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Zbl 1113.49031
Gombao, Sophie; Raymond, Jean-Pierre
Hamilton-Jacobi equations for control problems of parabolic equations. (English)
[J] ESAIM, Control Optim. Calc. Var. 12, 311-349 (2006). ISSN 1292-8119; ISSN 1262-3377/e

Summary: 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 $(A,D(A))$ 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.

MSC 2000:: *49L25 Viscosity solutions
35K20 Second order parabolic equations, boundary value problems
49K20 Optimal control problems with PDE (nec./ suff.)
49J20 Optimal control problems with PDE (existence)

Keywords: Hamilton-Jacobi-Bellman equation; boundary control; semilinear parabolic equations

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