This paper deals with state-constrained optimal control problems governed by semilinear parabolic equations. We establish a minimum principle of Pontryagin's type. To deal with the state constraints, we introduce a penalty problem by using Ekeland's principle. The key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined. Conditions for normality of optimality conditions are given.

This paper deals with a quadratic control problem for elliptic equations with pointwise state constraints. Existence and uniqueness ofthe solution is proved. Optimality conditions are given and regularity of the optimal solution is investigated.

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The control is the trace of the state on the boundary of the domain, which is assumed to be a convex, polygonal, open set in R 2 . Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the error estimates are of order O(h 1−1/p ) for some p > 2, which is consistent with the W 1−1/p,p (Γ)-regularity of the optimal control. 1. Introduction. In this paper we study an optimal control problem governed by a semilinear elliptic equation. The control is the Dirichlet datum on the boundary of the domain. Bound constraints are imposed on the control. The cost functional involves the control in a quadratic way and the state in a general way. The goal is to derive error estimates for the discretization of the control problem.There are not many papers devoted to the derivation of error estimates for the discretization of control problems governed by partial differential equations; see the pioneering works by Falk [19] and Geveci [21]. However, recently some papers have appeared, providing new methods and ideas. Arada, Casas, and Tröltzsch [1] derived error estimates for the controls in the L ∞ and L 2 norms for distributed control problems. Similar results for an analogous problem, but also including integral state constraints, were obtained by Casas [8]. The case of a Neumann boundary control problem has been studied by Casas, Mateos, and Tröltzsch [11]. The novelty of our paper with respect to the previous ones is twofold. First, here we deal with a Dirichlet problem, the control being the value of the state on the boundary. Second, we consider piecewise linear continuous functions to approximate the optimal control, which is necessary because of the Dirichlet nature of the control, but it introduces some new difficulties. In the previous papers the controls were always approximated by piecewise constant functions. In the present situation we have developed new methods, which can be used in the framework of distributed or Neumann controls to consider piecewise linear approximations. This could lead to better error estimates than those deduced for piecewise controls.As far as we know, there is another paper dealing with the numerical approximation of a Dirichlet control problem of Navier-Stokes equations, by Gunzburger, *

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Abstract. Semilinear elliptic optimal control problems involving the L 1 norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown. Error estimates for the variational discretization of the problem in the sense of [13] are also obtained. Numerical experiments confirm the convergence rates.Key words. optimal control of partial differential equations, non-differentiable objective, sparse controls, finite element discretization, a priori error estimates