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 l...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.展开更多
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 exogen...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.展开更多
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 strate...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.展开更多
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 vulnera...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.展开更多
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 so...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.展开更多
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, inc...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.展开更多
This paper studies a bounded discriminating domain for hybrid linear differential game with two players and two targets using viability theory. First of all, we prove that the convex hull of a closed set is also a dis...This paper studies a bounded discriminating domain for hybrid linear differential game with two players and two targets using viability theory. First of all, we prove that the convex hull of a closed set is also a discriminating domain if the set is a discriminating domain. Secondly, in order to determine that a bounded polyhedron is a discriminating domain, we give a result that it only needs to verify that the extreme points of the polyhedron meet the viability conditions. The difference between our result and the existing ones is that our result just needs to verify the finite points (extreme points) and the existing ones need to verify all points in the bounded polyhedron.展开更多
In this paper,we have proposed a differential game model to optimally solve the resource allocation problems in the edge-computing based wireless networks.In the proposed model,a wireless network with one cloud-comput...In this paper,we have proposed a differential game model to optimally solve the resource allocation problems in the edge-computing based wireless networks.In the proposed model,a wireless network with one cloud-computing center(CC)and lots of edge services providers(ESPs)is investigated.In order to provide users with higher services quality,the ESPs in the proposed wireless network should lease the computing resources from the CC and the CC can allocate its idle cloud computing resource to the ESPs.We will try to optimally allocate the edge computing resources between the ESPs and CC using the differential game and feedback control.Based on the proposed model,the ESPs can choose the amount of computing resources from the CC using feedback control,which is affected by the unit price of computing resources controlled by the CC.In the simulation part,the optimal allocated resources for users’services are obtained based on the Nash equilibrium of the proposed differential game.The effectiveness and correctness of the proposed scheme is also verified through the numerical simulations and results.展开更多
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 s...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.展开更多
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.
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 i...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.展开更多
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 mainl...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.展开更多
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 i...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.展开更多
The optimal guidance problem for an interceptor against a ballistic missile with active defense is investigated in this paper.A class of optimal guidance schemes are proposed based on linear quadratic differential gam...The optimal guidance problem for an interceptor against a ballistic missile with active defense is investigated in this paper.A class of optimal guidance schemes are proposed based on linear quadratic differential game method and numerical solution of Riccati differential equation.By choosing proper parameters, the proposed guidance schemes are able to drive the interceptor to the target and away from the defender simultaneously.Additionally, fuel cost, control saturation,chattering phenomenon and parameters selection were taken into account.Satisfaction of the proposed guidance schemes of the saddle point condition is proven theoretically.Finally, nonlinear numerical examples are included to demonstrate the effectiveness and performance of the developed guidance approaches.Comparison of control performance between different guidance schemes are presented and analysis.展开更多
Stackelberg differential game models have been used to study sequential decision making in noncooperative games in diverse fields. In this paper, we survey recent applications of Stackelberg differential game models t...Stackelberg differential game models have been used to study sequential decision making in noncooperative games in diverse fields. In this paper, we survey recent applications of Stackelberg differential game models to the supply chain management and marketing channels literatures. A common feature of these applications is the specification of the game structure: a decentralized channel composed of a manufacturer and independent retailers, and a sequential decision procedure with demand and supply dynamics and coordination issues. In supply chain management, Stackelberg differential games have been used to investigate inventory issues, wholesale and retail pricing strategies, and outsourcing in dynamic environments. The underlying demand typically has growth dynamics or seasonal variation. In marketing, Stackelberg differential games have been used to model cooperative advertising programs, store brand and national brand advertising strategies, shelf space allocation, and pricing and advertising decisions. The demand dynamics are usually extensions of the classical advertising capital models or sales-advertising response models. We begin by explaining the Stackelberg differential game solution methodology and then provide a description of the models and results reported in the literature.展开更多
This paper will present an approximate/adaptive dynamic programming(ADP) algorithm,that uses the idea of integral reinforcement learning(IRL),to determine online the Nash equilibrium solution for the two-player zerosu...This paper will present an approximate/adaptive dynamic programming(ADP) algorithm,that uses the idea of integral reinforcement learning(IRL),to determine online the Nash equilibrium solution for the two-player zerosum differential game with linear dynamics and infinite horizon quadratic cost.The algorithm is built around an iterative method that has been developed in the control engineering community for solving the continuous-time game algebraic Riccati equation(CT-GARE),which underlies the game problem.We here show how the ADP techniques will enhance the capabilities of the offline method allowing an online solution without the requirement of complete knowledge of the system dynamics.The feasibility of the ADP scheme is demonstrated in simulation for a power system control application.The adaptation goal is the best control policy that will face in an optimal manner the highest load disturbance.展开更多
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-eva...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.展开更多
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,wh...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.展开更多
For intercepting modern high maneuverable targets, a novel adaptive weighted differential game guidance law based on the game theory of mixed strategy is proposed, combining two guidance laws which are derived from th...For intercepting modern high maneuverable targets, a novel adaptive weighted differential game guidance law based on the game theory of mixed strategy is proposed, combining two guidance laws which are derived from the perfect and imperfect in- formation pattern, respectively. The weights vary according to the estimated error of the target's acceleration, the guidance law is generated by directly using the estimation of target's acceleration when the estimated error is small, and a differential game guidance law with adaptive penalty coefficient is implemented when the estimated error is large. The adaptive penalty coeffi- cients are not constants and they can be adjusted with current target maneuverability. The superior homing performance of the new guidance law is verified by computer simulations.展开更多
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. ...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.展开更多
基金supported by the Industry-University-Research Cooperation Fund Project of the Eighth Research Institute of China Aerospace Science and Technology Corporation (USCAST2022-11)Aeronautical Science Foundation of China (20220001057001)。
文摘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.
