Value Iteration-Based Cooperative Adaptive Optimal Control for Multi-Player Differential Games With Incomplete Information

This paper presents a novel cooperative value iteration(VI)-based adaptive dynamic programming method for multi-player differential game models with a convergence proof.The players are divided into two groups in the learning process and adapt their policies sequentially.Our method removes the dependence of admissible initial policies,which is one of the main drawbacks of the PI-based frameworks.Furthermore,this algorithm enables the players to adapt their control policies without full knowledge of others’ system parameters or control laws.The efficacy of our method is illustrated by three examples.

NUMERICAL METHOD BASED ON HAMILTON SYSTEM AND SYMPLECTIC ALGORITHM TO DIFFERENTIAL GAMES

The resolution of differential games often concerns the difficult problem of two points border value （TPBV）, then ascribe linear quadratic differential game to Hamilton system. To Hamilton system, the algorithm of symplectic geometry has the merits of being able to copy the dynamic structure of Hamilton system and keep the measure of phase plane. From the viewpoint of Hamilton system, the symplectic characters of linear quadratic differential game were probed; as a try, Symplectic-Runge-Kutta algorithm was presented for the resolution of infinite horizon linear quadratic differential game. An example of numerical calculation was given, and the result can illuminate the feasibility of this method. At the same time, it embodies the fine conservation characteristics of symplectic algorithm to system energy.

Open Loop Saddle Point on Linear Quadratic Stochastic Differential Games

In this paper, we deal with one kind of two-player zero-sum linear quadratic stochastic differential game problem. We give the existence of an open loop saddle point if and only if the lower and upper values exist.

Linear Quadratic Leader-Follower Stochastic Differential Games:Closed-Loop Solvability

In this paper,a leader-follower stochastic differential game is studied for a linear stochastic differential equation with quadratic cost functionals.The coefficients in the state equation and the weighting matrices in the cost functionals are all deterministic.Closed-loop strategies are introduced,which require to be independent of initial states;and such a nature makes it very useful and convenient in applications.The follower first solves a stochastic linear quadratic optimal control problem,and his optimal closed-loop strategy is characterized by a Riccati equation,together with an adapted solution to a linear backward stochastic differential equation.Then the leader turns to solve a stochastic linear quadratic optimal control problem of a forward-backward stochastic differential equation,necessary conditions for the existence of the optimal closed-loop strategy for the leader is given by a Riccati equation.Some examples are also given.

Cooperative differential games guidance laws for multiple attackers against an active defense target

This paper is concerned with a scenario of multiple attackers trying to intercept a target with active defense.Three types of agents are considered in the guidance:The multiple attackers,the target and the defender,where the attackers aim to pursuit the target from different directions and evade from the defender simultaneously.The guidance engagement is formulated in the framework of a zero-sum two-person differential game between the two opposing teams,such that the measurements on the maneuver of the target or estimations on the defending strategy of the defender can be absent.Cooperation of the attackers resides in two aspects:redundant interception under the threat of the defender and the relative intercept geometry with the target.The miss distances,the relative intercept angle errors and the costs of the agents are combined into a single performance index of the game.Such formulation enables a unitary approach to the design of guidance laws for the agents.To minimize the control efforts and miss distances for the attackers,an optimization method is proposed to find the best anticipated miss distances to the defender under the constraint that the defender is endowed with a capture radius.Numerical simulations with two cases are conducted to illustrate the effectiveness of the proposed cooperative guidance law.

Satellite proximate interception vector guidance based on differential games

This paper studies the proximate satellite interception guidance strategies where both the interceptor and target can perform orbital maneuvers with magnitude limited thrusts. This problem is regarded as a pursuit-evasion game since satellites in both sides will try their best to capture or escape. In this game, the distance of these two players is small enough so that the highly nonlinear earth-centered gravitational dynamics can be reduced to the linear Clohessy-Wiltshire（CW） equations. The system is then simplified by introducing the zero effort miss variables. Saddle solution is formulated for the pursuit-evasion game and time-to-go is estimated similarly as that for the exoatmospheric interception. Then a vector guidance is derived to ensure that the interception can be achieved in the optimal time. The proposed guidance law is validated by numerical simulations.

Infinite horizon linear quadratic differential games for discrete-time stochastic systems

This paper deals with the infinite horizon linear quadratic （LQ） differential games for discrete-time stochas- tic systems with both state and control dependent noise. The Popov-Belevitch-Hautus （PBH） criteria for exact observability and exact detectability of discrete-time stochastic systems are presented. By means of them, we give the optimal strategies （Nash equilibrium strategies） and the optimal cost values for infinite horizon stochastic differential games. It indicates that the infinite horizon LQ stochastic differential gaines are associated with four coupled matrix-valued equations. Further- more, an iterative algorithm is proposed to solve the four coupled equations. Finally, an example is given to demonstrate our results.

Infnite time horizon nonzero-sum linear quadratic stochastic differential games with state and control-dependent noise

This paper discusses the infinite time horizon nonzero-sum linear quadratic （LQ） differential games of stochastic systems governed by Itoe＇s equation with state and control-dependent noise. First, the nonzero-sum LQ differential games are formulated by applying the results of stochastic LQ problems. Second, under the assumption of mean-square stabilizability of stochastic systems, necessary and sufficient conditions for the existence of the Nash strategy are presented by means of four coupled stochastic algebraic Riccati equations. Moreover, in order to demonstrate the usefulness of the obtained results, the stochastic H-two/H-infinity control with state, control and external disturbance-dependent noise is discussed as an immediate application.

A BSDE Approach to Stochastic Differential Games Involving Impulse Controls and HJBI Equation

This paper focuses on zero-sum stochastic differential games in the framework of forwardbackward stochastic differential equations on a finite time horizon with both players adopting impulse controls.By means of BSDE methods,in particular that of the notion from Peng’s stochastic backward semigroups,the authors prove a dynamic programming principle for both the upper and the lower value functions of the game.The upper and the lower value functions are then shown to be the unique viscosity solutions of the Hamilton-Jacobi-Bellman-Isaacs equations with a double-obstacle.As a consequence,the uniqueness implies that the upper and lower value functions coincide and the game admits a value.

Stochastic Differential Games with Reflection and Related Obstacle Problems for Isaacs Equations

In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations （RBSDEs）. We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bettman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of E1 Karoui, Kapoudjian, Pardoux, Peng and Quenez （1997）, which turns out to be very useful because it allows us to estimate the LP-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.

Policy Iteration Algorithms for Zero-Sum Stochastic Differential Games with Long-Run Average Payoff Criteria

This paper studies the policy iteration algorithm(PIA)for zero-sum stochastic differential games with the basic long-run average criterion,as well as with its more selective version,the so-called bias criterion.The system is assumed to be a nondegenerate diffusion.We use Lyapunov-like stability conditions that ensure the existence and boundedness of the solution to certain Poisson equation.We also ensure the convergence of a sequence of such solutions,of the corresponding sequence of policies,and,ultimately,of the PIA.

LINEAR QUADRATIC NONZERO-SUM DIFFERENTIAL GAMES WITH RANDOM JUMPS

The existence and uniqueness of the solutions for one kind of forward- backward stochastic differential equations with Brownian motion and Poisson process as the noise source were given under the monotone conditions. Then these results were applied to nonzero-sum differential games with random jumps to get the explicit form of the open-loop Nash equilibrium point by the solution of the forward-backward stochastic differential equations.

Capturability of Pseudo Differential Games of Pursuit

This paper discusses the capturability with fixed time of a pseudo-linear differential game of pursuit. The pursuit set in which the pursuit will end once the initial state lies in this set is given by the method of integration of multi-valued function. The results obtained here solve an open problem of Pontrjagin’s on the linear differential game of pursuit. Meanwhile, the requirement of the convexity of the control set and other related Pontrjagin’s conditions are removed.

Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations,in which the coefficient contains not only the state process but also its marginal distribution,and the cost functional is also of mean-field type.It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions.We establish a necessary condition in the form of maximum principle and a verification theorem,which is a sufficient condition for Nash equilibrium point.We use the theoretical results to deal with a partial information linear-quadratic(LQ)game,and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.

How do international agreement constraints affect marine litter management strategies?A research based on differential game

Based on differential game theory,the decision-making problem of two homogeneous countries facing transboundary marine litter governance is studied.On the basis of assuming that the input of marine litter is an exogenous variable,the focus is on reducing the accumulation of marine litter through cleanup and transfer processing by both parties.Considering the constant and increasing input of marine litter,in the framework of international agreement constraints,the analysis of the game behavior of the players in the marine litter governance under the open-loop strategy(in the case of agreement constraints)and the Markov strategy(in the case of no agreement constraints)was compared and analyzed.The research results show that when the direct pollution cost of marine litter is high enough,both sides of the game adopt an open-loop strategy that complies with the constraints of the agreement,which can reduce the accumulation of marine litter and improve the environmental quality.However,when there is a high initial accumulation of marine litter,the Markov strategy without protocol constraints will be better than the open-loop strategy.In the case that marine litter does not need to be transferred,there will be no difference between the two sides of the game adopting the Markov strategy and adopting the open-loop strategy on the equilibrium growth path.

Equilibrium Strategy of the Pursuit-Evasion Game in Three-Dimensional Space

The pursuit-evasion game models the strategic interaction among players, attracting attention in many realistic scenarios, such as missile guidance, unmanned aerial vehicles, and target defense. Existing studies mainly concentrate on the cooperative pursuit of multiple players in two-dimensional pursuit-evasion games. However, these approaches can hardly be applied to practical situations where players usually move in three-dimensional space with a three-degree-of-freedom control. In this paper,we make the first attempt to investigate the equilibrium strategy of the realistic pursuit-evasion game, in which the pursuer follows a three-degree-of-freedom control, and the evader moves freely. First, we describe the pursuer's three-degree-of-freedom control and the evader's relative coordinate. We then rigorously derive the equilibrium strategy by solving the retrogressive path equation according to the Hamilton-Jacobi-Bellman-Isaacs(HJBI) method, which divides the pursuit-evasion process into the navigation and acceleration phases. Besides, we analyze the maximum allowable speed for the pursuer to capture the evader successfully and provide the strategy with which the evader can escape when the pursuer's speed exceeds the threshold. We further conduct comparison tests with various unilateral deviations to verify that the proposed strategy forms a Nash equilibrium.

Differential game strategy in three-player evasion and pursuit scenarios

A conflict of three players, including an attacker, a defender, and a target with bounded control is discussed based on the differential game theories in which the target and the defender use an optimal pursuit strategy. The current approach chooses the miss distance as the outcome of the conflict. Different optimal guidance laws are investigated, and feasible conditions are analyzed for the attacker to accomplish an attacking task. For some given conditions, the attacker cannot intercept the target by only using a one-to-one optimal pursuit guidance law; thus, a guidance law for the attacker to reach a critical safe value is investigated.Specifically, the guidance law is divided into two parts. Before the engagement time between the defender and the attacker, the attacker uses this derived guidance law to guarantee that the evasion distance from the defender is safe, and that the zero-effort-miss(ZEM) distance between the attacker and the target is the smallest.After that engagement time, the attacker uses the optimal one-toone guidance law to accomplish the pursuit task. The advantages and limited conditions of these derived guidance laws are also investigated by using nonlinear simulations.

A Non-Cooperative Differential Game-Based Security Model in Fog Computing

Fog computing is a new paradigm providing network services such as computing, storage between the end users and cloud. The distributed and open structure are the characteristics of fog computing, which make it vulnerable and very weak to security threats. In this article, the interaction between vulnerable nodes and malicious nodes in the fog computing is investigated as a non-cooperative differential game. The complex decision making process is reviewed and analyzed. To solve the game, a fictitious play-based algorithm is which the vulnerable node and the malicious nodes reach a feedback Nash equilibrium. We attain optimal strategy of energy consumption with Qo S guarantee for the system, which are conveniently operated and suitable for fog nodes. The system simulation identifies the propagation of malicious nodes. We also determine the effects of various parameters on the optimal strategy. The simulation results support a theoretical foundation to limit malicious nodes in fog computing, which can help fog service providers make the optimal dynamic strategies when different types of nodes dynamically change their strategies.

On the finite horizon Nash equilibrium solution in the differential game approach to formation control

The solvability of the coupled Riccati differential equations appearing in the differential game approach to the formation control problem is vital to the finite horizon Nash equilibrium solution.These equations(if solvable)can be solved numerically by using the terminal value and the backward iteration.To investigate the solvability and solution of these equations the formation control problem as the differential game is replaced by a discrete-time dynamic game.The main contributions of this paper are as follows.First,the existence of Nash equilibrium controls for the discretetime formation control problem is shown.Second,a backward iteration approximate solution to the coupled Riccati differential equations in the continuous-time differential game is developed.An illustrative example is given to justify the models and solution.

Resource Allocation for Network Security Risk Assessment：A Non-Cooperative Differential Game Based Approach

In this paper, we propose a non-cooperative differential game theory based resource allocation approach for the network security risk assessment. For the risk assessment, the resource will be used for risk assess, including response cost and response negative cost. The whole assessment process is considered as a differential game for optimal resource control. The proposed scheme can be obtained through the Nash Equilibrium. It is proved that the game theory based algorithm is applicable and the optimal resource level can be achieved based on the proposed algorithm.