaaGikaiaadshacaaIPaGaey4kaSIa amOuamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaadkeadaahaaWcbe qaaiaadsfaaaGccaaIOaGaamiDaiaaiMcacqaHhpWycaaIOaGaamiD aiaaiMcaaiaawIcacaGLPaaadaahaaWcbeqaaiaadsfaaaGccaWGsb GaaGikaiaadshacaaIPaWaaeWaaeaacaWG1bGaey4kaSIaamOuamaa CaaaleqabaGaeyOeI0IaaGymaaaakiaadkeadaahaaWcbeqaaiaads faaaGccaaIOaGaamiDaiaaiMcacaWGqbGaaGikaiaadshacaaIPaGa amiEaiaaiIcacaWG0bGaaGykaiabgUcaRiaadkfadaahaaWcbeqaai abgkHiTiaaigdaaaGccaWGcbWaaWbaaSqabeaacaWGubaaaOGaaGik aiaadshacaaIPaGaeq4XdmMaaGikaiaadshacaaIPaaacaGLOaGaay zkaaaacaGLBbGaayzxaaGaamizaiaadshacqGHRaWkaaa@8078@

+ x T (0)P(0)x(0)+2 x T (0)χ(0)+κ(0). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4kaSIaamiEamaaCaaaleqabaGaamivaaaakiaa iIcacaaIWaGaaGykaiaadcfacaaIOaGaaGimaiaaiMcacaWG4bGaaG ikaiaaicdacaaIPaGaey4kaSIaaGOmaiaadIhadaahaaWcbeqaaiaa dsfaaaGccaaIOaGaaGimaiaaiMcacqaHhpWycaaIOaGaaGimaiaaiM cacqGHRaWkcqaH6oWAcaaIOaGaaGimaiaaiMcacaaIUaaaaa@522C@

Здесь

κ ˙ = χ T Sχ ξ T Qξ2 f T χ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGafqOUdSMbaiaacaaI9aGaeq4Xdm2aaWbaaSqabeaa caWGubaaaOGaam4uaiabeE8aJjabgkHiTiabe67a4naaCaaaleqaba GaamivaaaakiaadgfacqaH+oaEcqGHsislcaaIYaGaamOzamaaCaaa leqabaGaamivaaaakiabeE8aJbaa@4B8F@

с условием на конце рассматриваемого промежутка κ( t f )=0, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOUdSMaaGikaiaadshadaWgaaWcbaGaamOzaaqa baGccaaIPaGaaGypaiaaicdacaaISaaaaa@3F6B@ т. е.

κ(0)= 0 t f χ T Sχ ξ T Qξ2 f T χ dt. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOUdSMaaGikaiaaicdacaaIPaGaaGypaiabgkHi TmaapehabeWcbaGaaGimaaqaaiaadshadaWgaaqaaiaadAgaaeqaaa qdcqGHRiI8aOWaamWaaeaacqaHhpWydaahaaWcbeqaaiaadsfaaaGc caWGtbGaeq4XdmMaeyOeI0IaeqOVdG3aaWbaaSqabeaacaWGubaaaO Gaamyuaiabe67a4jabgkHiTiaaikdacaWGMbWaaWbaaSqabeaacaWG ubaaaOGaeq4XdmgacaGLBbGaayzxaaGaamizaiaadshacaaIUaaaaa@5851@

Для доказательства этого факта достаточно использовать непосредственно проверяемое равенство

C(t)x(t)ξ(t) T Q(t) C(t)x(t))ξ(t) + u T (t)R(t)u(t)= MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaeWaaeaacaWGdbGaaGikaiaadshacaaIPaGaamiE aiaaiIcacaWG0bGaaGykaiabgkHiTiabe67a4jaaiIcacaWG0bGaaG ykaaGaayjkaiaawMcaamaaCaaaleqabaGaamivaaaakiaadgfacaaI OaGaamiDaiaaiMcadaqadaqaaiaadoeacaaIOaGaamiDaiaaiMcaca WG4bGaaGikaiaadshacaaIPaGaaGykaiabgkHiTiabe67a4jaaiIca caWG0bGaaGykaaGaayjkaiaawMcaaiabgUcaRiaadwhadaahaaWcbe qaaiaadsfaaaGccaaIOaGaamiDaiaaiMcacaWGsbGaaGikaiaadsha caaIPaGaamyDaiaaiIcacaWG0bGaaGykaiaai2daaaa@63C8@

= u+ R 1 B T (t)x(t)+ R 1 B T (t)χ(t) T R(t) u+ R 1 B T (t)x(t)+ R 1 B T (t)χ(t) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGypamaabmaabaGaamyDaiabgUcaRiaadkfadaah aaWcbeqaaiabgkHiTiaaigdaaaGccaWGcbWaaWbaaSqabeaacaWGub aaaOGaaGikaiaadshacaaIPaGaamiEaiaaiIcacaWG0bGaaGykaiab gUcaRiaadkfadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWGcbWaaW baaSqabeaacaWGubaaaOGaaGikaiaadshacaaIPaGaeq4XdmMaaGik aiaadshacaaIPaaacaGLOaGaayzkaaWaaWbaaSqabeaacaWGubaaaO GaamOuaiaaiIcacaWG0bGaaGykamaabmaabaGaamyDaiabgUcaRiaa dkfadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWGcbWaaWbaaSqabe aacaWGubaaaOGaaGikaiaadshacaaIPaGaamiEaiaaiIcacaWG0bGa aGykaiabgUcaRiaadkfadaahaaWcbeqaaiabgkHiTiaaigdaaaGcca WGcbWaaWbaaSqabeaacaWGubaaaOGaaGikaiaadshacaaIPaGaeq4X dmMaaGikaiaadshacaaIPaaacaGLOaGaayzkaaGaeyOeI0caaa@7116@

d dt x T (t)P(t)x(t)+2 x T (t)χ(t)+κ(t) . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOeI0YaaSaaaeaacaWGKbaabaGaamizaiaadsha aaWaaeWaaeaacaWG4bWaaWbaaSqabeaacaWGubaaaOGaaGikaiaads hacaaIPaGaamiuaiaaiIcacaWG0bGaaGykaiaadIhacaaIOaGaamiD aiaaiMcacqGHRaWkcaaIYaGaamiEamaaCaaaleqabaGaamivaaaaki aaiIcacaWG0bGaaGykaiabeE8aJjaaiIcacaWG0bGaaGykaiabgUca RiabeQ7aRjaaiIcacaWG0bGaaGykaaGaayjkaiaawMcaaiaai6caaa a@5815@

Легко видеть, что минимальное значение J opt MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOsamaaBaaaleaacaWGVbGaamiCaiaadshaaeqa aaaa@3BE0@ определяется равенством

J opt = x T (0)P(0)x(0)+2 x T (0)χ(0)+κ(0). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOsamaaBaaaleaacaWGVbGaamiCaiaadshaaeqa aOGaaGypaiaadIhadaahaaWcbeqaaiaadsfaaaGccaaIOaGaaGimai aaiMcacaWGqbGaaGikaiaaicdacaaIPaGaamiEaiaaiIcacaaIWaGa aGykaiabgUcaRiaaikdacaWG4bWaaWbaaSqabeaacaWGubaaaOGaaG ikaiaaicdacaaIPaGaeq4XdmMaaGikaiaaicdacaaIPaGaey4kaSIa eqOUdSMaaGikaiaaicdacaaIPaGaaGOlaaaa@55F8@

Пусть каким-либо способом построено субоптимальное управление

u s = R 1 B T ( P s x s + χ s ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyDamaaBaaaleaacaWGZbaabeaakiaai2dacqGH sislcaWGsbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaamOqamaaCa aaleqabaGaamivaaaakiaaiIcacaWGqbWaaSbaaSqaaiaadohaaeqa aOGaamiEamaaBaaaleaacaWGZbaabeaakiabgUcaRiabeE8aJnaaBa aaleaacaWGZbaabeaakiaaiMcaaaa@49C6@

