# American Institute of Mathematical Sciences

November  2003, 9(6): 1549-1570. doi: 10.3934/dcds.2003.9.1549

## Time optimal problems with Dirichlet boundary controls

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

Received  June 2002 Revised  May 2003 Published  September 2003

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).
Citation: N. Arada, J.-P. Raymond. Time optimal problems with Dirichlet boundary controls. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1549-1570. doi: 10.3934/dcds.2003.9.1549
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