Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations
type: Item
instance of: scholarly article
title: Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations
zbMATH Open document ID: 0922.49013
published in: Applied Mathematics and Optimization
publication date: 1999-09-21T00:00:00Z
review text: The system is \[ {\partial y \over \partial t} + Ay + f(t, x, y, u) = 0, \] where \(A\) is a second order elliptic operator in an \(m\)-dimensional domain \(\Omega\) with boundary \(\Gamma\) and \(0 \leq t \leq T\). Besides the distributed control \(u,\) there are controls \(v, w\) acting on the initial and the boundary condition, \[ y(0) = w \quad (x \in \Omega), \qquad {\partial y \over \partial n_A} + g(t, x, y, v) = 0 \quad(x \in \Gamma). \] Here \(u\) (resp. \(v, w)\) belongs to \(L^q((0, T) \times \Omega)\) (resp. \(L^r((0, T) \times \Gamma)\), \(L^\infty(\Omega))\). The controls are subject to the pointwise constraints \[ u(t, x) \in K_U, \quad v(t, x) \in K_V, \quad w(x) \in K_W. \] The cost functional undergoing minimization is \[ J(y, u, v, w)= \int_{(0, T) \times \Omega} F(t, x, y, u)dx dt+ \int_{(0, T) \times \Gamma} G(t, x, y, v) d\sigma dt + \int_\Omega \ell(x, y(T), w) dx. \] The authors define three Hamiltonians (resp. \textit{distributed}, \textit{boundary} and \textit{initial}) by \[ \begin{aligned} H_d(t, x, y, u, p) &= F(t, x, y, u) - pf(t, x, y, u),\\ H_b(t, x, y, v, p) &= G(t, x, y, v) - pg(t, x, y, v),\\ H_i(x, y, w, p) &= \ell(x, y, w) + pw,\end{aligned} \] and obtain necessary conditions for optimal controls \(\bar u(t, x),\) \(\bar v(t, x),\) \(\bar w(x)\) and optimal solutions \(\bar y(t, x)\) in the form of three Pontryagin's principles. These minimum principles are \textit{decoupled} in the sense that each one involves only one of the controls. Since both the distributed and the boundary control may be unbounded, the proof is technically complicated and needs new regularity results for linear and nonlinear parabolic equations.
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 1271357
zbMATH Keywords: Pontryagin's minimum principle
zbMATH Keywords: semilinear parabolic equations
zbMATH Keywords: distributed control
zbMATH Keywords: boundary control
zbMATH Keywords: unbounded controls
author: J-P Raymond
author: Housnaa Zidani
reviewed by: Hector O. Fattorini
MaRDI profile type: Publication
full work available at URL: s002459900102
OpenAlex ID: W2109168354
DOI: 10.1007/S002459900102
description: scientific article; zbMATH DE number 1271357