Applied Mathematics and Optimization ( IF 1.8 ) Pub Date : 2020-05-30 , DOI: 10.1007/s00245-020-09687-y Elena Bandini , Michèle Thieullen
In this paper we consider the optimal control of Hilbert space-valued infinite-dimensional Piecewise Deterministic Markov Processes (PDMP) and we prove that the corresponding value function can be represented via a Feynman–Kac type formula through the solution of a constrained Backward Stochastic Differential Equation. A fundamental step consists in showing that the corresponding integro-differential Hamilton–Jacobi–Bellman equation has a unique viscosity solution, by proving a suitable comparison theorem. We apply our results to the control of a PDMP Hodgkin-Huxley model with spatial component.
中文翻译:
无限维分段确定性马尔可夫过程的最优控制:BSDE方法。在兴奋细胞膜控制中的应用
在本文中,我们考虑了希尔伯特空间值无限维分段确定性马尔可夫过程(PDMP)的最优控制,并且证明了通过约束后向随机微分的解,可以通过Feynman-Kac型公式来表示相应的值函数。方程。一个基本步骤包括通过证明一个合适的比较定理,证明相应的积分微分哈密顿–雅各比–贝尔曼方程具有唯一的粘度解。我们将我们的结果应用于具有空间分量的PDMP Hodgkin-Huxley模型的控制。