Solving Nonzero-Sum Nash Differential Game via Variation and Pseudospectral Methods

A numerical methods are presented in this paper to solve the nonzero-sum N-player Nash differential game. Variation methods are used to convert the original game into a regular optimal control problem which consists of one objective function, the state equations of all the players with initial conditions and the other necessary conditions derived. Then the later optimization problem is interpolated through the Legendre-pseudospectral method(LPM) and solved by applying SNOPT algorithm to get the optimal state trajectory. As an illustration, the air combat between a superior fighter and an inferior fighter is formulated as a nonzero-sum differential game. The states, traces, costs of both and the relative distance between them are displayed. The results show that numerical solutions converge to the saddle-points successfully, which show the feasibility and effectiveness of the proposed method in solving the nonzero-sum differential game. Introduction Over the past few decades, many researchers have studied a variety of the Nash differential games, especially the pursuit-evasion game[1] in depth, and a large number of valuable results have been achieved. In the qualitative analysis, a tail-chase air combat problem was solved by using differential dynamic programming[2]. A coplanar pursuit-evasion game in the atmosphere was studied with assuming a constant speed for evader[3]. A pursuit–evasion problem between missile and aircraft with realistic dynamics was solved by using an indirect, multiple shooting method[4,5]. Iterating a direct method[6] and nonlinear MPC method[7] was also applied for the pursuit-evasion game. Besides, in recently, a variety of evolutionary algorithms have also been used to solve this kind of differential game problem[8,9]. In addition, a named semi-direct method was introduced for pursuit–evasion dynamic game[10] and applied for the games between missiles[11] and spacecrafts[12]. This research concentrates on solving the nonzero-sum N-player differential game. This is the first time for that game to be solved accurately by the variation method and the LPM method[13,14,15]. In the beginning of the content, the conversation by variation method is deduced in details. Then a nonzero-sum pursuit–evasion game with two Bolza-type objective functions is taken as an illustration. Simulations are performed at last to support the conclusions. Nonzero-sum Nash differential game and optimal control problem In this section, the nonzero-sum n-player Nash differential game is converted to the optimal control problem. With regard to that game, every player possesses one objective function. All of the objective functions can be described as the kind of the Bolza-type. Thus, the game can be depicted as: International Conference on Computational Science and Engineering (ICCSE 2015) © 2015. The authors Published by Atlantis Press 1   0 1, , ,.., , ( , , ) f t i i f n f f i i i t J t L t dt     x x x u .   1,..., i n  (1) subject to the uncoupled state equations:   i i i i x = f x ,u , t  .   1,..., i n  (2) with the initial conditions 0 ,0 ( ) i i t  x x .To state the necessary conditions, the Hamiltonians of n-1 players except the first one are introduced as follows:   , , , T i i i i i i i H t L  x u λ λ f  .   2,..., i n  (3) The adjoint equations of the n-1 players are: T i i i i i i i i H L            f λ λ x x x  .   2,..., i n  (4) The boundary constraints of the adjoint variables mentioned in relationship are:   ( ) i i f i f t t    λ x .   2,..., i n  (5) As unbounded control variables are assumed for the problem, the following first-order optimality and the second-order conditions must be satisfied by the control variables for the n-1 players: