Sparse optimal stochastic control

Kaito Ito, Takuya Ikeda, Kenji Kashima

Abstract: In this paper, we investigate a sparse optimal control of continuous-time stochastic systems. We adopt the dynamic programming approach and analyze the optimal control via the value function. Due to the non-smoothness of the L0<math><msup is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msup></math> 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 necessary and sufficient condition for the L0<math><msup is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msup></math> optimality, which immediately gives the optimal feedback map. Especially for control-affine systems, we consider the relationship with L1<math><msup is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msup></math> optimal control problem and show an equivalence theorem.