This paper concerns the numerical procedure for solving hybrid optimal control problems with sliding modes. A sliding mode is coped with differential-algebraic equations (DAEs) and that guarantees accurate tracking of the sliding motion surface. In the second part of the paper we demonstrate the correspondence between the discrete adjoint equations and the discretized version of the continuous adjoint equations in the case of system equations described by DAEs. We show that the discrete adjoint state trajectories converge to their continuous counterparts. Next, we describe the application of the proposed procedure to three optimal control problems. The first problem concerns optimal control of a simple mechanical system with dry friction. The second problem is related to the planning of a haemodialysis process. The third problem concerns the optimal steering of a racing car.

中文翻译：

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带有滑模的混合系统最优控制的数值程序，第二部分

本文涉及求解带有滑模的混合最优控制问题的数值程序。滑动模式可以解决微分代数方程（DAE），并确保精确跟踪滑动运动表面。在本文的第二部分中，我们演示了在DAE描述的系统方程的情况下，离散伴随方程与连续伴随方程的离散化版本之间的对应关系。我们表明，离散的伴随状态轨迹收敛于它们的连续对应关系。接下来，我们描述所提出的程序对三个最优控制问题的应用。第一个问题涉及具有干摩擦的简单机械系统的最佳控制。第二个问题与血液透析过程的计划有关。