Directional sparsity in optimal control of partial differential equations. (English) Zbl 1244.49038

Herzog, Roland; Stadler, Georg; Wachsmuth, Gerd

SIAM J. Control Optim. 50, No. 2, 943-963 (2012).

Summary: 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.

MSC:

49K20	Optimality conditions for problems involving partial differential equations
65K10	Numerical optimization and variational techniques
49M15	Newton-type methods
49J52	Nonsmooth analysis

Keywords:

directional sparsity; optimal control; partial differential equations; nondifferentiable cost functional; methods of Newton type; elliptic problems; parabolic problems

Software:

SLEP