Optimal control of conditioned processes with feedback controls
Optimal control of conditioned processes with feedback controls
复制标题
通过反馈控制对条件过程进行优化控制
DOI：
10.1016/j.matpur.2020.07.014
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
P. Lions
中科院分区：
文献类型：
--
作者：
Y. Achdou;M. Laurière;P. Lions
考虑了一类有限时间内的闭环随机最优控制问题，其中代价是一个期望，条件是过程不退出给定的有界区域.一个重要的困难是，制约策略的事件的概率随着时间的增长而衰减。最优性条件由一个偏微分方程组组成，包括一个Hamilton-Jacobi-Bellman方程（向后w.r.t.时间）和a（前向w.r.t.时间）福克-普朗克方程的规律的条件过程。这两个方程都补充了Dirichlet条件。接下来，我们讨论当时间范围趋于+∞时的渐近行为。这导致了一种新的最优控制问题驱动的特征值问题与边界上的Dirichlet条件的连续性方程。我们证明了后者的存在。我们还提出了数值方法和补充的各种理论方面的模拟。
We consider a class of closed loop stochastic optimal control problems in finite time horizon, in which the cost is an expectation conditional on the event that the process has not exited a given bounded domain. An important difficulty is that the probability of the event that conditionates the strategy decays as time grows. The optimality conditions consist of a system of partial differential equations, including a Hamilton-Jacobi-Bellman equation (backward w.r.t. time) and a (forward w.r.t. time) Fokker-Planck equation for the law of the conditioned process. The two equations are supplemented with Dirichlet conditions. Next, we discuss the asymptotic behavior as the time horizon tends to +∞. This leads to a new kind of optimal control problem driven by an eigenvalue problem related to a continuity equation with Dirichlet conditions on the boundary. We prove existence for the latter. We also propose numerical methods and supplement the various theoretical aspects with simulations.