Sparse optimal feedback control for continuous-time systems

In this paper, we investigate a sparse optimal control in a feedback framework. We adopt the dynamic programming approach and analyze the optimal control via the value function. Due to the non-smoothness of the $L^{0}$ cost functional, in general, the value function is not differentiable in the domain. Then, we characterize the value function as a viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation. Based on the result, we derive a sufficient and necessary condition for the $L^{0}$ optimality, which immediately gives the optimal feedback map. In addition, we consider the relationship with $L^{1}$ optimal control problem and show an equivalence theorem.

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