Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints

The goal of this paper is to prove the first and second order optimality conditions for some control problems governed by semilinear elliptic equations with pointwise control constraints and finitely many equality and inequality pointwise state constraints. To carry out the analysis we formulate a regularity assumption which is equivalent to the first order optimality conditions. Though the presence of pointwise state constraints leads to a discontinuous adjoint state, we prove that the optimal control is Lipschitz in the whole domain. Necessary and sufficient second order conditions are proved with a minimal gap between them

Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations

Optimal control problems with semilinear parabolic state equations are considered. The objective features one out of three different terms promoting various spatio-Temporal sparsity patterns of the control variable. For each problem, first-order necessary optimality conditions, as well as secondorder necessary and sufficient optimality conditions are proved. The analysis includes the case in which the objective does not contain the squared norm of the control.The first author was partially supported by Spanish Ministerio de Economía y Competitividad under the project MTM2011-22711

Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems

Optimal control problems for semilinear elliptic equations with control constraints and pointwise state constraints are studied. Several theoretical results are derived, which are necessary to carry out a numerical analysis for this class of control problems. In particular, sufficient second-order optimality conditions, some new regularity results on optimal controls and a sufficient condition for the uniqueness of the Lagrange multiplier associated with the state constraints are presented

Optimality conditions and error analysis of semilinear elliptic control problems with L1 cost functional

Semilinear elliptic optimal control problems involving the L1 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 [M. Hinze, Comput. Optim. Appl., 30 (2005), pp. 45--61] are also obtained. Numerical experiments confirm the convergence rates.This author was partially supported by the Spanish Ministerio de Economía y Competitividad under the project MTM2011-2271

Convergence analysis of the semismooth Newton method for sparse control problems governed by semilinear elliptic equations

We show that a second order sufficient condition for local optimality, along with a strict complementarity condition, is enough to get the superlinear convergence of the semismooth Newton method for an optimal control problem governed by a semilinear elliptic equation. The objective functional may include a sparsity promoting term and we allow for box control constraints.Comment: 18 page

Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints

This paper deals with necessary and sufficient optimality conditions for control problems governed by semilinear elliptic partial differential equations with finitely many equality and inequality state constraints. Some recent results on this topic for optimal control problems based upon results for abstract optimization problems are compared with some new results using methods adapted to the control problems. Meanwhile, the Lagrangian formulation is followed to provide the optimality conditions in the first case; the Lagrangian and Hamiltonian functions are used in the second statement. Finally, we prove the equivalence of both formulations

Approximation of boundary control problems on curved domains

In this paper we consider boundary control problems associated to a semilinear elliptic equation defined in a curved domain Ω. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of Ω by an appropriate domain Ωh (typically polygonal) is required. Here we do not consider the numerical approximation of the control problems. Instead, we formulate the corresponding infinite dimensional control problems in Ωh, and we study the influence of the replacement of Ω by Ωh on the solutions of the control problems. Our goal is to compare the optimal controls defined on Γ = ∂Ω with those defined on Γh = ∂Ωh and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates