Nonlinear Differential Games Research Articles

Incentive schemes using the follower's strategies in differential games

We deal with Stackelberg games in which a leader has access not only to the state information but also to information on the follower's strategy. We derive sufficient conditions for incentive schemes using information on the follower's strategies in both linear and non-linear differential games, and show that the incentive schemes using information on the follower's strategies depend on an initial state value. We also derive a sufficient condition for the incentive scheme using the follower's strategy in the linear quadratic differential game with infinite time interval.

A missile duel between two aircraft

A delicate differential game is formed by two aircraft performing a long-range missile duel. The duel starts for each aircraft with a semiagressive phase until the launch of the missile, when a pure evasion commences. In numerically solving the nonlinear differential game problem, an optimization algorithm is used consisting of a modified first-order differential dynamic programming method combined with an effective convergence-control technique. The three-dimensional spatial motion of point mass vehicles with realistic models is treated. The results of optimizing the problem point out the potential of new tactics. If the vehicles are free to maneuver in altitude, the contributions from the missiles will dominate the outcome of the game. Of particular importance is the desirable ability to enter the game at a higher altitude than that of the opponent.

Method of alternating directions in nonlinear differential games of evasion

Method of alternating directions in nonlinear differential games of evasion

Three-level incentive strategies in differential games

The three-level incentive strategy in a non-linear differential game ia defined. The separability in obtaining incentive strategies is shown. The sufficient condition for a three-level incentive strategy in a linear quadratic differential game is derived. It is shown that a gradient method is applicable for finding a three-level incentive strategy in a linear quadratic differential game.

A class of nonlinear differential games

A class of nonlinear differential games

Nonlinear differential games of evasion

Nonlinear differential games of evasion

Differential escape games with information lag: PMM vol. 43, no. 2, 1979, pp. 195–203

Nonlinear differential escape games with geometric and integral constraints on the controls are examined under the assumption that at each instant t the evader, when constructing his own control, uses the values of the phase vector z ( s) and of his opponent's control u ( s) for all s such that s ⩽ t — τ ( z ( t)), where τ ( z) = ρ ( z) / c ( z). The function o ( z) tends to zero if the distance between z and the terminal set tends to zero; c ( z) is the positive “rate of propagation of information”. Lower estimates are obtained for function c ( z), for which the fulfilment of the escape conditions in [1 – 3] guarantees the existence of an escape strategy for any initial position.

A nonlinear differential evasion game

The evasion problem for nonlinear differential games with a target set which is a linear subspace is considered. A sufficient condition for the possibility of avoidance of contact with all points not belonging to the target set is obtained. Examples are considered. This article borders on the investigations of L. S. Pontryagin, E. F. Mishchenko, and B. N. Pshenichnyi.

A transition matrix method for generating near-optimal closed-loop solutions to non-linear zero-sum differential games†

This paper presents a method for generating near optimal closed-loop solutions to zero-sum perfect information differential games. Such a near-optimal solution is generated by periodically updating the solution to the two-point boundary-value problem (TPBVP) obtained from the application of the necessary conditions for a saddle-point solution. This procedure is accomplished by updating the co-state vector at each updating time based on the state error from a reference TPBVP solution. The relationship between the required change in the co-state vector and the state error is obtained using the transition matrices for the linearized TPBVP. Between updating times the player using this method plays his open-loop control determined from the updated TPBVP solution. A number of examples are presented to illustrate the advantages and shortcomings of this method.

On a problem of l-escape : PMM vol. 40, n≗1, 1976, pp. 25–37

We give criteria for escape and l-escape in nonlinear differential games. The paper is closely related to the investigations in [1–8].

Deviation problems in nonlinear differential games

Deviation problems in nonlinear differential games

Differential Games and a Nash Equilibrium Searching Algorithm

Nonlinear nonzero sum differential games with fixed playing time are considered. They are transformed into a two-level hierarchical game, with a supremal referee optimizing an overall objective function in order to improve the respect of the Nash playing rule. We propose an algorithm, based on constraint coordination, in order to approximate open-loop Nash equilibrium controls. The theory illustrating this algorithm is presented.

Pursuit problems in nonlinear differential games

Pursuit problems in nonlinear differential games

Nonlinear differential pursuit games

Nonlinear differential pursuit games

Nonlinear variational inequalities and differential games with stopping times

Differential games with stopping times are introduced. The upper value satisfies a variational inequality of the first order. Next, stochastic differential games with stopping times are introduced and studied using nonlinear parabolic variational inequalities. The behavior of the value is studied as the coefficient of the diffusion matrix goes to zero.

On the pursuit problem in nonlinear differential games: PMM vol. 38, n≗ 1, 1974, pp. 38–44

We prove a sufficient condition for the termination of pursuit in nonlinear games. We indicate a class of games on a plane, for which this condition is satisfied, we introduce the notion of relative optimality, and we consider an example.

On encounter-evasion game problems: PMM vol. 38, n≗2, 1974, pp. 211–223

We consider a nonlinear encounter-evasion differential game on a finite time interval. To solve it we use an auxiliary program construction. The article is closely related to the investigations in [1–8].

On an encounter game problem under composite controls: PMM vol. 38, n≗ 3, 1974, pp. 402–408

We examine a nonlinear differential game of the encounter of a conflict-controlled phase point with a given set. We prove sufficient conditions for the successful termination of the game in the class of mixed strategies. These conditions are based on the extremal construction introduced in [1] and modified here to conform to the question being discussed.

Two numerical techniques to solve differential game problems †

In this paper two computational techniques are presented, to solve non-linear differential games which have a value. It is assumed that there may be interaction in the dynamics of the players and that the conflicting index of performance of the game is a scalar function of the final state of the players of the miss-distance type. The first of these techniques is based on the Newton-Raphson method in function space in association with one of the two following non-iterative procedures to solve linear differential games with quadratic pay-off function : the generalized Riccati transformation or the transition matrix method. The second computational technique presented is based on the gradient of the Hamiltonian function with respect to the strategies. Some numerical examples are given to illustrate the methods.

A real-time closed-loop solution method for a class of nonlinear differential games

This correspondence presents a method for generating near-optimal closed-loop solutions to fixed time differential games with unconstrained controls by periodically updating the solution to the two-point boundary-value problem obtained by the application of the necessary condition for a saddle-point solution. This method should permit the real-time closed-loop solution of this class of games using a digital or hybrid computer.