This paper proposes a novel optimal control design framework for hybrid nonlinear dynamical systems involving an interacting combination of continuous-time and discrete-time dynamics. Two numerical algorithms are proposed to approximate the continuous-time and discrete-time portions of the hybrid Hamilton-Jacobi-Bellman (HJB) equation. Galerkin’s spectral method is utilised to approximate the value function involved in the continuous-time HJB equation, thereby computing the optimal control gains between impulsive events. Employing the spectral collocation method, the discrete-time HJB equation is then approximated to find the optimal control gain vector at impulsive instants. These two algorithms are ultimately combined to obtain the desired hybrid nonlinear optimal control law. Describing practical considerations for implementing the algorithms, some illustrative examples are presented to evaluate the functionality of the proposed hybrid nonlinear optimal controller.

本文提出了一种新颖的优化控制设计框架，用于混合非线性动力系统，涉及连续时间和离散时间动态的相互作用组合。提出了两种数值算法来近似混合 Hamilton-Jacobi-Bellman (HJB) 方程的连续时间和离散时间部分。伽辽金谱法用于逼近连续时间HJB方程中涉及的值函数，从而计算脉冲事件之间的最优控制增益。采用谱配置方法，离散时间HJB方程然后被近似以找到脉冲时刻的最佳控制增益向量。这两种算法最终结合起来，得到所需的混合非线性最优控制律。描述实现算法的实际考虑，