Asymptotic solution of a cheap control problem with state delay
type: Item
instance of: scholarly article
title: Asymptotic solution of a cheap control problem with state delay
zbMATH Open document ID: 0958.49013
DOI: 10.1023/A:1026484201241
author: Valery Y. Glizer
published in: Dynamics and Control
publication date: 2001-04-09T00:00:00Z
review text: The author notes that cheap controls (that is with small cost of control), with either point or distributed delays, were considered in several publications by R. E. O'Malley, P. Sanuti and others. Changes in a small cost of control amounts to a singular perturbation in the Hamilton-Jacobi-Bellman equation, and in the corresponding Hamilton boundary value problem. As the cost of control approaches zero, the limit of optimal trajectory corresponds to a singular arc. Thus the cheap control is a subclass of singularly perturbed control problems. The author comments that there are few works on singularly perturbed problems with delays, with the exception of his own technical reports (Technion, Israel, 1996). Let \(x(\tau)= \varphi_x(\tau)\), \(y(\tau)= \varphi_y(\tau)\) be the state variables, \(-h\leq \tau\leq 0\), while the state \[ \begin{multlined} dx/dt= A_1(t) x(t)+ A_2(t) y(t)+ H_1(t) x(t- h)+ H_2(t) y(t- h)+\\ \int^0_h [G_1(t, \tau) x(t+\tau)+ G_2(t, \tau) y(t+ \tau)] d\tau,\end{multlined} \] \[ \begin{multlined} dy/dt= A_3(t) x(t)+ A_4(t) y(t)+ H_3(t) x(t- h)+ H_4(t) y(t- h)+\\ \int^0_{-h} [G_3(t, \tau)x(t+ \tau)+ G_4(t, \tau)y(t+ \tau)] dt+ u(t),\end{multlined} \] \(t>0\), where \(x\) and \(y\) are vector variables, \(h\) is a positive constant \(A\), \(H\), \(G\) are matrices, while \(\varphi_i\) are continuous. The cost function is: \[ J_\varepsilon= \int^T_0 [x'(t) D_1(t) x(t)+ y'(t) D_2(t) y(t)+ \varepsilon^2u'(t) N(t) u(t)] dt, \] where primes denote transposition. \(D_1\) is a symmetric positive semidefinite matrix, \(D_2\) and \(N\) are symmetric positive definite matrices, \(\varepsilon\) is a small parameter. Using known results of V. B. Kolmanovskij and others, the author obtains the following results for optimal performance: \[ u^*(t)= -\varepsilon^2 N^{-1}(t) B'\left[P(t)z(t))+ \int^0_{-h} [Q(t, \tau) z(t+ \tau) d\tau]\right], \] where \(B\) is a matrix composed of an \(n\times m\) zero matrix and \(m\times m\) identity matrix. \[ J^*_\varepsilon= \…
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 49K25
Mathematics Subject Classification ID: 49N10
zbMATH DE Number: 1389591
zbMATH Keywords: singular perturbations
zbMATH Keywords: asymptotic expansions
zbMATH Keywords: Hamilton-Jacobi-Bellman equation
zbMATH Keywords: cheap control
zbMATH Keywords: delays
zbMATH Keywords: optimal performance
zbMATH Keywords: Riccati equations
MaRDI profile type: Publication
full work available at URL: a:1026484201241
OpenAlex ID: W171932230
description: scientific article; zbMATH DE number 1389591