Maximum hands-off control with time-space sparsity

Abstract

This paper treats a novel maximum hands-off control problem in which two types of sparsity are considered. Our optimization problem is mathematically formulated as an L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> optimal control problem with an ℓ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> constraint. This optimal control has the minimum length of support and a small number of activated components among all control inputs that steer the system to a target state within a fixed time duration. Since the formulated problem is combinatorial, we introduce a convex relaxation problem for its computational tractability. We show a sufficient condition under which our sparse optimization problem boils down to the convex optimization problem and give an existence theorem of the optimal controls. The proposed control is illustrated through a numerical example.