Parabolic control problems in measure spaces with sparse solutions

Abstract

Optimal control problems in measure spaces lead to controls that have small support, which is desirable, e.g., in the context of optimal actuator placement. For problems governed by parabolic partial differential equations, well-posedness is guaranteed in the space of square-integrable measure-valued functions, which leads to controls with a spatial sparsity structure. A conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates. In particular, although the state is discretized, the control problem can still be formulated and solved in the measure space. Numerical examples illustrate the structural features of the optimal controls.