с соответствующими приближенными выражениями для вектора состояния ( x s MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiEamaaBaaaleaacaWGZbaabeaaaaa@3A24@ ), коэффициента усиления ( P s MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaWGZbaabeaaaaa@39FC@ ) и вектора χ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdmgaaa@39BA@ ( χ s MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaSbaaSqaaiaadohaaeqaaaaa@3ADE@ ) (вместо индекса subopt MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaSbaaSqaaiaadohacaWG1bGaamOyaiaad+gacaWG WbGaamiDaaqabaaaaa@3DEA@ используется индекс s MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaWaaSbaaSqaaiaadohaaeqaaaaa@3927@ ). Введем следующие обозначения:

Δx= x s x opt , ΔP= P s P opt , Δχ= χ s χ opt , ΔJ= J s J opt . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiLdqKaamiEaiaai2dacaWG4bWaaSbaaSqaaiaa dohaaeqaaOGaeyOeI0IaamiEamaaBaaaleaacaWGVbGaamiCaiaads haaeqaaOGaaGilaiaaiccacqqHuoarcaWGqbGaaGypaiaadcfadaWg aaWcbaGaam4CaaqabaGccqGHsislcaWGqbWaaSbaaSqaaiaad+gaca WGWbGaamiDaaqabaGccaaISaGaaGiiaiabfs5aejabeE8aJjaai2da cqaHhpWydaWgaaWcbaGaam4CaaqabaGccqGHsislcqaHhpWydaWgaa WcbaGaam4BaiaadchacaWG0baabeaakiaaiYcacaaIGaGaeuiLdqKa amOsaiaai2dacaWGkbWaaSbaaSqaaiaadohaaeqaaOGaeyOeI0Iaam OsamaaBaaaleaacaWGVbGaamiCaiaadshaaeqaaOGaaGOlaaaa@6763@

Отсюда следует, что если вместо оптимального управления используется приближенное (субоптимальное) управление

u= R 1 B T ( P s x s + χ s ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyDaiaai2dacqGHsislcaWGsbWaaWbaaSqabeaa cqGHsislcaaIXaaaaOGaamOqamaaCaaaleqabaGaamivaaaakiaaiI cacaWGqbWaaSbaaSqaaiaadohaaeqaaOGaamiEamaaBaaaleaacaWG ZbaabeaakiabgUcaRiabeE8aJnaaBaaaleaacaWGZbaabeaakiaaiM cacaaISaaaaa@494E@

то возникающая при этом погрешность функционала качества ΔJ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiLdqKaamOsaaaa@3A38@ представима в следующем виде:

ΔJ= J s J opt = 0 t f [( R 1 B T (ΔP x s +Δχ )) T R( R 1 B T (ΔP x s +Δχ))]dt+ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiLdqKaamOsaiaai2dacaWGkbWaaSbaaSqaaiaa dohaaeqaaOGaeyOeI0IaamOsamaaBaaaleaacaWGVbGaamiCaiaads haaeqaaOGaaGypamaapehabeWcbaGaaGimaaqaaiaadshadaWgaaqa aiaadAgaaeqaaaqdcqGHRiI8aOGaaG4waiaaiIcacaWGsbWaaWbaaS qabeaacqGHsislcaaIXaaaaOGaamOqamaaCaaaleqabaGaamivaaaa kiaaiIcacqqHuoarcaWGqbGaamiEamaaBaaaleaacaWGZbaabeaaki abgUcaRiabfs5aejabeE8aJjaaiMcacaaIPaWaaWbaaSqabeaacaWG ubaaaOGaamOuaiaaiIcacaWGsbWaaWbaaSqabeaacqGHsislcaaIXa aaaOGaamOqamaaCaaaleqabaGaamivaaaakiaaiIcacqqHuoarcaWG qbGaamiEamaaBaaaleaacaWGZbaabeaakiabgUcaRiabfs5aejabeE 8aJjaaiMcacaaIPaGaaGyxaiaadsgacaWG0bGaey4kaScaaa@6DB9@

+ x s T (0) P s (0) x s (0)+2 x s T (0) χ s (0)+ κ s (0) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaey4kaSIaamiEamaaDaaaleaacaWGZbaabaGaamiv aaaakiaaiIcacaaIWaGaaGykaiaadcfadaWgaaWcbaGaam4Caaqaba GccaaIOaGaaGimaiaaiMcacaWG4bWaaSbaaSqaaiaadohaaeqaaOGa aGikaiaaicdacaaIPaGaey4kaSIaaGOmaiaadIhadaqhaaWcbaGaam 4CaaqaaiaadsfaaaGccaaIOaGaaGimaiaaiMcacqaHhpWydaWgaaWc baGaam4CaaqabaGccaaIOaGaaGimaiaaiMcacqGHRaWkcqaH6oWAda WgaaWcbaGaam4CaaqabaGccaaIOaGaaGimaiaaiMcacqGHsislaaa@5909@

[ x 0 T ( P s (0)ΔP(0)) x 0 +2 x 0 T ( χ s (0)Δχ(0))+ κ s (0)Δ κ s (0)]. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOeI0IaaG4waiaadIhadaqhaaWcbaGaaGimaaqa aiaadsfaaaGccaaIOaGaamiuamaaBaaaleaacaWGZbaabeaakiaaiI cacaaIWaGaaGykaiabgkHiTiabfs5aejaadcfacaaIOaGaaGimaiaa iMcacaaIPaGaamiEamaaBaaaleaacaaIWaaabeaakiabgUcaRiaaik dacaWG4bWaa0baaSqaaiaaicdaaeaacaWGubaaaOGaaGikaiabeE8a JnaaBaaaleaacaWGZbaabeaakiaaiIcacaaIWaGaaGykaiabgkHiTi abfs5aejabeE8aJjaaiIcacaaIWaGaaGykaiaaiMcacqGHRaWkcqaH 6oWAdaWgaaWcbaGaam4CaaqabaGccaaIOaGaaGimaiaaiMcacqGHsi slcqqHuoarcqaH6oWAdaWgaaWcbaGaam4CaaqabaGccaaIOaGaaGim aiaaiMcacaaIDbGaaGOlaaaa@6920@

Для краткости аргументы у функций под знаком интеграла опущены.

Полагая x s (0)= x 0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiEamaaBaaaleaacaWGZbaabeaakiaaiIcacaaI WaGaaGykaiaai2dacaWG4bWaaSbaaSqaaiaaicdaaeqaaaaa@3EF7@ и используя выражение для κ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqOUdSgaaa@39B5@ , получаем

ΔJ=0tf(ΔPxs+Δχ))TS(ΔPxs+Δχ))]dt++x0TΔP(0)x0+2x0TΔχ(0)+0tf2χTSΔχ2fTΔχdt. (9)

Следует заметить, что полученная формула не связана с конкретным выбором приближений и может применяться для оценки погрешности при применении как асимптотических, так и численных методов приближенного анализа.

Если, например, рассмотреть случай регулярной зависимости матричных и векторных функций, входящих в (1), (2) и (3) от малого параметра, в предположении, что эти функции достаточное число раз дифференцируемы по своим аргументам, то можно применить эту формулу для оценки погрешности функционала при применении простейшего варианта метода малого параметра. При этом проявляется некоторое отличие от задач оптимального управления, связанное с зависимостью погрешности от Δχ MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiLdqKaeq4Xdmgaaa@3B20@ .

