Optimal guaranteed control of linear systems under disturbances
type: Item
instance of: scholarly article
title: Optimal guaranteed control of linear systems under disturbances
zbMATH Open document ID: 0956.49017
DOI: 10.1007/BF02305178
published in: Mathematical Notes
publication date: 2001-02-20T00:00:00Z
review text: The author considers the following linear systems \[ \dot x(t)=A(t)x(t)-B(t)u(t)-D(t)w(t)+C(t)v(t), \tag{1} \] \(t\in [t^0,\theta], \;x(t^0)=x^0\), where \(A(t), B(t), C(t), D(t)\) are continuous matrices of dimensions \(n\times n\), \(n\times m\), \(n\times l\), \(n\times k\), respectively. Parameters \(u\) and \(w\) are controls, and \(v\) is a disturbance. It is assumed that \[ u(t)\in U(t), \tag{2} \] where \(t\mapsto U(t)\subset R_m\) is a continuous convex- and compact-valued mapping. The performance (cost) functional has the form \[ I=\gamma(x(\theta))+\int_\theta^{t^0} ( w_T (t)G(t)w(t)v_T (t)H(t)v(t)) dt, \tag{3} \] where \[ \;\gamma(x)=\min_{y\in M^0} (x-y)_T R^0 (x-y), \tag{4} \] \(G(t), H(t)\) are symmetric positive semidefinite continuous matrices, \(R^0\) is a positive definite matrix, \(M^0 \subset R_n\) is the convex closed target set. The aim of controls is minimizing the cost functional, the disturbance tries to maximize the cost functional. It is proven that the differential game has the value function, i.e., there exists the equilibrium in the classes of feedbacks: Borel measurable controls \(u(t,x)\) plus continuous controls \(w(t,x)\) and continuous disturbances \(v(t,x)\). In the paper, simple formulas for calculating the value function and optimal guaranteed feedbacks are obtained.
Mathematics Subject Classification ID: 49K35
Mathematics Subject Classification ID: 49N10
Mathematics Subject Classification ID: 49N70
Mathematics Subject Classification ID: 93C73
Mathematics Subject Classification ID: 91A23
zbMATH DE Number: 1158886
zbMATH Keywords: optimal control under disturbances
zbMATH Keywords: minimax control
zbMATH Keywords: optimal guaranteed control
zbMATH Keywords: geometric control constraints
zbMATH Keywords: performance functional
author: G. E. Ivanov
reviewed by: N. N. Subbotina
MaRDI profile type: Publication
cites work: Q3890468
cites work: Q3994690
cites work: CONVEXITY OF VALUES OF VECTOR INTEGRALS, THEOREMS ON MEASURABLE CHOICE AND VARIATIONAL PROBLEMS
description: scientific article; zbMATH DE number 1158886