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

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 tosolve the problem of transferring the trajectory of the system from an arbitrary initial position tothe smallest possible neighborhood of a given target set on a fixed time interval by selecting anappropriate feedback control. To solve this problem, it is proposed to construct a continuouspiecewise cubic function of a special kind. The level sets of this function correspond to internalestimates of the solvability sets of the system. Using this function, it is also possible to construct afeedback control function that solves the target control problem on a fixed time interval. Thepaper proposes formulas for calculating the values of the piecewise cubic function, examines itsproperties, and considers an algorithm for searching for parameters defining this function.