Spatially sparse optimization problems in fractional order Sobolev spaces

TL;DR

This work addresses time-dependent optimal control with spatial sparsity by introducing an auxiliary spatial variable $w$ in $H^s(\Omega)$ to promote sparsity of $u$ via the $L^p$-pseudo norm with $0<p<1$. A smoothing/penalization framework yields a tractable auxiliary problem whose solutions converge to a stationary system for the original non-smooth formulation, featuring multipliers associated with the sparsity term and the inequality constraints. An iterative majorize-minimize scheme coupled with a semismooth Newton method solves the discretized subproblems, and the authors prove existence, regularity, and convergence results while demonstrating numerically that smaller $p$ and larger $\gamma$ yield stronger spatial sparsity. The approach avoids computing the fractional Laplacian on the time-space cylinder by working with an auxiliary function on the spatial domain, enabling feasible numerical treatment and revealing effective sparsity patterns in both 1D and 2D examples. Overall, the paper advances theory and computation for spatially sparse, time-dependent control problems in fractional Sobolev spaces with practical implications for actuator placement and sparse control design.

Abstract

We investigate time-dependent optimization problems in fractional Sobolev spaces with the sparsity promoting $L^p$-pseudo norm for $0<p<1$ in the objective functional. In order to avoid computing the fractional Laplacian on the time-space cylinder $I\times Ω$, we introduce an auxiliary function $w$ on $Ω$ that is an upper bound for the function $u\in L^2(I\timesΩ)$. We prove existence and regularity results and derive a necessary optimality condition. This is done by smoothing the $L^p$-pseudo norm and by penalizing the inequality constraint regarding $u$ and $w$. The problem is solved numerically with an iterative scheme whose weak limit points satisfy a weaker form of the necessary optimality condition.

Figures (2)

• Figure 1: Section \ref{['sec:1dExample']}: Plot of $u_d$ and the computed solutions $w_\mathrm{ref}$ and $u_\mathrm{ref}$ for $\gamma = 5$.
• Figure 2: Section \ref{['sec:2dExample']}: Plots of $u_d$ and numerical solutions $u$ and $w$ for $p=0.3$.

Theorems & Definitions (31)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 3.2
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Theorem 4.4
• Lemma 4.5