Estimate of the Capture Time and Construction of the Pursuer's Strategy in a Nonlinear Two-Person Differential Game

In a finite-dimensional Euclidean space, we consider a differential game of two persons-a pursuer and an evader-described by a nonlinear autonomous controlled system of differential equations in normal form the right-hand side of which is the sum of two functions, one of which depends only on the state variable and the pursuer's control and the other, only on the state variable and the evader's control. The set of values of the pursuer's control is finite, and the set of values of the evader's control is compact. The goal of the pursuer is to bring the trajectory of the system from the initial position to any predetermined neighborhood of zero in finite time. The pursuer strategy is constructed as a piecewise constant function with values in a given finite set. To construct the pursuer control, it is allowed to use only information about the value of the current state coordinates. The evader's control is a measurable function for the construction of which there are no constraints on available information. It is shown that, to transfer the system to any predetermined neighborhood of zero, it is sufficient for the pursuer to use a strategy with a constant step of partitioning the time interval. The value of the fixed partitioning step is found in closed form. A class of systems is singled out for which an estimate of the transfer time from an arbitrary initial position to a given neighborhood of zero is obtained. The estimate is sharp in some well-defined sense. The solution essentially uses the notion of a positive basis in a vector space.