This paper concerns the numerical procedure for solving hybrid optimal control problems with sliding modes. The proposed procedure has several features which distinguishes it from the other procedures for the problem. First of all a sliding mode is coped with differential-algebraic equations (DAEs) and that guarantees accurate tracking of the sliding motion surface. The second important feature is the calculation of cost and constraints functions gradients with the help of adjoint equations. The adjoint equations presented in the paper take into account sliding motion and exhibit jump conditions at transition instants. The procedure uses the discretization of system equations by Radau IIA Runge--Kutta scheme and the evaluation of optimization functions gradients with the help of the adjoint equations stated for discretized system equations. In the first 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 ODEs. We show that the discrete adjoint state trajectories converge to their continuous counterparts in the case of ODEs.

翻译：本文涉及用滑动模式解决混合最佳控制问题的数字程序。 所拟议的程序有若干特点, 使它有别于问题的其他程序。 首先, 滑动模式是用不同相位方程式( DAEs) 处理的, 保证了对滑动运动表面的准确跟踪。 第二个重要特征是借助双向方程式计算成本和制约梯度。 本文所呈现的双向方程式考虑到滑动动作和在过渡时显示跳跃条件。 该程序使用Radau IIA Runge- Kutta 方案对系统方程式的离散化和对优化函数梯度的评价, 借助为离散式系统方程式说明的双向方程式。 在文件第一部分, 我们演示离散的联式方程式与离散式组合方程式之间的对应关系。 我们显示, 离异方方方方方方方程式在ODs 中与连续对等方方程式的对应方程式相交汇。