Evasion in a certain class of distributed control systems

Abstract

The differential game of evasion in a system described by a control partial differential equation containing the second time derivative and an elliptic operator is considered. The control parameters of the players are additive terms on the right-hand side of the equation. New spaces depending on a nonnegative parameter are introduced by using generalized eigenvalues and generalized eigenfunctions of the given elliptic operator. Games are introduced and studied on the entire scale of these spaces. Sufficient conditions for the possibility of evasion in games obtained under various constraints on the control parameters are given.

Solutions to the problem of differential games with electric oscillations by finite difference methods

“…Proof of Theorem 1. From ( 8) and ( 9) it follows that there are such 4) and applying formula (7), we obtain Hence, the projection operator  and from equalities (10), (11), we have…”

“…Proof. Substituting (7) into equation ( 4) and taking into account equality ( 5) and ( 6), we obtain ( ) The lemma is proven.…”

“…We partially use the methods seen in [4][5][6][7][8] and their analogies. And adjoins the works [9][10][11][12][13][14][15][16][17][18][19][20].…”