An optimal control problem with unbounded control operator and unbounded observation operator where the algebraic Riccati equation is satisfied as a Lyapunov equation
type: Item
instance of: scholarly article
title: An optimal control problem with unbounded control operator and unbounded observation operator where the algebraic Riccati equation is satisfied as a Lyapunov equation
zbMATH Open document ID: 0906.49001
DOI: 10.1016/S0893-9659(97)00018-9
published in: Applied Mathematics Letters
publication date: 1998-05-13T00:00:00Z
review text: The author considers the control system \[ \begin{aligned} y_t & =-y_x -\alpha y,\quad t>0,\;x>0, \\ y(0,x) & =y_0(x),\quad t\geq 0, \\ y(t,0) & =u(t),\quad t\geq 0,\end{aligned} \] the quadratic cost function \(J(u,y) = \int_0^\infty (| y(t,1)| ^2+| u(t)|^2)dt\) and the associated optimal control problem. Thus, the generator \(A\), the control operator \(B\) and the observation operator \(R\) are unbounded. It is shown that in despite of these bad properties, the problem reveals some striking features: (a) the unbounded gain operator \(B^\ast P\) vanishes on \(D(A)\); (b) the algebraic Riccati equation is satisfied on \(D(A)\) by \(P\), indeed as a Lyapunov equation.
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49N10
zbMATH DE Number: 1085342
zbMATH Keywords: unbounded control
zbMATH Keywords: unbounded observation
zbMATH Keywords: Riccati equation
zbMATH Keywords: Lyapunov equation
zbMATH Keywords: optimal control
author: R. Triggiani
reviewed by: Ovidiu Cârjă
MaRDI profile type: Publication
cites work: Q3139178
cites work: Q4954010
cites work: Representation and control of infinite dimensional systems. Volume I
full work available at URL: s0893-9659(97)00018-9
OpenAlex ID: W1972911168
description: scientific article; zbMATH DE number 1085342