Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. (English) Zbl 0953.49026

Raymond, J.-P.

Discrete Contin. Dyn. Syst. 3, No. 3, 341-370 (1997).

Summary: We consider optimal control problems governed by semilinear parabolic equations with nonlinear boundary conditions and pointwise constraints on the state variable. In Robin boundary conditions considered here, the nonlinear term is neither necessarily monotone nor Lipschitz with respect to the state variable. We derive optimality conditions by means of a Lagrange multiplier theorem in Banach spaces. The adjoint state must satisfy a parabolic equation with Radon measures in Robin boundary conditions, in the terminal condition and in the distributed term. We give a precise meaning to the adjoint equation with measures as data and prove the existence of a unique weak solution for this equation in an appropriate space.

MSC:

49K20	Optimality conditions for problems involving partial differential equations
35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
49J20	Existence theories for optimal control problems involving partial differential equations

Keywords:

nonlinear boundary controls; pointwise state constraints; parabolic equations with measures as data; optimal control; semilinear parabolic equations; Robin boundary conditions; optimality conditions; existence