Resolution of a linear-quadratic optimal control problem based on finite-dimensional models

We consider a linear-quadratic optimal control problem with indefinite matrices and the interval control constraint. The problem also has a regularization parameter in the functional. The approximate solution of the problem is carried out on subsets of admissible controls, which are formed using linear combinations of special functions with an orientation to the optimal control structure due to the maximum principle. As a result of this procedure, a finite-dimensional quadratic optimization problem with the interval constraint on variables is obtained.
The following relations between the variational problem and its finite-dimensional model are established: the convexity property of the optimal control problem is preserved for finite-dimensional model; a nonconvex optimal control problem under a certain condition on the regularization parameter (estimate from below) is approximated by a convex quadratic problem, which is solved in a finite number of operations; a special non-convex optimal control problem with an upper bound on the regularization parameter passes into the problem of minimizing a concave function on a finite set of points. A special case of a non-convex optimal control problem for the maximum of the norm of the final state is distinguished. Two procedures for improving the extreme points of finite-dimensional model are constructed, which reduce the computational costs for the global solution of the problem within the framework of the linearization method.