Sparse optimal control for infinite-dimensional linear systems with applications to graphon control

TL;DR

This paper develops a method for sparse optimal control of infinite-dimensional systems, leveraging graphon theory to approximate large-scale networked systems efficiently, with practical numerical demonstrations.

Contribution

It introduces a sufficient condition for deriving sparse controls via L1 optimization and connects finite network controls to their graphon limits for scalable solutions.

Findings

Sparse control can be obtained through L1 optimization under certain conditions.

Approximation of large network controls by graphon-based controls is effective.

Numerical examples demonstrate the approach's practicality.

Abstract

Large-scale networked systems typically operate under resource constraints, and it is also difficult to exactly obtain the network structure between nodes. To address these issues, this paper investigates a sparse optimal control for infinite-dimensional linear systems and its application to networked systems where the network structure is represented by a limit function called a graphon that captures the overall connection pattern. The contributions of this paper are twofold: (i) To reduce computational complexity, we derive a sufficient condition under which the sparse optimal control can be obtained by solving its corresponding L1 optimization problem. Furthermore, we introduce a class of non-convex optimal control problems such that the optimal solution always coincides with a sparse optimal control, provided that the non-convex problems admit optimal solutions. (ii) We show that the sparse optimal control for large-scale finite-dimensional networked systems can be approximated by that of the corresponding limit graphon system, provided that the underlying graph is close to the limit graphon in the cut-norm topology. The effectiveness of the proposed approach is illustrated through numerical examples.