Time optimal problems with Dirichlet boundary controls

N. Arada; J.-P. Raymond

doi:10.3934/dcds.2003.9.1549

Article Contents

2003, Volume 9, Issue 6: 1549-1570. Doi: 10.3934/dcds.2003.9.1549

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Time optimal problems with Dirichlet boundary controls

N. Arada^1, and
J.-P. Raymond^2,

1.
UTL IST, Dep. de Matematica, Av. Rovisco Pais, 1049-001, Lisboa
2.
Laboratoire MIP, UMR 5640, Université Paul Sabatier, 31062 Toulouse Cedex 4

Received: June 2002

Revised: May 2003

Published: November 2003

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Abstract

We consider time optimal control problems governed by semilinear parabolic equations with Dirichlet boundary controls in the presence of a target state constraint. To establish optimality conditions for the terminal time $T$, we define a new Hamiltonian functional. Due to regularity results for the state and the adjoint state variables, this Hamiltonian belongs to $L_{l o c}^r(0,T)$ for some $r>1$. By proving that it satisfies a differential equation corresponding to an optimality condition for $T$, we deduce that it belongs to $W^{1,1}(0,T)$. This result answers to the question: how to define Hamiltonian functionals for infinite dimensional problems with variable endpoints (see [10], p. 282 and p. 595).

Keywords:

Time control problem,
optimal control,
Dirichlet controls,
semilinear parabolic equations,
state constraints,
parabolic equations with measures as data.

Mathematics Subject Classification: 49K20, 93C20, 93C50.

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