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

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.