A dynamic programming approach for L⁰ optimal control design

The present work investigates the optimal control problems with L⁰-control cost. The value function is characterized as the unique viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. The sparsity properties of optimal controllers induced by L⁰-penalty is analyzed under different cases of control constraints. The existence of optimal controllers is discussed for the time-discretized problem. The value function and the optimal control are computed by solving the corresponding HJB equation. Numerical examples are presented under different types of control constraints and different penalization parameters with special attention to the sparsity. Comparisons between L⁰-controller and other types of controllers are also illustrated.