When solving optimal impulse control problems, one can use the dynamic programming approach in two different ways: at each time moment, one can make the decision whether to apply a particular type of impulse, leading to the instantaneous change of the state, or apply no impulses at all; or, otherwise, one can plan an impulse after a certain interval, so that the length of that interval is to be optimised along with the type of that impulse. The first method leads to the differential Bellman equation, while the second method leads to the integral Bellman equation. The target of the current article is to prove the equivalence of those Bellman equations in many specific models. Those include abstract dynamical systems, controlled ordinary differential equations, piece-wise deterministic Markov processes and continuous-time Markov decision processes.

在求解最优脉冲控制问题时，可以通过两种不同的方式使用动态规划方法：在每个时刻，可以决定是否施加特定类型的脉冲，导致状态的瞬时变化，或者不施加完全没有冲动；或者，可以在某个时间间隔后规划一个脉冲，以便根据该脉冲的类型优化该时间间隔的长度。第一种方法导致微分贝尔曼方程，而第二种方法导致积分贝尔曼方程。本文的目标是证明那些贝尔曼方程在许多特定模型中的等价性。这些包括抽象动态系统、受控常微分方程、分段确定性马尔可夫过程和连续时间马尔可夫决策过程。