Structural properties of minimax controllers for a class of differential games arising in nonlinear \(H^ \infty\) control
type: Item
instance of: scholarly article
title: Structural properties of minimax controllers for a class of differential games arising in nonlinear \(H^ \infty\) control
zbMATH Open document ID: 0804.90151
DOI: 10.1016/0167-6911(93)90048-B
author: Garry Didinsky
author: Tamer Başar
author: Pierre Bernhard
published in: Systems \& Control Letters
publication date: 1995-01-19T00:00:00Z
review text: The authors deal with a minimax control problem described by the equation \[ \dot x= f(t,x,u,w),\quad x(0)= x_ 0,\quad t\in [0,T],\tag{1} \] where \(x\in \mathbb{R}^ n\) is a state vector, \(u\in U\subset\mathbb{R}^ m\) is a control. It is assumed that \[ (x_ 0,w(\cdot))\in \Omega\subset\mathbb{R}^ n\times {\mathcal L}_ 2([0,T],\mathbb{R}^ I), \] i.e., both the initial state \(x_ 0\) and the disturbance \(w(\cdot)\) are not known. The controller choosing \(u\) has access to only a noise-corrupted output \(y\) in \(\mathbb{R}^ p\), given by \(y(t)= h(t,x(t))+ v(t)\), where \(v(\cdot)\in {\mathcal V}\subset {\mathcal L}_ 2([0,T],\mathbb{R}^ p)\) is part of general disturbance. Here \(\Omega\) and \(\mathcal V\) are some specified sets. An admissible control strategy is defined as a nonanticipatory (casual) map \(y(\cdot)\mapsto \mu(y(\cdot))\) such that the differential equation (1) with \(u(\cdot)= \mu(y(\cdot))\) has a solution for every \((x_ 0,w(\cdot))\in \Omega\) and every \(v(\cdot)\in {\mathcal V}\). The cost functional \(J\) is given by \[ J= q_ T(x(T))+ \int^ T_ 0 [q(t,x(t),u(t),w(t))- r(t,v(t))]dt- q_ 0(x_ 0). \] The objective is to find a minimax strategy for the controller that minimizes the supremum of \(J\) over all possible disturbances. This problem is considered under certain assumptions. In particular, it is assumed that there exists a solution \(V\) of class \(C^ 1\) of a Cauchy problem for Isaacs' equation \[ -{\partial V\over \partial t}= \inf_ u \sup_ w \left\{{\partial V\over \partial x} f(t,x,u,w)+ q(t,x,u,w)\right\},\quad V(T,x)= q_ T(x). \] It is also assumed the uniqueness of solutions of some extremal problems. Examples satisfying these requirements are not considered.
Mathematics Subject Classification ID: 91A23
Mathematics Subject Classification ID: 93B36
zbMATH DE Number: 550117
zbMATH Keywords: certainty equivalence
zbMATH Keywords: minimax control
zbMATH Keywords: noise-corrupted output
zbMATH Keywords: admissible control strategy
zbMATH Keywords: Cauchy problem for Isaacs' equation
reviewed by: N. N. Subbotina
MaRDI profile type: Publication
cites work: Dynamic noncooperative game theory
cites work: \(H^ \infty\) - optimal control and related minimax design problems. A dynamic game approach
cites work: Q5534710
cites work: State-space solutions to standard H/sub 2/ and H/sub infinity / control problems
cites work: Disturbance attenuation and H/sub infinity /-control via measurement feedback in nonlinear systems
cites work: On a state space approach to nonlinear \(H_ \infty\) control
full work available at URL: 0167-6911(93)90048-b
OpenAlex ID: W2044032991
description: scientific article; zbMATH DE number 550117