Dirichlet boundary control of semilinear parabolic equations. II: Problems with pointwise state constraints
type: Item
instance of: scholarly article
title: Dirichlet boundary control of semilinear parabolic equations. II: Problems with pointwise state constraints
zbMATH Open document ID: 1037.49023
published in: Applied Mathematics and Optimization
publication date: 2002-06-10T00:00:00Z
review text: The systems are described by the parabolic equation \[ {\partial y(t, x) \over \partial t} + Ay(t, x) + \Phi(t, x, y(t, x), u(t, x)) = 0 \] in a cylinder \((0, T) \times \Omega,\) \(\Omega\) an \(n\)-dimensional domain with boundary \(\Gamma,\) with an initial condition on the base of the cylinder and the Dirichlet boundary condition \(y(t, x) = v(t, x)\) on \((0, T) \times \Gamma;\) \(A\) is a second order elliptic operator. The optimal problem is that of minimizing an integral cost functional under a pointwise state constraint. In order to obtain a version of Pontryagin's maximum principle for this kind of problem one needs to pick multipliers in the dual of \(L^\infty((0, T) \times \Omega).\) These multipliers are finitely additive measures and produce an adjoint equation not interpretable in the sense of distributions. The authors tackle this problem by proving a theorem on decomposition of finitely additive measures into a regular part (a Radon measure) and a finitely additive part. Only the regular measure intervenes in the adjoint equation; the other measure plays a role in the part of Pontryagin's principle involving the boundary control \(v(t, x).\) [Cf. also Part I in Appl. Math. Optimization 45, No.~2, 125--143 (2002; Zbl 1005.49016)].
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35B37
zbMATH DE Number: 1753056
zbMATH Keywords: Dirichlet boundary control
zbMATH Keywords: pointwise state constraints
zbMATH Keywords: semilinear parabolic equations
zbMATH Keywords: Pontryagin's maximum principle
zbMATH Keywords: Stone-Čech compactification
author: Nadir Arada
author: J-P Raymond
reviewed by: Hector O. Fattorini
MaRDI profile type: Publication
full work available at URL: s00245-001-0036-4
OpenAlex ID: W2029606802
DOI: 10.1007/S00245-001-0036-4
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description: scientific article; zbMATH DE number 1753056