Asymptotic solution of the singularly perturbed infinite dimensional Riccati equation
type: Item
instance of: scholarly article
title: Asymptotic solution of the singularly perturbed infinite dimensional Riccati equation
zbMATH Open document ID: 0916.49028
author: Valery Y. Glizer
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-07-14T00:00:00Z
review text: The main problem discussed by the author is the singularly perturbed linear-quadratic control posed in a Hilbert space setting. The author notes that if the corresponding operators are bounded then the problem can be modeled by Barbashin's type integro-differential equations. He considers a linear system: \[ \begin{aligned} dx/dt & = A_1(t)x+ A_2(t)y+ B_1(t)u,\quad x(0)= x^0,\\ \varepsilon dy/dt & = A_3(t)x+ A_4(t)y+ B_2(t)u,\quad y(0)= y^0,\quad t\in[0, T].\end{aligned} \] Here, \(x\), \(y\) are state variables in Hilbert spaces \(H_x\) and \(H_y\), respectively, \(u\) is the control in the Hilbert space \(H_u\), and \(\varepsilon> 0\) is a ``small'' parameter. The reviewer is somewhat troubled by author's identification of the tangent space with the integral curves \((A:H_{x\to}H_x)\), etc\dots, but hopefully no harm is done in the present discussion. In the same spirit, the author introduces the Hilbert space \(H_z\), which is the direct sum of mutually orthogonal spaces \(H_x\) and \(H_y\). The resulting equations are \(dz/dt= A(t,\varepsilon)z+ B(t,\varepsilon)u\), \(z(0)= z^0\). The cost functional is \[ J= (z(T), F(z(T)))+ \int^T_0 \{\langle z,D(t)z\rangle_{H_z}+ \langle u,Ru\rangle_{H_u}\}dt, \] where \(\langle,\rangle\) denotes an inner product, \(F\) is a positive definite and selfadjoint operator: \[ F=\begin{matrix} \lceil F_1, & F_2\rceil \\ \lfloor F^*_2, & F_3\rfloor\end{matrix}. \] Then \(u\) can be represented as \(u= -R^{-1} B^*K(t,\varepsilon)z\). The operator \(K\) satisfies the differential equation: \[ dK/dt= -KA(t,\varepsilon)- A^*(t,\varepsilon)K= KS(t,\varepsilon)K- D(t), \] and \(S(t,\varepsilon)= BR^{-1}B^*\). (Here, the author does not specify what kind of derivative of the operator \(K\) is considered, even if it is easy to guess.) To avoid singularities, the author assumes that the solutions are of the form: \[ K(t,\varepsilon)=\begin{matrix} \lceil K_1,& \varepsilon K_2\rceil \\ \lfloor\varepsilon K^*_2, & \varepsilon K_3\rfloor\end{ma…
Mathematics Subject Classification ID: 49N10
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 34H05
Mathematics Subject Classification ID: 49K40
Mathematics Subject Classification ID: 93C15
zbMATH DE Number: 1115502
zbMATH Keywords: Riccati equations
zbMATH Keywords: singularly perturbed linear-quadratic control
zbMATH Keywords: Hilbert space
zbMATH Keywords: asymptotic solutions
zbMATH Keywords: existence of an optimal control
reviewed by: Vadim Komkov
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5586
OpenAlex ID: W2050866979
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DOI: 10.1006/JMAA.1997.5586
description: scientific article; zbMATH DE number 1115502