В рассматриваемом случае формула (9) с учетом выражений

ΔP x s = Δ P 1 x 1s +εΔ P 2 x 2s εΔ P 2 x 1s +εΔ P 3 x 2s , Δχ= Δ χ 1 εΔ χ 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiLdqKaamiuaiaadIhadaWgaaWcbaGaam4Caaqa baGccaaI9aWaaeWaaeaafaqabeGabaaabaGaeuiLdqKaamiuamaaBa aaleaacaaIXaaabeaakiaadIhadaWgaaWcbaGaaGymaiaadohaaeqa aOGaey4kaSIaeqyTduMaeuiLdqKaamiuamaaBaaaleaacaaIYaaabe aakiaadIhadaWgaaWcbaGaaGOmaiaadohaaeqaaaGcbaGaeqyTduMa euiLdqKaamiuamaaBaaaleaacaaIYaaabeaakiaadIhadaWgaaWcba GaaGymaiaadohaaeqaaOGaey4kaSIaeqyTduMaeuiLdqKaamiuamaa BaaaleaacaaIZaaabeaakiaadIhadaWgaaWcbaGaaGOmaiaadohaae qaaaaaaOGaayjkaiaawMcaaiaaiYcacaaIGaGaaGiiaiabfs5aejab eE8aJjaai2dadaqadaqaauaabeqaceaaaeaacqqHuoarcqaHhpWyda WgaaWcbaGaaGymaaqabaaakeaacqaH1oqzcqqHuoarcqaHhpWydaWg aaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaaaaa@6F15@

имеет вид

ΔJ= 0 t f Δ 2 T S Δ 2 dt+ x 10 T Δ P 1 (0) x 10 +ε( x 20 T Δ P 2 (0) T x 10 + x 10 T Δ P 2 (0) x 20 + + x 20 T Δ P 3 (0) x 20 )+2 x 10 T Δ χ 1 (0)+2ε x 20 T Δ χ 2 (0)+2 0 t f ( χ 2 T SΔ χ 2 ε f T Δ χ 2 )dt. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaqbaeGabiWaaaqaaiabfs5aejaadQeacaaI9aWaa8qC aeqaleaacaaIWaaabaGaamiDamaaBaaabaGaamOzaaqabaaaniabgU IiYdGccqqHuoardaqhaaWcbaGaaGOmaaqaaiaadsfaaaGccaWGtbGa euiLdq0aaSbaaSqaaiaaikdaaeqaaOGaamizaiaadshacqGHRaWkca WG4bWaa0baaSqaaiaaigdacaaIWaaabaGaamivaaaakiabfs5aejaa dcfadaWgaaWcbaGaaGymaaqabaGccaaIOaGaaGimaiaaiMcacaWG4b WaaSbaaSqaaiaaigdacaaIWaaabeaakiabgUcaRiabew7aLjaaiIca caWG4bWaa0baaSqaaiaaikdacaaIWaaabaGaamivaaaakiabfs5aej aadcfadaWgaaWcbaGaaGOmaaqabaGccaaIOaGaaGimaiaaiMcadaah aaWcbeqaaiaadsfaaaGccaWG4bWaaSbaaSqaaiaaigdacaaIWaaabe aakiabgUcaRiaadIhadaqhaaWcbaGaaGymaiaaicdaaeaacaWGubaa aOGaeuiLdqKaamiuamaaBaaaleaacaaIYaaabeaakiaaiIcacaaIWa GaaGykaiaadIhadaWgaaWcbaGaaGOmaiaaicdaaeqaaOGaey4kaSca baaabaaabaGaey4kaSIaamiEamaaDaaaleaacaaIYaGaaGimaaqaai aadsfaaaGccqqHuoarcaWGqbWaaSbaaSqaaiaaiodaaeqaaOGaaGik aiaaicdacaaIPaGaamiEamaaBaaaleaacaaIYaGaaGimaaqabaGcca aIPaGaey4kaSIaaGOmaiaadIhadaqhaaWcbaGaaGymaiaaicdaaeaa caWGubaaaOGaeuiLdqKaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaOGaaG ikaiaaicdacaaIPaGaey4kaSIaaGOmaiabew7aLjaadIhadaqhaaWc baGaaGOmaiaaicdaaeaacaWGubaaaOGaeuiLdqKaeq4Xdm2aaSbaaS qaaiaaikdaaeqaaOGaaGikaiaaicdacaaIPaGaey4kaSIaaGOmamaa pehabeWcbaGaaGimaaqaaiaadshadaWgaaqaaiaadAgaaeqaaaqdcq GHRiI8aOGaaGikaiabeE8aJnaaDaaaleaacaaIYaaabaGaamivaaaa kiaadofacqqHuoarcqaHhpWydaWgaaWcbaGaaGOmaaqabaGccqGHsi slcqaH1oqzcaWGMbWaaWbaaSqabeaacaWGubaaaOGaeuiLdqKaeq4X dm2aaSbaaSqaaiaaikdaaeqaaOGaaGykaiaadsgacaWG0bGaaGOlaa qaaaqaaaaaaaa@B307@ (10)

Нетрудно видеть, что пренебрежение регулярными членами порядка O( ε 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4taiaaiIcacqaH1oqzdaahaaWcbeqaaiaaikda aaGccaaIPaaaaa@3CD6@ и правыми пограничными функциями, содержащими в качестве множителя малый параметр, в представлении переменных P 1 , P 2 , P 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG qbWaaSbaaSqaaiaaikdaaeqaaOGaaGilaiaadcfadaWgaaWcbaGaaG 4maaqabaaaaa@3EBA@ и χ 1 , χ 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaOGaaGilaiab eE8aJnaaBaaaleaacaaIYaaabeaaaaa@3E00@ приводит к погрешности порядка O( ε 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4taiaaiIcacqaH1oqzdaahaaWcbeqaaiaaikda aaGccaaIPaaaaa@3CD6@ в функционале качества.

2. Декомпозиция системы уравнений Риккати

Будем предполагать, что все собственные числа матрицы H MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamisaaaa@38D0@ на рассматриваемом отрезке имеют положительные вещественные части. Полагая в последних двух уравнениях системы матричных дифференциальных уравнений (6) малый параметр равным нулю, получим уравнения

0= P 1 P 2 H A T P 3 + P 2 S P 3 , 0= P 3 H H T P 3 + P 3 S P 3 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGimaiaai2dacqGHsislcaWGqbWaaSbaaSqaaiaa igdaaeqaaOGaeyOeI0IaamiuamaaBaaaleaacaaIYaaabeaakiaadI eacqGHsislcaWGbbWaaWbaaSqabeaacaWGubaaaOGaamiuamaaBaaa leaacaaIZaaabeaakiabgUcaRiaadcfadaWgaaWcbaGaaGOmaaqaba GccaWGtbGaamiuamaaBaaaleaacaaIZaaabeaakiaaiYcacaaIGaGa aGiiaiaaicdacaaI9aGaeyOeI0IaamiuamaaBaaaleaacaaIZaaabe aakiaadIeacqGHsislcaWGibWaaWbaaSqabeaacaWGubaaaOGaamiu amaaBaaaleaacaaIZaaabeaakiabgUcaRiaadcfadaWgaaWcbaGaaG 4maaqabaGccaWGtbGaamiuamaaBaaaleaacaaIZaaabeaakiaai6ca aaa@5B2D@

