Pontryagin's principle for state-constrained control problems governed by parabolic equations

Here we are concerned with a state-constrained boundary control problem governed by a semilinear parabolic equation. The aim of the paper is to present a general Pontryagin's principle. In the last years, some papers have appeared which extend the results for finite dimensional systems to distributed parameter systems. Many difficulties have arisen. in, the way of this extension, mainly for state-constrained problems. The so-called variational principle of Ekeland has shown. to be the most powerful mathematical tool to deal with these constraints. A second crucial tool has been provided by Li and Yao. The idea of Li and Yao is to replace the classical spike perturbations of the controls, used firstly by Pontryagin, with some variations concentrated also in. a small domain, but instead of being localized around a point, they are distributed in, the whole domain. Here, we present these perturbations in a way that we believe is the simplest and the most general one and which definitively lets to understand their true nature. We will distinguish two different Pontryagin's principles, called weak and strong, respectively. The difference between both principles is that the second one is formulated in a qualified form, which makes necessary an additional assumption. In the absence of state equality constraints, this assumption is formulated here in terms of a stability condition of the minimum cost functional with respect to small perturbations of the feasible state set. We also prove that ''almost all'' problems are stable.