CLOT norm minimization for continuous hands-off control

In this paper, we propose optimal control that is both sparse and continuous, unlike previously proposed alternatives to maximum hands-off control. The maximum hands-off control is the L0-optimal (or sparsest) control among all feasible controls that are bounded by a specified value and transfer the state from a given initial state to the origin within a fixed time duration. The proposed control is obtained via minimization of the CLOT (Combined L-One and Two) norm of the control input along with the constraints on the state variable. The constraints on the state variable ensures that the states are not blown up while achieving the optimal control. By using the non-smooth maximum principle, we prove that the CLOT-norm optimal control is unique, and it is continuous in the time variable. We show by numerical simulations that the CLOT control is continuous unlike L0-optimal control (or maximum hands-off control) and much sparser (i.e. has longer time duration on which the control equals 0) than the conventional EN (elastic net) control, which is a convex combination of L1 and squared L2 norms.