Отсюда следует, что медленное интегральное многообразие этой системы имеет вид

P 2 =Φ( P 1 ,t,ε)= P 1 H 1 +ε Φ 1 ( P 1 ,t)+ ε 2 , P 3 =εΨ( P 1 ,t,ε)=ε Ψ 1 ( P 1 ,t)+ ε 2 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIYaaabeaakiaai2dacqqH MoGrcaaIOaGaamiuamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG0b GaaGilaiabew7aLjaaiMcacaaI9aGaeyOeI0IaamiuamaaBaaaleaa caaIXaaabeaakiaadIeadaahaaWcbeqaaiabgkHiTiaaigdaaaGccq GHRaWkcqaH1oqzcqqHMoGrdaWgaaWcbaGaaGymaaqabaGccaaIOaGa amiuamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG0bGaaGykaiabgU caRiabew7aLnaaCaaaleqabaGaaGOmaaaakiablAciljaaiYcacaaI GaGaaGiiaiaadcfadaWgaaWcbaGaaG4maaqabaGccaaI9aGaeqyTdu MaeuiQdKLaaGikaiaadcfadaWgaaWcbaGaaGymaaqabaGccaaISaGa amiDaiaaiYcacqaH1oqzcaaIPaGaaGypaiabew7aLjabfI6aznaaBa aaleaacaaIXaaabeaakiaaiIcacaWGqbWaaSbaaSqaaiaaigdaaeqa aOGaaGilaiaadshacaaIPaGaey4kaSIaeqyTdu2aaWbaaSqabeaaca aIYaaaaOGaeSOjGSKaaGOlaaaa@75F8@

Приравнивая в соответствующих уравнениях инвариантности члены, содержащие множителем первую степень малого параметра, получим соотношения для определения матричных функций Φ 1 ( P 1 ,t) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuOPdy0aaSbaaSqaaiaaigdaaeqaaOGaaGikaiaa dcfadaWgaaWcbaGaaGymaaqabaGccaaISaGaamiDaiaaiMcaaaa@3F48@

F 1 ( P 1 , P 1 H 1 ,t,0) P 1 d dt H 1 = Φ 1 H A T Ψ 1 P 1 H 1 S Ψ 1 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeyOeI0IaamOramaaBaaaleaacaaIXaaabeaakiaa iIcacaWGqbWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiabgkHiTiaadc fadaWgaaWcbaGaaGymaaqabaGccaWGibWaaWbaaSqabeaacqGHsisl caaIXaaaaOGaaGilaiaadshacaaISaGaaGimaiaaiMcacqGHsislca WGqbWaaSbaaSqaaiaaigdaaeqaaOWaaSaaaeaacaWGKbaabaGaamiz aiaadshaaaWaaeWaaeaacaWGibWaaWbaaSqabeaacqGHsislcaaIXa aaaaGccaGLOaGaayzkaaGaaGypaiabgkHiTiabfA6agnaaBaaaleaa caaIXaaabeaakiaadIeacqGHsislcaWGbbWaaWbaaSqabeaacaWGub aaaOGaeuiQdK1aaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaamiuamaa BaaaleaacaaIXaaabeaakiaadIeadaahaaWcbeqaaiabgkHiTiaaig daaaGccaWGtbGaeuiQdK1aaSbaaSqaaiaaigdaaeqaaOGaaGilaaaa @646A@

и Ψ 1 ( P 1 ,t) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiQdK1aaSbaaSqaaiaaigdaaeqaaOGaaGikaiaa dcfadaWgaaWcbaGaaGymaaqabaGccaaISaGaamiDaiaaiMcaaaa@3F5D@

0= Ψ 1 H H T Ψ 1 + P 1 H 1 + P 1 H 1 T . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaaGimaiaai2dacqGHsislcqqHOoqwdaWgaaWcbaGa aGymaaqabaGccaWGibGaeyOeI0IaamisamaaCaaaleqabaGaamivaa aakiabfI6aznaaBaaaleaacaaIXaaabeaakiabgUcaRiaadcfadaWg aaWcbaGaaGymaaqabaGccaWGibWaaWbaaSqabeaacqGHsislcaaIXa aaaOGaey4kaSYaaeWaaeaacaWGqbWaaSbaaSqaaiaaigdaaeqaaOGa amisamaaCaaaleqabaGaeyOeI0IaaGymaaaaaOGaayjkaiaawMcaam aaCaaaleqabaGaamivaaaakiaai6caaaa@5101@ (11)

Последнее равенство представляет собой однозначно разрешимое матричное уравнение Ляпунова. После подстановки найденного решения Ψ 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiQdK1aaSbaaSqaaiaaigdaaeqaaaaa@3A79@ в предыдущее уравнение матрица Φ 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuOPdy0aaSbaaSqaaiaaigdaaeqaaaaa@3A64@ находится путем умножения на H 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamisamaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@3AA5@ соответствующих слагаемых, т. е.

Φ 1 = F 1 ( P 1 , P 1 H 1 ,t,0)+ P 1 d dt H 1 A T Ψ 1 P 1 H 1 S Ψ 1 H 1 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuOPdy0aaSbaaSqaaiaaigdaaeqaaOGaaGypamaa bmaabaGaamOramaaBaaaleaacaaIXaaabeaakiaaiIcacaWGqbWaaS baaSqaaiaaigdaaeqaaOGaaGilaiabgkHiTiaadcfadaWgaaWcbaGa aGymaaqabaGccaWGibWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaG ilaiaadshacaaISaGaaGimaiaaiMcacqGHRaWkcaWGqbWaaSbaaSqa aiaaigdaaeqaaOWaaSaaaeaacaWGKbaabaGaamizaiaadshaaaWaae WaaeaacaWGibWaaWbaaSqabeaacqGHsislcaaIXaaaaaGccaGLOaGa ayzkaaGaeyOeI0IaamyqamaaCaaaleqabaGaamivaaaakiabfI6azn aaBaaaleaacaaIXaaabeaakiabgkHiTiaadcfadaWgaaWcbaGaaGym aaqabaGccaWGibWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaam4uai abfI6aznaaBaaaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiaadIea daahaaWcbeqaaiabgkHiTiaaigdaaaGccaaIUaaaaa@65EF@ (12)

При необходимости аналогичным образом определяются соответствующие матричные коэффициенты при более высоких степенях малого параметра.

Важно отметить, что при t= t 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiDaiaai2dacaWG0bWaaSbaaSqaaiaaigdaaeqa aaaa@3BA3@ матричные функции P 1 , P 2 , P 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG qbWaaSbaaSqaaiaaikdaaeqaaOGaaGilaiaadcfadaWgaaWcbaGaaG 4maaqabaaaaa@3EBA@ обращаются в нуль. Отсюда следует, что правые пограничные функции Z 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwamaaBaaaleaacaaIYaaabeaaaaa@39CA@ , Z 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwamaaBaaaleaacaaIZaaabeaaaaa@39CB@ в представлении матриц P 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIYaaabeaaaaa@39C0@ , P 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIZaaabeaaaaa@39C1@ должны содержать в качестве множителя малый параметр. Это означает, что если при построении закона управления пренебречь правыми пограничными функциями, то в силу формулы (10) погрешность функционала качества не превысит величину O( ε 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4taiaaiIcacqaH1oqzdaahaaWcbeqaaiaaikda aaGccaaIPaaaaa@3CD6@ .

