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Stochastic impulse control of non-smooth dynamics with partial observation and execution delay: application to an environmental restoration problem
arXiv - CS - Systems and Control Pub Date : 2020-06-29 , DOI: arxiv-2006.16034
Hidekazu Yoshioka, Yuta Yaegashi

Non-smooth dynamics driven by stochastic disturbance arise in a wide variety of engineering problems. Impulsive interventions are often employed to control stochastic systems; however, the modeling and analysis subject to execution delay have been less explored. In addition, continuously receiving information of the dynamics is not always possible. In this paper, with an application to an environmental restoration problem, a continuous-time stochastic impulse control problem subject to execution delay under discrete and random observations is newly formulated and analyzed. The dynamics have a non-smooth coefficient modulated by a Markov chain, and eventually attain an undesirable state like a depletion due to the non-smoothness. The goal of the control problem is to find the most cost-efficient policy that can prevent the dynamics from attaining the undesirable state. We demonstrate that finding the optimal policy reduces to solving a non-standard system of degenerate elliptic equations, the optimality equation, which is rigorously and analytically verified in a simplified case. The associated Fokker-Planck equation for the controlled dynamics is derived and solved explicitly as well. The model is finally applied to numerical computation of a recent river environmental restoration problem. The optimality and Fokker-Planck equations are successfully computed, and the optimal policy and the probability density functions are numerically obtained. The impacts of execution delay are discussed to deeper analyze the model.

中文翻译:

具有部分观察和执行延迟的非光滑动力学的随机脉冲控制:在环境恢复问题中的应用

由随机扰动驱动的非光滑动力学出现在各种各样的工程问题中。冲动性干预通常用于控制随机系统;然而,受执行延迟影响的建模和分析的探索较少。此外,连续接收动态信息并不总是可能的。在本文中,通过对环境恢复问题的应用,新制定和分析了在离散和随机观测下受执行延迟影响的连续时间随机脉冲控制问题。动力学具有由马尔可夫链调制的非平滑系数,最终由于非平滑性而达到耗尽等不良状态。控制问题的目标是找到可以防止动态达到不良状态的最具成本效益的策略。我们证明了找到最优策略可以简化为求解退化椭圆方程的非标准系统,即最优方程,在简化的情况下经过严格的分析验证。受控动力学的相关 Fokker-Planck 方程也被明确导出和求解。该模型最终应用于近期河流环境恢复问题的数值计算。成功地计算了最优性和Fokker-Planck方程,并通过数值获得了最优策略和概率密度函数。讨论执行延迟的影响以更深入地分析模型。我们证明了找到最优策略可以简化为求解退化椭圆方程的非标准系统,即最优方程,在简化的情况下经过严格的分析验证。受控动力学的相关 Fokker-Planck 方程也被明确导出和求解。该模型最终应用于近期河流环境恢复问题的数值计算。成功地计算了最优性和Fokker-Planck方程,并通过数值获得了最优策略和概率密度函数。讨论执行延迟的影响以更深入地分析模型。我们证明了找到最优策略可以简化为求解退化椭圆方程的非标准系统,即最优方程,在简化的情况下经过严格的分析验证。受控动力学的相关 Fokker-Planck 方程也被明确导出和求解。该模型最终应用于近期河流环境恢复问题的数值计算。成功地计算了最优性和Fokker-Planck方程,并通过数值获得了最优策略和概率密度函数。讨论执行延迟的影响以更深入地分析模型。受控动力学的相关 Fokker-Planck 方程也被明确导出和求解。该模型最终应用于近期河流环境恢复问题的数值计算。成功地计算了最优性和Fokker-Planck方程,并通过数值获得了最优策略和概率密度函数。讨论执行延迟的影响以更深入地分析模型。受控动力学的相关 Fokker-Planck 方程也被明确导出和求解。该模型最终应用于近期河流环境恢复问题的数值计算。成功地计算了最优性和Fokker-Planck方程,并通过数值获得了最优策略和概率密度函数。讨论执行延迟的影响以更深入地分析模型。
更新日期:2020-06-30
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