On Piecewise Cubic Estimates of the Value Function in a Target Control Problem for a Nonlinear System

Abstract

A nonlinear system of ordinary differential equations with control parameters is considered. Pointwise restrictions are imposed on the possible values of these parameters. It is required to solve the problem of transferring the trajectory of the system from an arbitrary initial position to the smallest possible neighborhood of a given target set on a fixed time interval by selecting an appropriate feedback control. To solve this problem, it is proposed to construct a continuous piecewise cubic function of a special kind. The level sets of this function correspond to internal estimates of the solvability sets of the system. Using this function, it is also possible to construct a feedback control function that solves the target control problem on a fixed time interval. The paper proposes formulas for calculating the values of the piecewise cubic function, examines its properties, and considers an algorithm for searching for parameters defining this function.