The synthesis of approximate optimal control of systems with polinomial nonlinearities by expansion over block impulse functions

A problem of optimal control of a system with quadratic quality fucntional, fixed time and fixed left edge is considered. The system is described by the vector-matrix differential equation with nonlinear elements of polynomial type. The restrictions in the form of inequalities at the vectors of control and state are imposed. A simple orthogonal basis of piece-wise constant fucntions which are called block impulse fucntions (BIF) is introduced, and the continious problem is reduced to the mathematical programming problem by the Ritz–Galerkin method. The nonlinear algebraic system decomposes in the systems of not high order that allows to simplify essentially the solving problem of mathematical programming. A peculiarity of BIF basis in case of possibility of state vector measurement permits to construct an algorithm of stabilization of system movement near the programmed trajectory.