, Volume 24, Issue 1, pp 197-220

A boundary-value problem for Hamilton-Jacobi equations in hilbert spaces

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We study a Hamilton-Jacobi equation in infinite dimensions arising in optimal control theory for problems involving both exit times and state-space constraints. The corresponding boundary conditions for the Hamilton-Jacobi equation, of mixed nature, have been derived and investigated in [19], [2], [5], and [15] in the finite-dimensional case. We obtain a uniqueness result for viscosity solutions of such a problem and then prove the existence of a solution by showing that the value function is continuous.

The work of P. Cannarsa was partially supported by the Italian National Project “Equazioni Differenziali e Calcolo delle Variazioni”. H. M. Soner's work was supported by National Science Foundation Grant DMS-90-02249.