Stabilization and random linear regulator problem for random linear control processes
type: Item
instance of: scholarly article
title: Stabilization and random linear regulator problem for random linear control processes
zbMATH Open document ID: 0853.93102
author: Mahesh Nerurkar
author: Russell Johnson
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-08-29T00:00:00Z
review text: The authors consider the feedback stabilization problem for linear time varying control systems by interpreting the system matrices as being defined by a dynamical system: Let \(C\) and \(D\) be spaces of matrix functions with appropriate topologies, and define the shift \(\tau_t : C \times D \to C \times D\) by \(\tau_t (a,b) (s) = (a (t + s), b(t + s))\) for \((a, b) \in C \times D\). Let \(\Omega \subset C \times D\) be a compact, shift-invariant set and let \(A : \Omega \to M(n,n)\), \(B : \Omega \to M (n,m)\) be matrix valued functions. Then a (family of) linear, varying control systems is given by \[ \dot x = A \bigl( \tau_t (w) \bigr) x + B \bigl( \tau_t (w) \bigr) u, \quad w \in \Omega. \] The linear quadratic regulator problem for these systems is solved in the following way: Assume the (suitable modified) controllability assumptions for the system and its adjoint, construct an exponential dichotomy over \(\Omega\) via properties of the rotation numbers of the associated Hamiltonian system, interpret the stable bundle of the dichotomy as the solution of a Riccati equation, and define the optimal feedback in the usual way, using this solution. The advantage of this approach is that the resulting feedback inherits smoothness, robustness, and invariance properties (w.r.t. te shift \(\tau)\) from the stable bundle, uniformly, in \(\Omega\). Choosing the \(Q\)-matrix of the LQR problem as \(\delta \cdot\) identity, the authors show that the Lyapunov exponents of the system lie to the left of \(- \delta \cdot\) constant, uniformly in \(w \in Q\). This result can be considered as a time varying version of the pole placement theorem.
reviewed by: Wolfgang Kliemann
Mathematics Subject Classification ID: 93E15
Mathematics Subject Classification ID: 93E20
zbMATH DE Number: 902462
zbMATH Keywords: stabilization
zbMATH Keywords: linear
zbMATH Keywords: time varying
zbMATH Keywords: linear quadratic regulator
zbMATH Keywords: exponential dichotomy
zbMATH Keywords: rotation numbers
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: shift
zbMATH Keywords: Lyapunov exponents
zbMATH Keywords: pole placement
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0042
OpenAlex ID: W1972010115
DOI: 10.1006/JMAA.1996.0042
description: scientific article; zbMATH DE number 902462