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

Abstract. We study optimal control problems in which controls with certain sparsity patterns are preferred. For time-dependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval. The approach uses a nondifferentiable cost functional to implement the sparsity requirements; additionally, bound constraints for the optimal controls can be included. We study the resulting problem in appropriate function spaces and present two solution methods of Newton type, based on different formulations of the optimality system. Using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms.

Optimality systems and their linearizations arising in optimal control of partial differential equations with pointwise control and (regularized) state constraints are considered. The preconditioned conjugate gradient (pcg) method in a non-standard inner product is employed for their efficient solution. Preconditioned condition numbers are estimated for problems with pointwise control constraints, mixed control-state constraints, and of Moreau-Yosida penalty type. Numerical results for elliptic problems demonstrate the performance of the pcg iteration. Regularized stateconstrained problems in 3D with more than 750,000 variables are solved.

MSC (2000)49-M05, 49-M37, 76-D55, 90-C06, Some first and second order algorithmic approaches for the solution of PDE-constrained optimization problems are reviewed. An optimal control problem for the stationary Navier-Stokes system with pointwise control constraints serves as an illustrative example. Some issues in treating inequality constraints for the state variable and alternative objective functions are also discussed.

Equations of linear and nonlinear infinitesimal elasticity with mixed boundary conditions are considered. The bounded domain is assumed to have a Lipschitz boundary and to satisfy additional regularity assumptions. W 1,p regularity for the displacements and L p regularity for the stresses are proved for some p > 2.