带Markov跳的离散时间随机控制系统的最大值原理
A maximum principle for optimal control of discrete-time stochastic systems with Markov jump
蔺香运 1王鑫瑞 1张维海2
作者信息
- 1. 山东科技大学数学与系统科学学院,山东青岛 266590
- 2. 山东科技大学电气与自动化工程学院,山东青岛 266590
- 折叠
摘要
本文研究一类同时含有Markov跳过程和乘性噪声的离散时间非线性随机系统的最优控制问题,给出并证明了相应的最大值原理.首先,利用条件期望的平滑性,通过引入具有适应解的倒向随机差分方程,给出了带有线性差分方程约束的线性泛函的表示形式,并利用Riesz定理证明其唯一性.其次,对带Markov跳的非线性随机控制系统,利用针状变分法,对状态方程进行一阶变分,获得其变分所满足的线性差分方程.然后,在引入Hamilton函数的基础上,通过一对由倒向随机差分方程刻画的伴随方程,给出并证明了带有Markov跳的离散时间非线性随机最优控制问题的最大值原理,并给出该最优控制问题的一个充分条件和相应的Hamilton-Jacobi-Bellman方程.最后,通过一个实际例子说明了所提理论的实用性和可行性.
Abstract
The maximum principle(MP)of the discrete-time nonlinear stochastic optimal control problem is proved,in which the control systems are driven by both Markov jumps and multiplicative noise.Firstly,based on the adapted solutions of the backward stochastic difference equation,the linear functional with the constraint of a linear difference equation is represented.The Riesz theorem is used to prove the uniqueness of such representation.Secondly,the spike variation method is extend to the nonlinear stochastic difference equation with Markov jumps.The variation equation of such state equation is obtained.Thirdly,by introducing a Hamiltonian function,a necessary condition of the discrete-time nonlinear stochastic optimal control system with Markov jump is obtained.It is proved that the adjoint equation of the maximum principle of the system is a pair of backward stochastic difference equations.Moreover,a sufficient condition is also given and the corresponding Hamilton-Jacobi-Bellman equation is derived.Finally,a practical example is given to illustrate the practicability and feasibility of the proposed theory.
关键词
最大值原理/最优控制/Markov跳/倒向随机差分方程/Hamilton-Jacobi-Bellman方程Key words
maximum principle/optimal control/Markov jump/backward stochastic difference equations/Hamilton-Jacobi-Bellman equations引用本文复制引用
基金项目
国家自然科学基金(62273212)
国家自然科学基金(61973198)
山东省"泰山学者"研究项目()
山东省自然科学基金(62273212)
山东省自然科学基金(61973198)
出版年
2024
国家科技期刊平台
NSTL
万方数据