ИСТИНА

A terminal control problem with linear dynamics on a fixed time interval and a moving right end of a state trajectory is considered. The right end implicitly sets the terminal condition, which is defined as a solution of a boundary value problem of convex programming. The left end of the state trajectory is fixed. The set of controls is convex and closed. The main feature and difficulty of the problem under consideration is the presence of state constraints. State constraints are defined over the entire time interval. The talk addresses the development of evidence-based and reasonable methods for solving problems of terminal control in the presence of state constraints. The problem is investigated in the Hilbert function space. To obtain sufficient optimality conditions, the Lagrangian is used instead of the Hamiltonian. Slater's regularity condition is required for the existence of saddle points. But the interpretation of the Slater condition depends on the topology of the functional space and this creates additional difficulties. In the Hilbert function space, Slater's condition for state constraints is not satisfied. To overcome these difficulties, we introduce a discretization of state constraints. Cross-sectional state constraints generate polyhedra, on the basis of which convex (linear) programming problems are formed. Constraints (in sections) approximate a continuous “tube” of state constraints. The more sections, the more accurate is the approximation of a continuous tube. A saddle-point method of the extragradient type is proposed, in which continuous state and conjugate trajectories are recalculated (shifted) along the gradient only at the discretization points. The convergence of the proposed method is proved for all components of the optimal control problem. Namely, convergence in controls is weak, convergence in state, conjugate trajectories and in terminal variables is strong. The limiting state trajectory at the discretization points passes through all the constraints (sections) of finite-dimensional convex problems.