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Optimal control of piecewise deterministic nonlinear systems with controlled transitions: viscosity solutions, their existence and uniqueness

机译：具有控制转移的分段确定性非线性系统的最优控制：粘度解，存在性和唯一性

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The paper studies viscosity solutions of two sets of Hamilton-Jacob-Bellman (HJB) equations (one for finite horizon and the other one for infinite horizon) which arise in the optimal control of nonlinear piecewise deterministic systems where the controls could be unbounded. The controls enter through the system dynamics as well as the transitions for the underlying Markov chain process, and have access to both the continuous state and the current state of the Markov chain. The two HJB equations associated with this problem are coupled partial differential equations, as a result of which their Hamiltonian structures are different from the standard ones. The paper establishes the existence and uniqueness of their viscosity solutions, and derives explicit structures for the optimum controllers by using such viscosity solutions.
机译：本文研究了两套Hamilton-Jacob-Bellman（HJB）方程的粘性解决方案（一组用于有限水平，另一组用于无限水平），这些方程是在非线性分段确定性系统的最优控制中产生的，其中控制可以是无界的。控件通过系统动力学以及基础马尔可夫链过程的过渡进入，并且可以访问马尔可夫链的连续状态和当前状态。与该问题相关的两个HJB方程是耦合的偏微分方程，其哈密顿结构与标准方程不同。本文建立了它们的粘度解的存在性和唯一性，并通过使用这种粘度解推导了最优控制器的明确结构。

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