Hamilton-Jacobi equations for control problems of parabolic equations

Gombao, Sophie; Raymond, Jean-Pierre

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 311-349.

We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers’ type in 2D. To deal with a control acting in a boundary condition a fractional power ${(-A)}^{\beta }$ - where $(A,D(A\left)\right)$ is an unbounded operator in a Hilbert space $X$ - is contained in the hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power ${(-A)}^{\beta }$ appears in another nonlinear term whose behavior is different from the one of the hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets in $X$, but only in bounded subsets in a space $Y\hookrightarrow X$. To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets in $Y$ to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are Hölder continuous with respect to the time variable.

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