3. Декомпозиция линейной системы уравнений

Обратимся к системе (7) MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ (8) и сначала рассмотрим соответствующую однородную систему, не содержащую правых пограничных функций Z 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwamaaBaaaleaacaaIXaaabeaaaaa@39C9@ , Z 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwamaaBaaaleaacaaIYaaabeaaaaa@39CA@ , Z 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwamaaBaaaleaacaaIZaaabeaaaaa@39CB@ и членов порядка o(ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4BaiaaiIcacqaH1oqzcaaIPaaaaa@3C03@ у регулярных матричных функций V 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvamaaBaaaleaacaaIYaaabeaaaaa@39C6@ , V 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvamaaBaaaleaacaaIZaaabeaaaaa@39C7@ :

χ ˙ 1 = (AS V 2 T ) T χ 2 , ε χ ˙ 2 = χ 1 (H V 3 ) T χ 2 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGafq4XdmMbaiaadaWgaaWcbaGaaGymaaqabaGccaaI 9aGaeyOeI0IaaGikaiaadgeacqGHsislcaWGtbGaamOvamaaDaaale aacaaIYaaabaGaamivaaaakiaaiMcadaahaaWcbeqaaiaadsfaaaGc cqaHhpWydaWgaaWcbaGaaGOmaaqabaGccaaISaGaaGiiaiaaiccacq aH1oqzcuaHhpWygaGaamaaBaaaleaacaaIYaaabeaakiaai2dacqGH sislcqaHhpWydaWgaaWcbaGaaGymaaqabaGccqGHsislcaaIOaGaam isaiabgkHiTiaadAfadaWgaaWcbaGaaG4maaqabaGccaaIPaWaaWba aSqabeaacaWGubaaaOGaeq4Xdm2aaSbaaSqaaiaaikdaaeqaaOGaaG ilaaaa@5BC0@

где V 2 =Φ( V 1 ,t,ε)= V 1 H 1 +ε Φ 1 ( V 1 ,t))+O( ε 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvamaaBaaaleaacaaIYaaabeaakiaai2dacqqH MoGrcaaIOaGaamOvamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG0b GaaGilaiabew7aLjaaiMcacaaI9aGaeyOeI0IaamOvamaaBaaaleaa caaIXaaabeaakiaadIeadaahaaWcbeqaaiabgkHiTiaaigdaaaGccq GHRaWkcqaH1oqzcqqHMoGrdaWgaaWcbaGaaGymaaqabaGccaaIOaGa amOvamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG0bGaaGykaiaaiM cacqGHRaWkcaWGpbGaaGikaiabew7aLnaaCaaaleqabaGaaGOmaaaa kiaaiMcaaaa@59B6@ , V 3 =εΨ( V 1 ,t,ε)=ε Ψ 1 ( V 1 ,t)+O( ε 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvamaaBaaaleaacaaIZaaabeaakiaai2dacqaH 1oqzcqqHOoqwcaaIOaGaamOvamaaBaaaleaacaaIXaaabeaakiaaiY cacaWG0bGaaGilaiabew7aLjaaiMcacaaI9aGaeqyTduMaeuiQdK1a aSbaaSqaaiaaigdaaeqaaOGaaGikaiaadAfadaWgaaWcbaGaaGymaa qabaGccaaISaGaamiDaiaaiMcacqGHRaWkcaWGpbGaaGikaiabew7a LnaaCaaaleqabaGaaGOmaaaakiaaiMcaaaa@548E@ .

Для декомпозиции этой линейной системы можно применить известный метод приведения к блочно-диагональной форме. С этой целью сначала вводится новая быстрая переменная y 2 = χ 2 l χ 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyEamaaBaaaleaacaaIYaaabeaakiaai2dacqaH hpWydaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGSbGaeq4Xdm2aaS baaSqaaiaaigdaaeqaaaaa@41DF@ . Используемая в этой формуле матричная функция l=l(t,ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiBaiaai2dacaWGSbGaaGikaiaadshacaaISaGa eqyTduMaaGykaaaa@3F67@ удовлетворяет несимметричному матричному дифференциальному уравнению Риккати

ε l ˙ +εl[ (AS Φ T ) T l]=I (HεΨS) T l, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTduMabmiBayaacaGaey4kaSIaeqyTduMaamiB aiaaiUfacqGHsislcaaIOaGaamyqaiabgkHiTiaadofacqqHMoGrda ahaaWcbeqaaiaadsfaaaGccaaIPaWaaWbaaSqabeaacaWGubaaaOGa amiBaiaai2facaaI9aGaamysaiabgkHiTiaaiIcacaWGibGaeyOeI0 IaeqyTduMaeuiQdKLaam4uaiaaiMcadaahaaWcbeqaaiaadsfaaaGc caWGSbGaaGilaaaa@55B8@

из которого она может быть легко найдена в виде разложения по степеням малого параметра

l= l 0 (t)+ε l 1 (t)+ ε 2 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiBaiaai2dacaWGSbWaaSbaaSqaaiaaicdaaeqa aOGaaGikaiaadshacaaIPaGaey4kaSIaeqyTduMaamiBamaaBaaale aacaaIXaaabeaakiaaiIcacaWG0bGaaGykaiabgUcaRiabew7aLnaa CaaaleqabaGaaGOmaaaakiablAciljaaiYcaaaa@4A17@

где

l 0 = ( H T ) 1 , l 1 =( H T ) 1 d l 0 dt l 0 ( A T + ( H T ) 1 V 1 S) l 0 Ψ 1 S . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiBamaaBaaaleaacaaIWaaabeaakiaai2dacqGH sislcaaIOaGaamisamaaCaaaleqabaGaamivaaaakiaaiMcadaahaa WcbeqaaiabgkHiTiaaigdaaaGccaaISaGaaGiiaiaaiccacaWGSbWa aSbaaSqaaiaaigdaaeqaaOGaaGypaiaaiIcacaWGibWaaWbaaSqabe aacaWGubaaaOGaaGykamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaa dmaabaWaaSaaaeaacaWGKbGaamiBamaaBaaaleaacaaIWaaabeaaaO qaaiaadsgacaWG0baaaiabgkHiTiaadYgadaWgaaWcbaGaaGimaaqa baGccaaIOaGaamyqamaaCaaaleqabaGaamivaaaakiabgUcaRiaaiI cacaWGibWaaWbaaSqabeaacaWGubaaaOGaaGykamaaCaaaleqabaGa eyOeI0IaaGymaaaakiaadAfadaWgaaWcbaGaaGymaaqabaGccaWGtb GaaGykaiaadYgadaWgaaWcbaGaaGimaaqabaGccqGHsislcqqHOoqw daWgaaWcbaGaaGymaaqabaGccaWGtbaacaGLBbGaayzxaaGaaGOlaa aa@66C9@

Для переменных χ 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaSbaaSqaaiaaigdaaeqaaaaa@3AA1@ , y 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyEamaaBaaaleaacaaIYaaabeaaaaa@39E9@ получаем систему

χ ˙ 1 =[ (AS Φ T ) T ](l χ 1 + y 2 ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGafq4XdmMbaiaadaWgaaWcbaGaaGymaaqabaGccaaI 9aGaaG4waiabgkHiTiaaiIcacaWGbbGaeyOeI0Iaam4uaiabfA6agn aaCaaaleqabaGaamivaaaakiaaiMcadaahaaWcbeqaaiaadsfaaaGc caaIDbGaaGikaiaadYgacqaHhpWydaWgaaWcbaGaaGymaaqabaGccq GHRaWkcaWG5bWaaSbaaSqaaiaaikdaaeqaaOGaaGykaiaaiYcaaaa@4E44@

ε y ˙ 2 =[(HεΨS ) T +εl (AS Φ T ) T ] y 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTduMabmyEayaacaWaaSbaaSqaaiaaikdaaeqa aOGaaGypaiabgkHiTiaaiUfacaaIOaGaamisaiabgkHiTiabew7aLj abfI6azjaadofacaaIPaWaaWbaaSqabeaacaWGubaaaOGaey4kaSIa eqyTduMaamiBaiaaiIcacaWGbbGaeyOeI0Iaam4uaiabfA6agnaaCa aaleqabaGaamivaaaakiaaiMcadaahaaWcbeqaaiaadsfaaaGccaaI DbGaamyEamaaBaaaleaacaaIYaaabeaaaaa@544A@

или после выполнения транспонирования

χ ˙ 1 =[( A T ΦS)](l χ 1 + y 2 ), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGafq4XdmMbaiaadaWgaaWcbaGaaGymaaqabaGccaaI 9aGaaG4waiabgkHiTiaaiIcacaWGbbWaaWbaaSqabeaacaWGubaaaO GaeyOeI0IaeuOPdyKaam4uaiaaiMcacaaIDbGaaGikaiaadYgacqaH hpWydaWgaaWcbaGaaGymaaqabaGccqGHRaWkcaWG5bWaaSbaaSqaai aaikdaaeqaaOGaaGykaiaaiYcaaaa@4D34@

