Generalized optimal control of linear systems with distributed parameters
type: Item
instance of: scholarly article
title: Generalized optimal control of linear systems with distributed parameters
zbMATH Open document ID: 1030.49001
author: S. I. Lyashko
published in: Applied Optimization
publication date: 2002-12-03T00:00:00Z
review text: The book represents an attempt to give a unified approach for a special kind of optimal control problems governed by linear partial differential equations with impulse-like controls in the right-hand side. As a characteristic prototype one can consider the minimization of the functional \[ I(u):= \int_Q a(x,t)(u(x, t)- g(x,t))^2 dx dt \] over solutions \(u\) of the parabolic equation \[ \begin{multlined} (Lu)(x,t):= u_t(x,t)- \sum^n_{i,j=1} {\partial\over\partial x_i} (a_{ij}(x) u_{x_j})\\ = f(x,t)+ \sum^N_{s=1} \delta(t- \tau_s) \varphi_s(x)+ \sum^N_{k=1} \delta(x_1- \xi_k) \psi_k(x_2,\dots, x_n, t)\\ := (f+ F(h))(x, t),\qquad (x,t)\in Q,\end{multlined}\tag{1} \] \[ u|_{t=0}= 0,\qquad u|_{\partial\Omega}= 0, \] when the control \(h:= \{\tau_1,\dots, \tau_N; \varphi_1,\dots, \varphi_N; \xi_1,\dots, \xi_M;\psi_1,\dots, \psi_M\}\) runs over a given set \(U_0\) from an appropriate space \(U\). Here \(Q= \Omega\times (0, T)\) is a cylinder with \(\Omega\subset \mathbb{R}^n\) being a bounded set with a smooth boundary \(\partial\Omega\) and \(\delta\) is the Dirac function. Because the right-hand side of (1) does not belong to Lebesgue spaces one has to consider generalized solutions of (1). This problem in the book is solved by reformulating (1) as an equation in appropriate Hilbert spaces: find \(u\in H_1\) such that \[ \langle L^+v,u\rangle_{H_1}= \langle f+F(h),v\rangle_{H_2} \forall v\in H_2: L^+v\in H^-_1,\tag{2} \] where \(L^+\) is the algebraically adjoint to the operator \(L\) and by \(H^-\) is denoted the dual space to a Hilbert space \(H\). The most important part of the book are various, more or less standard, estimates, which ensure that the equation (2) is uniquely solvable and that the mapping \(L^{-1}\), \(L^{-1}(f+ F(h))= u\), is continuous from \(H^-_2\) to \(H_1\). That is done for specific types of operators \(L\) of the kind: parabolic (as in (1)), hyperbolic, pseudoparabolic, pseudohyperbolic and Sobolev type operators. The essence here is the right…
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49N25
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 1840598
zbMATH Keywords: impulse control
zbMATH Keywords: optimal control
zbMATH Keywords: linear partial differential equations
zbMATH Keywords: weak continuity
zbMATH Keywords: weak lower semicontinuity
zbMATH Keywords: cost functional
reviewed by: Uldis Raitums
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1840598