Maximum Principle for State - Constrained Optimal

In this paper, the authors study an optimal control problem for quasilinear elliptic PDEs with pointwise state constraints. Weak and strong optimality conditions of Pontryagin maximum principle type are derived. In proving these results, we penalized the state constraints and respectively use the Ekeland variational principle and an exact penalization method. In this paper, our aim is to prove Pontryagin's principle for pointwise state-constrained optimal control problems governed by very general quasilinear elliptic equations. The control is distributed and takes values in a bounded subset, not necessarily convex, of some Euclidean space. The cost functional is Lagrange type. Standard results of optimal control problems for linear elliptic equations with convex control set and convex functional can be found in 16]. In 1,5], the results were extended to linear or semilinear equations with state constraints. In the framework of semilinear elliptic equations, the Pontryagin type principle was rst proved in 2] for problems without