Dirichlet boundary control of semilinear parabolic equations - Part 1: Problems with no state constraints

This paper is concerned with distributed and Dirichlet boundary controls of semilinear parabolic equations, in the presence of pointwise state constraints. The paper is divided into two parts. In the first part we define solutions of the state equation as the limit of a sequence of solutions for equations with Robin boundary conditions. We establish Taylor expansions for solutions of the state equation with respect to perturbations of boundary control (Theorem 5.2). For problems with no state constraints, we prove three decoupled Pontryagin's principles, one for the distributed control, one for the boundary control, and the last one for the control in the initial condition (Theorem 2.1). Tools and results of Part 1 are used in the second part to derive Pontryagin's principles for problems with pointwise state constraints.