ε y ˙ 2 =[( H T εSΨ)+εl(AΦS)] y 2 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTduMabmyEayaacaWaaSbaaSqaaiaaikdaaeqa aOGaaGypaiabgkHiTiaaiUfacaaIOaGaamisamaaCaaaleqabaGaam ivaaaakiabgkHiTiabew7aLjaadofacqqHOoqwcaaIPaGaey4kaSIa eqyTduMaamiBaiaaiIcacaWGbbGaeyOeI0IaeuOPdyKaam4uaiaaiM cacaaIDbGaamyEamaaBaaaleaacaaIYaaabeaakiaai6caaaa@52EC@

На следующем шаге вводится новая медленная переменная y 1 = χ 1 εp y 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyEamaaBaaaleaacaaIXaaabeaakiaai2dacqaH hpWydaWgaaWcbaGaaGymaaqabaGccqGHsislcqaH1oqzcaWGWbGaam yEamaaBaaaleaacaaIYaaabeaaaaa@42D0@ . При этом матричная функция p=p(t,ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiCaiaai2dacaWGWbGaaGikaiaadshacaaISaGa eqyTduMaaGykaaaa@3F6F@ удовлетворяет линейному матричному дифференциальному уравнению

ε p ˙ p[( H T εSΨ)+εl(AΦS)]=( A T ΦS)ε( A T ΦS)lp, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTduMabmiCayaacaGaeyOeI0IaamiCaiaaiUfa caaIOaGaamisamaaCaaaleqabaGaamivaaaakiabgkHiTiabew7aLj aadofacqqHOoqwcaaIPaGaey4kaSIaeqyTduMaamiBaiaaiIcacaWG bbGaeyOeI0IaeuOPdyKaam4uaiaaiMcacaaIDbGaaGypaiabgkHiTi aaiIcacaWGbbWaaWbaaSqabeaacaWGubaaaOGaeyOeI0IaeuOPdyKa am4uaiaaiMcacqGHsislcqaH1oqzcaaIOaGaamyqamaaCaaaleqaba GaamivaaaakiabgkHiTiabfA6agjaadofacaaIPaGaamiBaiaadcha caaISaaaaa@634F@

из которого она может быть легко найдена в виде разложения по степеням малого параметра p= p 0 (t)+ε p 1 (t)+ ε 2 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiCaiaai2dacaWGWbWaaSbaaSqaaiaaicdaaeqa aOGaaGikaiaadshacaaIPaGaey4kaSIaeqyTduMaamiCamaaBaaale aacaaIXaaabeaakiaaiIcacaWG0bGaaGykaiabgUcaRiabew7aLnaa CaaaleqabaGaaGOmaaaakiablAciljaaiYcaaaa@4A23@ где p 0 (t)=( A T ΦS) H 1 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiCamaaBaaaleaacaaIWaaabeaakiaaiIcacaWG 0bGaaGykaiaai2dacqGHsislcaaIOaGaamyqamaaCaaaleqabaGaam ivaaaakiabgkHiTiabfA6agjaadofacaaIPaGaamisamaaCaaaleqa baGaeyOeI0IaaGymaaaakiaai6caaaa@47D8@

В результате получаются две независимые подсистемы

y ˙ 1 =[( A T ΦS)l] y 1 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGabmyEayaacaWaaSbaaSqaaiaaigdaaeqaaOGaaGyp aiaaiUfacqGHsislcaaIOaGaamyqamaaCaaaleqabaGaamivaaaaki abgkHiTiabfA6agjaadofacaaIPaGaamiBaiaai2facaWG5bWaaSba aSqaaiaaigdaaeqaaOGaaGilaaaa@478B@

ε y ˙ 2 =[( H T εSΨ)+εl(AΦS)] y 2 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTduMabmyEayaacaWaaSbaaSqaaiaaikdaaeqa aOGaaGypaiabgkHiTiaaiUfacaaIOaGaamisamaaCaaaleqabaGaam ivaaaakiabgkHiTiabew7aLjaadofacqqHOoqwcaaIPaGaey4kaSIa eqyTduMaamiBaiaaiIcacaWGbbGaeyOeI0IaeuOPdyKaam4uaiaaiM cacaaIDbGaamyEamaaBaaaleaacaaIYaaabeaakiaai6caaaa@52EC@

Для матричных функций l MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiBaaaa@38F4@ и p MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiCaaaa@38F8@ , пренебрегая членами второго и более высоких порядков в разложении по степеням малого параметра, получаем следующие представления:

l= l 0 +ε l 1 +O( ε 2 ), p= p 0 +O(ε). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiBaiaai2dacaWGSbWaaSbaaSqaaiaaicdaaeqa aOGaey4kaSIaeqyTduMaamiBamaaBaaaleaacaaIXaaabeaakiabgU caRiaad+eacaaIOaGaeqyTdu2aaWbaaSqabeaacaaIYaaaaOGaaGyk aiaaiYcacaaIGaGaaGiiaiaadchacaaI9aGaamiCamaaBaaaleaaca aIWaaabeaakiabgUcaRiaad+eacaaIOaGaeqyTduMaaGykaiaai6ca aaa@50E1@

Здесь

p 0 = [A+S V 1 H 1 ] T ( H T ) 1 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiCamaaBaaaleaacaaIWaaabeaakiaai2dacqGH sislcaaIBbGaamyqaiabgUcaRiaadofacaWGwbWaaSbaaSqaaiaaig daaeqaaOGaamisamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaai2fa daahaaWcbeqaaiaadsfaaaGccaaIOaGaamisamaaCaaaleqabaGaam ivaaaakiaaiMcadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaaIUaaa aa@4B49@

Применение преобразования

y 2 = χ 2 l χ 1 , y 1 = χ 1 εp y 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyEamaaBaaaleaacaaIYaaabeaakiaai2dacqaH hpWydaWgaaWcbaGaaGOmaaqabaGccqGHsislcaWGSbGaeq4Xdm2aaS baaSqaaiaaigdaaeqaaOGaaGilaiaaiccacaaIGaGaamyEamaaBaaa leaacaaIXaaabeaakiaai2dacqaHhpWydaWgaaWcbaGaaGymaaqaba GccqGHsislcqaH1oqzcaWGWbGaamyEamaaBaaaleaacaaIYaaabeaa aaa@4EC0@ (13)

к неоднородной системе (7), (8) приводит к уравнениям

y ˙ 1 =[ (A+SΦ( V 1 ,t,ε)) T l] y 1 + f 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGabmyEayaacaWaaSbaaSqaaiaaigdaaeqaaOGaaGyp aiaaiUfacqGHsislcaaIOaGaamyqaiabgUcaRiaadofacqqHMoGrca aIOaGaamOvamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG0bGaaGil aiabew7aLjaaiMcacaaIPaWaaWbaaSqabeaacaWGubaaaOGaamiBai aai2facaWG5bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamOzamaa BaaaleaacaaIXaaabeaaaaa@50BB@ (14)

и

ε y ˙ 2 =[(HεS Ψ 1 ) T +ε l 0 (AS Φ 0 (t) T ) T ] y 2 + f 2 . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeqyTduMabmyEayaacaWaaSbaaSqaaiaaikdaaeqa aOGaaGypaiabgkHiTiaaiUfacaaIOaGaamisaiabgkHiTiabew7aLj aadofacqqHOoqwdaWgaaWcbaGaaGymaaqabaGccaaIPaWaaWbaaSqa beaacaWGubaaaOGaey4kaSIaeqyTduMaamiBamaaBaaaleaacaaIWa aabeaakiaaiIcacaWGbbGaeyOeI0Iaam4uaiabfA6agnaaBaaaleaa caaIWaaabeaakiaaiIcacaWG0bGaaGykamaaCaaaleqabaGaamivaa aakiaaiMcadaahaaWcbeqaaiaadsfaaaGccaaIDbGaamyEamaaBaaa leaacaaIYaaabeaakiabgUcaRiaadAgadaWgaaWcbaGaaGOmaaqaba GccaaIUaaaaa@5CFA@

