Measure Valued Directional Sparsity for Parabolic Optimal Control Problems

Abstract

A directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems. It allows for controls which are localized in space, where the spatial support is independent of time. Well-posedness of the optimal control problems is established and the optimality system is derived. It is used to establish structural properties of the minimizer. An a priori error analysis for finite element discretization is obtained, and numerical results illustrate the effects of proposed cost functional and the convergence results.

Optimal Control of Semilinear Parabolic Equations by BV-Functions

“…This agrees to some extent with previous contributions on optimal control with measures, cf. [3,4,15,18], where it is also observed that this error decays faster than linear. The error ȳ − y σ (ū) L 2 (Q) converges quadratically, which is in accordance with Proposition 4.2.…”

“…Since g : M(0, T ) −→ R is convex and continuous and D t : BV (0, T ) −→ M(0, T ) is a linear and continuous mapping, we can apply the chain rule [11, Chapter I, Proposition 5.7] to deduce that ∂(g • D t )(u) = D * t ∂g(u ), which immediately leads to (15).…”

“…From the above inequality, the definition of the subdifferential of a convex function, and using (15) it follows that…”

Parabolic control problems in space-time measure spaces

“…A priori error estimates for finite element approximation of linear elliptic optimal control problems with measure valued controls were investigated in [16]. The parabolic case was considered in [6,13] with different measure valued topologies to enhance directional spatial sparsity. The terminology directional sparsity was introduces in [12].…”

Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations

“…Corresponding optimal control problems with linear elliptic and parabolic equations have been studied in [11], and the term directionally sparse controls was coined there. An extension of this work to measure-valued controls can be found in [13].…”