The flight problem

Abstract

In [1] and [2] a special differential-game problem the flight problem is considered. It consists in explaining sufficient conditions, under which the second player can conclude the game in his own favor, i.e., divert the trajectory from the terminal set out o f an arbitrary initial point, not lying in that set. The main purpose of this paper is to obtain flight conditions, generalizing the results of [1] and [2] to the nonlinear case. 1. STATEMENT OF THE PROBLEM Let an object be described by a system of differential equations z = f(z, u, v), (1) where z E E n, u E U , v E E s, [ : En • E" X E s ~ E ". Terminal set M is a subspace in E n and its orthogonal complement L has a dimension greater or equal to 2. In the sequel we shall also assume the following. 1. f(z, u, v) has derivatives o f all needed orders with respect to z and these derivatives satisfy the Lipschitz condit ion in every compact set with respect to all arguments. 2. Controls u and v vary in compact sets U and V, where V is convex and has interior points in E s. 3. For all u E U and v E V l ( z , f ( z , u , v ) l . i 2 and a number k such that none of the functions ep~ = az, (p~(z)= V~tp '-~ (z), i = 1 . . . . . /~ ~ 1 depend on u, v. Here rr is the matrix for the orthogonal projection on W. Remark. From (2) i t is clear that an application of operator "V~ leads, generally speaking, to a function which depends on u and v. I t is also clear that by the permutabil i ty of the constant matrix ~r and the differentiation operator all the functions soi(z) map E n into W, i.e., ~' (z)E W . Assumption 2. Funct ion r ~ (z, u, v)=---V.~-J(z)depends on u and v. Moreover, i f we denote F ( z ) = ~ f~(z, u, V) t~U then there exist continuous functions qD~(z):E"-+E"and e (z ) :E ' -~ s that e(z) > 0 and