Q1329990
type: Item
instance of: scholarly article
title: Linear-quadratic control with and without stability subject to general implicit continuous-time systems: Coordinate-free interpretations of the optimal costs in terms of dissipation inequality and linear matrix inequality; existence and uniqueness of optimal controls and state trajectories
zbMATH Open document ID: 0814.49031
DOI: 10.1016/0024-3795(94)90216-X
author: Ton Geerts
published in: Linear Algebra and its Applications
publication date: 1994-08-15T00:00:00Z
review text: The linear quadratic control problem for the class of singular, finite- dimensional, time-invariant systems of the form \[ \begin{aligned} E\dot x(t) & = Ax(t)+ Bu(t)\\ y(t) & = Cx(t)+ Du(t),\qquad x(0)= x_ 0\in \mathbb{R}^ n\end{aligned} \] with cost functional \[ J(x_ 0, u(\cdot))= \int^ \infty_ 0 y(t)^ T y(t)dt \] is investigated. Partially, the results are also new for regular systems, i.e., \(\text{det }E\neq 0\). The optimal costs are related to a dissipation inequality and a linear matrix inequality, the latter depending on \(E\), \(A\), \(B\), \(C\), \(D\). Finally, necessary and sufficient conditions for solvability of the optimal control problem and of the optimal controls are presented.
Mathematics Subject Classification ID: 49N10
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 93C05
zbMATH DE Number: 614164
zbMATH Keywords: linear quadratic control
zbMATH Keywords: dissipation inequality
zbMATH Keywords: matrix inequality
zbMATH Keywords: solvability
zbMATH Keywords: optimal control
zbMATH Keywords: time-invariant
zbMATH Keywords: continuous-time
reviewed by: Achim Ilchmann
MaRDI profile type: Publication
cites work: Dissipative dynamical systems. I: General theory
cites work: The time-invariant linear-quadratic optimal control problem
cites work: The role of the dissipation matrix in singular optimal control
cites work: Q4009242
cites work: Singular Optimal Control: A Geometric Approach
cites work: A necessary and sufficient condition for solvability of the linear- quadratic control problem without stability
cites work: All optimal controls for the singular linear-quadratic problem without stability; a new interpretation of the optimal cost
cites work: Q4183808
cites work: System structure and singular control
cites work: Q3940107
cites work: Q3254327
cites work: A generalized state-space for singular systems
cites work: Descriptor variable systems and optimal state regulation
cites work: The linear-quadratic optimal regulator for descriptor systems
cites work: Regularity and singularity in linear-quadratic control subject to implicit continuous-time systems
cites work: The autonomous linear quadratic control problem. Theory and numerical solution
cites work: Structural properties of linear dynamical systems
cites work: Q4692849
cites work: Geometric structure and feedback in singular systems
cites work: Singular systems: A new approach in the time domain
cites work: Feedback and pole placement in descriptor variable Systems†
cites work: Invariant subspaces and invertibility properties for singular systems: The general case
cites work: Q4115007
cites work: Solvability conditions, consistency, and weak consistency for linear differential-algebraic equations and time-invariant singular systems: The general case
cites work: Q5601127
cites work: Generalized linear systems: Geometric and structural approaches
cites work: Q3975073
cites work: Q3241581
cites work: Q4401837
cites work: Nonnegativity of a Quadratic Functional
cites work: Poles and zeros of linear multivariable systems : a survey of the algebraic, geometric and complex-variable theory
label: Linear-quadratic control with and without stability subject to general implicit continuous-time systems: Coordinate-free interpretations of the optimal costs in terms of dissipation inequality and linear matrix inequality; existence and uniqueness of optimal controls and state trajectories
description: scientific article; zbMATH DE number 614164