Pontryagin's principle for local solutions of control problems with mixed control-state constraints (Q2706149)

The paper considers optimal control problems for the equation NEWLINE\[NEWLINE\begin{aligned} {\partial y\over\partial t} &- \text{div}(A(x,t)\nabla y+\overline a(x,t) y)+ \langle\overline b(x,t),\nabla y\rangle+ f(x,t,y)= 0\quad\text{in }\Omega\times (0,T),\\ {\partial y\over\partial\nu} &+ g(x,t,y,u(x, t))= 0\quad\text{on }\partial\Omega\times (0,T),\quad y|_{t=0}= y_0,\end{aligned}\tag{1}NEWLINE\]NEWLINE with additional pointwise state constraints and integral state-control constraints.NEWLINENEWLINENEWLINEHere \(\Omega\subset \mathbb{R}^n\), \(n\geq 2\), is a bounded domain with Lipschitz boundary \(\partial\Omega\). The set \(U\), in general unbounded, of admissible controls \(u\) belongs to \(L_r(\partial\Omega\times (0,T); R^s)\), \(r> n+ 1\), consists of all measurable selections of a measurable set-valued mapping with nonempty closed values, and on elements of \(U\) can be imposed some integral constraints. Under some, rather general, assumptions on the coefficients of (1) the authors show that the mapping \(u\to y_u\) (\(y_u\) is the solution of (1) corresponding to a chosen \(u\in U\)) is continuous from \(U\) to \(C(\overline\Omega\times [0,T])\). After that by using penalization method and Ekeland's variational principle the necessary optimality conditions in the form of the Pontryagin's minimum principle are derived for global and local solutions of the corresponding optimal control problem.