About an algorithm for solving the speed problem in linear systems with convex restrictions on phase variables and control

The problem optimal speed control is investigated in the case when the process is described by a system of linear ordinary differential equations with nonlinear convex restrictions on phase variables and control. By moving from n-dimensional Euclidean space to Hilbert space, the optimal control problem with restrictions on phase variables and control is reduced to an optimal speed problem without restrictions. It is shown that the reachability region in the new space is a convex set. To solve the resulting problem, a modified method of separating hyperplanes is used. One of the key points of this method, on which the convergence speed of the algorithm depends, is finding the normal to the separating hyperplane. In this work, this normal at each iteration is constructed by minimizing a distance-type functional on the convex hull of points supporting the reachability set obtained at previous iterations. After finding the normal to the separating hyperplane, a hyperplane supporting the reachable region is constructed, which is then continuously transferred in increasing time and the first moment in time is found at which the supporting hyperplane reaches the given end point. This moment is taken as the next approximation to the performance time. A theorem is formulated on the convergence of successive approximations in time to the value of the performance time and on the weak convergence of a sequence of controls to an optimal control. The algorithm is tested by solving the problem of external heating of an unlimited plate to a given temperature in a minimal time, taking into account restrictions on tensile and compressive thermal stresses. The results of a computational experiment are presented.