Dynamics, phase constraints, and linear programming

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.