Здесь

f 1 =(I+ε p 0 L 0 )( C T QξΦ( V 1 ,t,ε)f) p 0 (ε Ψ 1 ( V 1 ,t)f), MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzamaaBaaaleaacaaIXaaabeaakiaai2dacaaI OaGaamysaiabgUcaRiabew7aLjaadchadaWgaaWcbaGaaGimaaqaba GccaWGmbWaaSbaaSqaaiaaicdaaeqaaOGaaGykaiaaiIcacaWGdbWa aWbaaSqabeaacaWGubaaaOGaamyuaiabe67a4jabgkHiTiabfA6agj aaiIcacaWGwbWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiaadshacaaI SaGaeqyTduMaaGykaiaadAgacaaIPaGaeyOeI0IaamiCamaaBaaale aacaaIWaaabeaakiaaiIcacqGHsislcqaH1oqzcqqHOoqwdaWgaaWc baGaaGymaaqabaGccaaIOaGaamOvamaaBaaaleaacaaIXaaabeaaki aaiYcacaWG0bGaaGykaiaadAgacaaIPaGaaGilaaaa@6339@

f 2 =ε Ψ 1 ( V 1 ,t,ε)fε L 0 ( C T Qξ Φ 0 (t)f). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzamaaBaaaleaacaaIYaaabeaakiaai2dacqGH sislcqaH1oqzcqqHOoqwdaWgaaWcbaGaaGymaaqabaGccaaIOaGaam OvamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG0bGaaGilaiabew7a LjaaiMcacaWGMbGaeyOeI0IaeqyTduMaamitamaaBaaaleaacaaIWa aabeaakiaaiIcacaWGdbWaaWbaaSqabeaacaWGubaaaOGaamyuaiab e67a4jabgkHiTiabfA6agnaaBaaaleaacaaIWaaabeaakiaaiIcaca WG0bGaaGykaiaadAgacaaIPaGaaGOlaaaa@5966@

В этих уравнениях и выражениях для функций f 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzamaaBaaaleaacaaIXaaabeaaaaa@39D5@ и f 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzamaaBaaaleaacaaIYaaabeaaaaa@39D6@ опущены правые пограничные функции Z 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwamaaBaaaleaacaaIXaaabeaaaaa@39C9@ , Z 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwamaaBaaaleaacaaIYaaabeaaaaa@39CA@ , Z 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOwamaaBaaaleaacaaIZaaabeaaaaa@39CB@ и члены порядка o(ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4BaiaaiIcacqaH1oqzcaaIPaaaaa@3C03@ у регулярных функций. Принимая во внимание, что функция f 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOzamaaBaaaleaacaaIYaaabeaaaaa@39D6@ содержит малый параметр в качестве множителя, получаем следующее приближенное выражение для y 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyEamaaBaaaleaacaaIYaaabeaaaaa@39E9@ :

y 2 =ε H 1 Ψ 1 ( V 1 ,t)f L 0 ( C 1 T Qξ Φ 0 (t)f) , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyEamaaBaaaleaacaaIYaaabeaakiaai2dacqGH sislcqaH1oqzcaWGibWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaae WaaeaacqqHOoqwdaWgaaWcbaGaaGymaaqabaGccaaIOaGaamOvamaa BaaaleaacaaIXaaabeaakiaaiYcacaWG0bGaaGykaiaadAgacqGHsi slcaWGmbWaaSbaaSqaaiaaicdaaeqaaOGaaGikaiaadoeadaqhaaWc baGaaGymaaqaaiaadsfaaaGccaWGrbGaeqOVdGNaeyOeI0IaeuOPdy 0aaSbaaSqaaiaaicdaaeqaaOGaaGikaiaadshacaaIPaGaamOzaiaa iMcaaiaawIcacaGLPaaacaaISaaaaa@5A63@

в котором учтены только регулярные члены порядка O(ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4taiaaiIcacqaH1oqzcaaIPaaaaa@3BE3@ , а регулярные члены более высоких порядков и правые пограничные функции, которые содержат в качестве множителя малый параметр, опущены. В рассматриваемом случае формула для оптимального управления принимает вид

u opt = R 1 B T [ P 2 x+ P 3 x ˙ + χ 2 ]. MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyDamaaBaaaleaacaWGVbGaamiCaiaadshaaeqa aOGaaGypaiabgkHiTiaadkfadaahaaWcbeqaaiabgkHiTiaaigdaaa GccaWGcbWaaWbaaSqabeaacaWGubaaaOGaaG4waiaadcfadaWgaaWc baGaaGOmaaqabaGccaWG4bGaey4kaSIaamiuamaaBaaaleaacaaIZa aabeaakiqadIhagaGaaiabgUcaRiabeE8aJnaaBaaaleaacaaIYaaa beaakiaai2facaaIUaaaaa@4ED9@

Чтобы получить погрешность порядка O( ε 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4taiaaiIcacqaH1oqzdaahaaWcbeqaaiaaikda aaGccaaIPaaaaa@3CD6@ при вычислении значения функционала качества для субоптимального управления, следует использовать приближенное выражение

P 2 =Φ( V 1 ,t,ε) Φ 0 +ε Φ 1 = V 1 H 1 +ε Φ 1 ( V 1 ,t), P 3 ε Ψ 1 ( V 1 ,t). MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIYaaabeaakiaai2dacqqH MoGrcaaIOaGaamOvamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG0b GaaGilaiabew7aLjaaiMcarqqr1ngBPrgifHhDYfgaiuaacqWFdjYo cqqHMoGrdaWgaaWcbaGaaGimaaqabaGccqGHRaWkcqaH1oqzcqqHMo GrdaWgaaWcbaGaaGymaaqabaGccaaI9aGaeyOeI0IaamOvamaaBaaa leaacaaIXaaabeaakiaadIeadaahaaWcbeqaaiabgkHiTiaaigdaaa GccqGHRaWkcqaH1oqzcqqHMoGrdaWgaaWcbaGaaGymaaqabaGccaaI OaGaamOvamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG0bGaaGykai aaiYcacaaIGaGaaGiiaiaadcfadaWgaaWcbaGaaG4maaqabaGccqWF djYocqaH1oqzcqqHOoqwdaWgaaWcbaGaaGymaaqabaGccaaIOaGaam OvamaaBaaaleaacaaIXaaabeaakiaaiYcacaWG0bGaaGykaiaai6ca aaa@6F1D@ (15)

Что касается χ 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaSbaaSqaaiaaikdaaeqaaaaa@3AA2@ , использование представления

χ 2 =l y 1 +(I+εlp) y 2 , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaSbaaSqaaiaaikdaaeqaaOGaaGypaiaa dYgacaWG5bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaaGikaiaadM eacqGHRaWkcqaH1oqzcaWGSbGaamiCaiaaiMcacaWG5bWaaSbaaSqa aiaaikdaaeqaaOGaaGilaaaa@487D@

которое вытекает из (13), и полученное выше выражение для y 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyEamaaBaaaleaacaaIYaaabeaaaaa@39E9@ позволяет применять следующее приближенное выражение:

