论文标题
向后可达的方法，用于跳跃扩散系统的状态约束随机最佳控制问题
Backward Reachability Approach to State-Constrained Stochastic Optimal Control Problems for Jump Diffusion Systems
论文作者
论文摘要
在本文中，我们考虑了具有状态约束的跳跃扩散系统的随机最佳控制问题。通常，此类问题的价值函数是汉密尔顿 - 雅各比 - 贝尔曼（HJB）方程的不连续粘度解，因为不能在状态约束的边界上保证规律性。通过适应\ cite {Bokanowski_sicon_2016}和随机目标理论的方法，我们获得了原始值函数作为后向接触集的等效表示。然后，我们表明，这种向后的可触及可以以无约束的随机控制问题的辅助值函数的零级集合为特征，该辅助值函数函数包括Martingale代表定理，其中包括两个附加的无界控件。我们证明，辅助值函数是相关HJB方程的唯一连续粘度解，这是二阶非线性integro-Partial-Partial Dinailial方程（IPDE）。我们的论文提供了一种表征原始（可能是不连续的）值函数的明确方法，它是辅助HJB方程的连续解决方案的零级集合。由于无界的控制集以及HJB方程非本地运算符中相应的lévy度量的奇异性，存在的存在和唯一性的证明需要一项新技术。
In this paper, we consider the stochastic optimal control problem for jump diffusion systems with state constraints. In general, the value function of such problems is a discontinuous viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation, since the regularity cannot be guaranteed at the boundary of the state constraint. By adapting approaches of \cite{Bokanowski_SICON_2016} and the stochastic target theory, we obtain an equivalent representation of the original value function as the backward reachable set. We then show that this backward reachable can be characterized by the zero-level set of the auxiliary value function for the unconstrained stochastic control problem, which includes two additional unbounded controls as a consequence of the martingale representation theorem. We prove that the auxiliary value function is a unique continuous viscosity solution of the associated HJB equation, which is the second-order nonlinear integro-partial differential equation (IPDE). Our paper provides an explicit way to characterize the original (possibly discontinuous) value function as a zero-level set of the continuous solution of the auxiliary HJB equation. The proof of the existence and uniqueness requires a new technique due to the unbounded control sets, and the presence of the singularity of the corresponding Lévy measure in the nonlocal operator of the HJB equation.
