(untitled) (Journal of Mathematics Mechanics and Computer Science, 2013)
The methods of building program and positional controls for processes described by ordinary differential equations in the presence of boundary conditions with the restrictions on the control are developed. An algorithm for solving problem of optimal fast action based on the solution of the controllability problem is elaborated. Two problems are solved: the existence of the controllability problem’s solution and the construction of the set of all controls, each element of which transfers trajectory of the system from any initial state to a given final state. The basis of the proposed methods of constructing program and positional control is a Fredholm integral equation of the first kind. The necessary and sufficient condition for existence of the solution of the integral equation was received. A general solution of one class of Fredholm integral equation of the first kind was found. It is shown that the solutions of problems of controllability of linear and nonlinear control systems can be reduced to the solution of the initial problem of optimal control of a special type. Algorithms for minimizing sequences and estimation of their rate of convergence are given.