This work addresses a switching control problem under which the cost associated with the changes of regimes is allowed to have discontinuities in time. Our main contribution is to show several characterizations of the optimal cost function as well as the existence of ε-optimal control policies. As a by-product, we also study the existence and uniqueness of solutions of a system of backward stochastic differential equations whose barriers (or obstacles) are discontinuous (in fact of càdlàg type) and depend itself on the unknown solution. In the last part of the paper, we study the case when an underlying diffusion is part of the dynamic of the system. In this special case, the optimal payoff becomes a weak solution of the HJB system of PDEs with discontinuous obstacles which is of quasi-variational type. This paper is somehow a continuation of the papers [B. Djehiche, S. Hamadène, and A. Popier, A finite horizon optimal multiple switching problem, SIAM J. Control Optim. 48 (2009), pp. 2751–2770; S. Hamadène and M.A. Morlais, Viscosity solutions of systems of PDEs with interconnected Obstacle and switching problem, Appl. Math. Optim. 67 (2013), pp. 163–196.] that consider continuous costs.

这项工作解决了一个切换控制问题，在该问题下，允许与制度变化相关的成本在时间上具有不连续性。我们的主要贡献是展示了最优成本函数的几个特征以及ε的存在——最优控制策略。作为副产品，我们还研究了障碍（或障碍）是不连续的（实际上是 càdlàg 类型）并依赖于未知解的后向随机微分方程系统的解的存在性和唯一性。在论文的最后一部分，我们研究了潜在扩散是系统动态的一部分的情况。在这种特殊情况下，最优收益成为具有不连续障碍物的准变分类型 PDE 的 HJB 系统的弱解。这篇论文在某种程度上是论文的延续 [B. Djehiche、S. Hamadène 和 A. Popier，有限范围最优多重切换问题，SIAM J. Control Optim。48 (2009)，第 2751-2770 页；S. Hamadène 和 MA Morlais，具有互连障碍和切换问题的偏微分方程系统的粘度解，Appl。数学。优化。67 (2013), pp. 163–196.] 考虑持续成本。