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 $L^0$ 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 necessary and sufficient condition for the $L^0$ optimality, which immediately gives the optimal feedback map. Especially for control-affine systems, we consider the relationship with $L^1$ optimal control problem and show an equivalence theorem.

在本文中，我们研究了连续时间随机系统的稀疏最优控制。我们采用动态编程方法，并通过值函数分析最优控制。由于不光滑${\mathrm{大号}}^0$成本函数，通常，价值函数在领域中是不可微的。然后，我们将值函数表征为相关汉密尔顿-雅各比-贝尔曼（HJB）方程的粘度解。根据结果​​，我们得出了${\mathrm{大号}}^0$最优性，立即给出最优反馈图。特别是对于仿射系统，我们考虑与${\mathrm{大号}}^{\mathrm{1个}}$ 最优控制问题并显示等价定理。