Directional Sparsity in Optimal Control of Partial Differential Equations

Abstract

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.

Annular and sectorial sparsity in optimal control of elliptic equations

“…The reason is that these constraints are not compatible with the sparsity structure induced by the iterated norm in the objective. This was observed already in Herzog et al [2012]. Following Lemma 4.1 in that paper, we reformulate the optimality system (2.6) as a non-smooth equation.…”

“…The term involving the · L 2 (µ ϕ ) in (3.8b) is treated in the same way as in (2.8). The derivation of the discrete semismooth Newton system proceeds in the same way as in the unconstrained case, and we refer to [Herzog et al, 2012, Section 4] for details. As was observed already there, two modifications should be applied to the Newton system in order to ensure its well-posedness in each step as well as the global convergence of the method in practice.…”

“…This issue was first addressed in Herzog et al [2012], where striped (directional) sparsity patterns were enforced by way of the iterated norm L 1 (L 2 ) in place of the plain L 1 norm. An alternative setting which involves the norm L 2 (0, T; M(Ω)) was studied in Casas et al [2013] for problems governed by linear parabolic equations, where M(Ω) is the space of regular Borel measures on Ω.…”

“…To achieve this, we use an iterated norm as in Herzog et al [2012], but switch to a polar coordinate system. To be precise, we consider an optimal control problem with the following objective, The parameters α and β are positive constants, and y d represents a desired state.…”

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].…”