Gombao, Sophie; Raymond, Jean-Pierre Hamilton-Jacobi equations for control problems of parabolic equations. (English) Zbl 1113.49031 ESAIM, Control Optim. Calc. Var. 12, 311-349 (2006). 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. Cited in 1 Document MSC: 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 35K20 Initial-boundary value problems for second-order parabolic equations 49K20 Optimality conditions for problems involving partial differential equations 49J20 Existence theories for optimal control problems involving partial differential equations Keywords:Hamilton-Jacobi-Bellman equation; boundary control; semilinear parabolic equations × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] H. Amann , Linear and quasilinear parabolic problems . Vol. I, Abstract linear theory. Birkhäuser Boston Inc., Boston, MA. Monographs Math. 89 ( 1995 ). MR 1345385 | Zbl 0819.35001 · Zbl 0819.35001 [2] V. Barbu and G. Da Prato , Hamilton-Jacobi equations in Hilbert spaces , Pitman (Advanced Publishing Program), Boston, MA Res. Notes Math. 86 ( 1983 ). MR 704182 | Zbl 0508.34001 · Zbl 0508.34001 [3] A. Bensoussan , G. Da Prato , M.C. 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