Abstract

A new approach to solving terminal control problems with phase constraints, using sufficient optimality conditions, is considered. The approach is based on the Lagrangian formalism and duality theory. Linear controlled dynamics under phase constraints is studied. The section of phase constraints at certain time points (on a discrete grid) leads to new intermediate optimal control problems without phase constraints. These problems generate intermediate solutions in intermediate spaces. The combination of all intermediate problems, in turn, leads to the original problem on the entire time interval. In each intermediate space, we have a polyhedral set obtained as a result of a section of the phase constraints. Based on this set, a linear programming problem is formed. Thus, on each small interval between two points of the section, a full-scale intermediate optimal control problem with a fixed left end and a moving right end of the phase trajectory is formed. The right end generates the reachability set and, at the same time, it is a solution of an intermediate boundary-value linear programming problem. The solution obtained, in turn, is the initial condition for the next intermediate optimal control problem. To solve the intermediate optimal control problem, an extragradient saddle-point method is proposed. The convergence of the method to the solution of the optimal control problem in all variables is proved. The convergence guarantees obtaining a solution of the problem with a given accuracy.

中文翻译：

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动力学，相位约束和线性规划

摘要

考虑了使用足够的最优性条件来解决具有相位约束的终端控制问题的新方法。该方法基于拉格朗日形式主义和对偶理论。研究了相位约束下的线性控制动力学。在某些时间点（在离散网格上）的相位约束部分会导致新的中间最优控制问题，而没有相位约束。这些问题在中间空间中产生中间解决方案。所有中间问题的组合又导致整个时间间隔上的原始问题。在每个中间空间中，由于一部分相位约束而获得了一个多面体集。基于该集合，形成线性规划问题。因此，在该部分的两点之间的每个小间隔上，形成了具有轨迹轨迹的左端固定和右端移动的全尺寸中间最优控制问题。右端生成可达性集，同时它是中间边界值线性规划问题的一种解决方案。反过来，获得的解决方案是下一个中间最优控制问题的初始条件。为了解决中间最优控制问题，提出了一种超梯度鞍点法。证明了该方法对所有变量最优控制问题的求解具有收敛性。收敛保证以给定的精度获得问题的解决方案。它是一个中间边界值线性规划问题的解决方案。反过来，获得的解决方案是下一个中间最优控制问题的初始条件。为了解决中间最优控制问题，提出了一种超梯度鞍点法。证明了该方法对所有变量最优控制问题的求解具有收敛性。收敛保证以给定的精度获得问题的解决方案。它是一个中间边界值线性规划问题的解决方案。反过来，获得的解决方案是下一个中间最优控制问题的初始条件。为了解决中间最优控制问题，提出了一种超梯度鞍点法。证明了该方法对所有变量最优控制问题的求解具有收敛性。收敛保证以给定的精度获得问题的解决方案。证明了该方法对所有变量最优控制问题的求解具有收敛性。收敛保证以给定的精度获得问题的解决方案。证明了该方法对所有变量最优控制问题的求解具有收敛性。收敛保证以给定的精度获得问题的解决方案。