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

Abstract

Abstract. Optimal control problems with semilinear parabolic state equations are considered. The objective features one out of three different terms promoting various spatio-temporal sparsity patterns of the control variable. For each problem, first-order necessary optimality conditions, as well as secondorder necessary and sufficient optimality conditions are proved. The analysis includes the case in which the objective does not contain the squared norm of the control.Mathematics Subject Classification. 49K20, 49J52, 35K58, 65K10.

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

“…We can deduce -like in the continuous case, see [8,Remark 2.10], or the case of piecewise constant approximations, see [10,Remark 4]-the existence of a critical value µc > 0 such thatūσ ≡ 0 for all µ > µc and all σ. Indeed, using the uniform boundness of {ūσ} in L 2 (Q h ) proved in Lemma 6.2 and applying the standard stability estimates to the discretization of the state equation, we conclude the existence of C > 0 such that…”

“…Next we state the properties of the nondifferentiable part of the functional. The proof of the following result can be found in [8].…”

“…Sparse time-dependent problems have been investigated in [1, 5, 8-10, 12, 15, 16]. In this type of problems, there are different possible formulations leading to optimal controls that have different sparse structures; see [8]. Here, we are interested in controls that are sparse in space, not necessarily in time.…”

Stabilization by Sparse Controls for a Class of Semilinear Parabolic Equations

“…The reader is referred to [6,Proposition 2.8] for the proof of this result where the role of x and t are reversed. Now we are prepared to establish the optimality conditions for a local solution of (P) in the sense of L 2 (0, ∞; L 2 (ω)).…”

A Hybrid Finite-Dimensional RHC for Stabilization of Time-Varying Parabolic Equations

“…Further, the choice of | · | * = | · | 1 leads to a nonsmooth infinite horizon problem. Finite-horizon optimal control problems with nonsmooth structure have been well-studied for both finite-and infinite-dimensional controlled systems, see e.g., [2,16,17,20,21,33,43,48]. On the other hand, there is very little research dealing with infinite horizon nonsmmoth problems, see e.g., [18,35].…”