Convergence and regularization results for optimal control problems with sparsity functional

Abstract

Abstract. Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L 1 -norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments.Mathematics Subject Classification. 49M05, 65N15, 65N30, 49N45.

A semismooth Newton method for a class of semilinear optimal control problems with box and volume constraints

“…with λ = 0 fixed) are established in [19]. The convergence of the solutions when ε → 0 is studied in [29] for linear problems including an L 1 control cost. (2) Noticing that T (2) (s, t) = s|t| + min(0, t), it has been proposed in [2] the function…”

“…A continuation technique is then necessary to obtain the solution of the nonregularized dual problem. Alternatively, the problem can be regularized by adding the L 2 -norm of the control to the functional to be minimized, without loosing the sparse properties of the L 1 -norm; see [9,25,29] for details.…”

Annular and sectorial sparsity in optimal control of elliptic equations

“…A priori and a posteriori error estimates for this case were provided in Wachsmuth and Wachsmuth [2011]. In a sequence of papers Casas et al [2012b,c], the authors proved second-order necessary and sufficient optimality conditions for the non-convex case governed by a semilinear elliptic equation, and provided a priori finite element error estimates for different choices of the control discretization.…”

Approximation of sparse controls in semilinear equations by piecewise linear functions

“…First, the L 1 norm of the control is often a natural measure of the control cost, as was observed for instance in [21, Section 6.1]. Second, this term leads to sparsely supported optimal controls, which are desirable, for instance, in actuator placement problems [20,14,11,22].…”

“…The first error estimates for problems with an L 1 norm were given in [22] for piecewise linear control discretizations in the case of a linear elliptic equation with the standard objective involving L(x, y) = 2 . The authors obtained convergence of order h in the L 2 norm.…”