χ 2 =( l 0 +ε l 1 ) y 1 ε H 1 Ψ 1 ( V 1 ,t)f L 0 ( C T Qξ Φ 0 (t)f) . MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeq4Xdm2aaSbaaSqaaiaaikdaaeqaaOGaaGypaiaa iIcacaWGSbWaaSbaaSqaaiaaicdaaeqaaOGaey4kaSIaeqyTduMaam iBamaaBaaaleaacaaIXaaabeaakiaaiMcacaWG5bWaaSbaaSqaaiaa igdaaeqaaOGaeyOeI0IaeqyTduMaamisamaaCaaaleqabaGaeyOeI0 IaaGymaaaakmaabmaabaGaeuiQdK1aaSbaaSqaaiaaigdaaeqaaOGa aGikaiaadAfadaWgaaWcbaGaaGymaaqabaGccaaISaGaamiDaiaaiM cacaWGMbGaeyOeI0IaamitamaaBaaaleaacaaIWaaabeaakiaaiIca caWGdbWaaWbaaSqabeaacaWGubaaaOGaamyuaiabe67a4jabgkHiTi abfA6agnaaBaaaleaacaaIWaaabeaakiaaiIcacaWG0bGaaGykaiaa dAgacaaIPaaacaGLOaGaayzkaaGaaGOlaaaa@6403@

Таким образом, система (6) имеет медленное интегральное многообразие, которое с точностью до членов порядка O(ε) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4taiaaiIcacqaH1oqzcaaIPaaaaa@3BE3@ включительно описывается уравнениями (15), где Ψ 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuiQdK1aaSbaaSqaaiaaigdaaeqaaaaa@3A79@ является решением уравнения Ляпунова (11), Φ 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaeuOPdy0aaSbaaSqaaiaaigdaaeqaaaaa@3A64@ задается формулой (12), а матрица V 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvamaaBaaaleaacaaIXaaabeaaaaa@39C5@ представляет собой решение матричного дифференциального уравнения

P ˙ 1 = P 2 A A T P 2 T + P 2 S P 2 T M, P 1 ( t 1 )=0, MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGabmiuayaacaWaaSbaaSqaaiaaigdaaeqaaOGaaGyp aiabgkHiTiaadcfadaWgaaWcbaGaaGOmaaqabaGccaWGbbGaeyOeI0 IaamyqamaaCaaaleqabaGaamivaaaakiaadcfadaqhaaWcbaGaaGOm aaqaaiaadsfaaaGccqGHRaWkcaWGqbWaaSbaaSqaaiaaikdaaeqaaO Gaam4uaiaadcfadaqhaaWcbaGaaGOmaaqaaiaadsfaaaGccqGHsisl caWGnbGaaGilaiaaiccacaaIGaGaamiuamaaBaaaleaacaaIXaaabe aakiaaiIcacaWG0bWaaSbaaSqaaiaaigdaaeqaaOGaaGykaiaai2da caaIWaGaaGilaaaa@54AE@

в котором P 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamiuamaaBaaaleaacaaIYaaabeaaaaa@39C0@ задается выражением (15). Через v 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamODamaaBaaaleaacaaIYaaabeaaaaa@39E6@ обозначим выражение

v 2 =( l 0 +ε l 1 ) v 1 ε H 1 Ψ 1 ( V 1 ,t)f l 0 ( C 1 T Qξ Φ 0 (t)f) , MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamODamaaBaaaleaacaaIYaaabeaakiaai2dacaaI OaGaamiBamaaBaaaleaacaaIWaaabeaakiabgUcaRiabew7aLjaadY gadaWgaaWcbaGaaGymaaqabaGccaaIPaGaamODamaaBaaaleaacaaI XaaabeaakiabgkHiTiabew7aLjaadIeadaahaaWcbeqaaiabgkHiTi aaigdaaaGcdaqadaqaaiabfI6aznaaBaaaleaacaaIXaaabeaakiaa iIcacaWGwbWaaSbaaSqaaiaaigdaaeqaaOGaaGilaiaadshacaaIPa GaamOzaiabgkHiTiaadYgadaWgaaWcbaGaaGimaaqabaGccaaIOaGa am4qamaaDaaaleaacaaIXaaabaGaamivaaaakiaadgfacqaH+oaEcq GHsislcqqHMoGrdaWgaaWcbaGaaGimaaqabaGccaaIOaGaamiDaiaa iMcacaWGMbGaaGykaaGaayjkaiaawMcaaiaaiYcaaaa@641D@ (16)

в котором в качестве v 1 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamODamaaBaaaleaacaaIXaaabeaaaaa@39E5@ следует взять решение уравнения (14) с граничным условием y 1 ( t 1 )=0 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyEamaaBaaaleaacaaIXaaabeaakiaaiIcacaWG 0bWaaSbaaSqaaiaaigdaaeqaaOGaaGykaiaai2dacaaIWaaaaa@3EC2@ .

Суммируя вышесказанное, приходим к следующему утверждению.

Теорема. Применение субоптимального управления

u s = R 1 B T [ V 2 x+ V 3 x ˙ + v 2 ], MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamyDamaaBaaaleaacaWGZbaabeaakiaai2dacqGH sislcaWGsbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaamOqamaaCa aaleqabaGaamivaaaakiaaiUfacaWGwbWaaSbaaSqaaiaaikdaaeqa aOGaamiEaiabgUcaRiaadAfadaWgaaWcbaGaaG4maaqabaGcceWG4b GbaiaacqGHRaWkcaWG2bWaaSbaaSqaaiaaikdaaeqaaOGaaGyxaiaa iYcaaaa@4C3D@

где V 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvamaaBaaaleaacaaIYaaabeaaaaa@39C6@ и V 3 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamOvamaaBaaaleaacaaIZaaabeaaaaa@39C7@ заданы выражениями (15), а v 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaamODamaaBaaaleaacaaIYaaabeaaaaa@39E6@ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaaruWqHXwAIjxAaGqbaKqzGfaeaaaaaaaaa8GacaWFtaca aa@3C72@ выражением (16), приводит к погрешности порядка O( ε 2 ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcqqaaaaaaaaGqSf 2yRbWdbeaapaGaam4taiaaiIcacqaH1oqzdaahaaWcbeqaaiaaikda aaGccaaIPaaaaa@3CD6@ в функционале (5).

Выводы

В статье обсуждается возможность применения метода декомпозиции для понижения размерности задачи оптимального слежения с сингулярными и случайными возмущениями. Традиционные методы решения задач оптимального управления с сингулярными возмущениями для таких задач неприменимы, так как основываются на предположении о гладкости правых частей, которое входит в противоречие с наличием случайных возмущений. Метод декомпозиции позволяет избежать этой трудности и получить формулу для субоптимального управления.

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About the authors

Vladimir A. Sobolev

Samara National Research University

Author for correspondence.
Email: v.sobolev@ssau.ru
ORCID iD: 0000-0001-7327-7340

Doctor of Physical and Mathematical Sciences, Professor, Department of Differential Equations and Control Theory

Russian Federation, Samara

References

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  2. Vasil’eva A.B., Dmitriev M.G. Singular perturbations in optimal control problems. Journal of Soviet Mathematics, 1986, vol. 34, issue 4, pp. 1579–1629. DOI: https://doi.org/10.1007/BF01262406. EDN: https://www.elibrary.ru/xorosl. (In Russ.)
  3. Dmitriev M.G., Kurina G.A. Singular perturbations in control problems. Automation and Remote Control, 2006, vol. 67, no. 1, pp. 1–43. DOI: http://doi.org/10.1134/S0005117906010012. EDN: https://www.elibrary.ru/ljogdl. (in English; Russian original).
  4. Naidu D.S. Singular Perturbations and Time Scales in Control Theory and Applications: An Overview. Dynamics of Continuous, Discrete and Impulsive Systems. Series B: Applications & Algorithms, 2002, vol. 9, issue 2, pp. 233–278. Available at: https://www.d.umn.edu/ dsnaidu/Naidu_Survey_DCDISJournal_2002.pdf.
  5. Sobolev V.A. Integral manifolds and decomposition of singularly perturbed systems. System and Control Letters, 1984, vol. 5, issue 3, pp. 169–179. DOI: http://doi.org/10.1016/S0167-6911(84)80099-7.

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