# American Institute of Mathematical Sciences

July  1997, 3(3): 341-370. doi: 10.3934/dcds.1997.3.341

## Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints

 1 Laboratoire MIP, UMR 5640, Université Paul Sabatier, 31062 Toulouse Cedex 4

Received  August 1996 Published  April 1997

We consider optimal control problems governed by semilinear par- abolic 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 we prove the existence of a unique weak solution for this equation in an appropriate space.
Citation: J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341
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