基金supported by the Qihang Project of Zhejiang University(Grant No.202016)。
文摘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.
基金supported by the National Natural Science Foundation of China(11672093)
文摘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.
基金supported by the National Science Foundation Project of P. R. China (No. 61501026,61572072)Fundamental Research Funds for the Central Universities (No. FRF-TP-15-032A1)
文摘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.
文摘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.
基金supported by the China Postdoctoral Science Foundation(No.2015M570936)National Science Foundation Project of P.R.China(No.61501026,61272506)Fundamental Research Funds for the Central Universities(No.FRF-TP-15032A1)
文摘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.
基金supported by National Science Foundation of China(11171221)Doctoral Program Foundation of Institutions of Higher Education of China(20123120110004)+2 种基金Natural Science Foundation of Shanghai(14ZR1429200)Innovation Program of Shanghai Municipal Education Commission(15ZZ073)Key Research Project Plan of Institutions of Higher of Henan Province(17A120010)
文摘This paper studies a bounded discriminating domain for hybrid linear differential game with two players and two targets using viability theory. First of all, we prove that the convex hull of a closed set is also a discriminating domain if the set is a discriminating domain. Secondly, in order to determine that a bounded polyhedron is a discriminating domain, we give a result that it only needs to verify that the extreme points of the polyhedron meet the viability conditions. The difference between our result and the existing ones is that our result just needs to verify the finite points (extreme points) and the existing ones need to verify all points in the bounded polyhedron.
基金This paper is supported by Natural Science Foundation of China,No.61971032the Fundamental Research Funds for the Central Universities,No.FRF-TP-18-008A3.
文摘In this paper,we have proposed a differential game model to optimally solve the resource allocation problems in the edge-computing based wireless networks.In the proposed model,a wireless network with one cloud-computing center(CC)and lots of edge services providers(ESPs)is investigated.In order to provide users with higher services quality,the ESPs in the proposed wireless network should lease the computing resources from the CC and the CC can allocate its idle cloud computing resource to the ESPs.We will try to optimally allocate the edge computing resources between the ESPs and CC using the differential game and feedback control.Based on the proposed model,the ESPs can choose the amount of computing resources from the CC using feedback control,which is affected by the unit price of computing resources controlled by the CC.In the simulation part,the optimal allocated resources for users’services are obtained based on the Nash equilibrium of the proposed differential game.The effectiveness and correctness of the proposed scheme is also verified through the numerical simulations and results.
基金Project supported by the National Aeronautics Base Science Foundation of China (No.2000CB080601)the National Defence Key Pre-research Program of China during the 10th Five-Year Plan Period (No.2002BK080602)
文摘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.
基金The Young Research Foundation(201201130) of Jilin Provincial Science&Technology DepartmentResearch Foundation(2011LG17) of Changchun University of Technology
文摘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.
基金This work was supported by National Key Research&Development Program of China under Grant No.2022YFA1006104National Natural Science Foundations of China under Grant Nos.11971266,11831010Shandong Provincial Natural Science Foundations under Grant Nos.ZR2022JQ01,ZR2020ZD24,ZR2019ZD42.
文摘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.
基金supported in part by the Strategic Priority Research Program of Chinese Academy of Sciences(XDA27030100)National Natural Science Foundation of China(72293575, 11832001)。
文摘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.
文摘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.
文摘The optimal guidance problem for an interceptor against a ballistic missile with active defense is investigated in this paper.A class of optimal guidance schemes are proposed based on linear quadratic differential game method and numerical solution of Riccati differential equation.By choosing proper parameters, the proposed guidance schemes are able to drive the interceptor to the target and away from the defender simultaneously.Additionally, fuel cost, control saturation,chattering phenomenon and parameters selection were taken into account.Satisfaction of the proposed guidance schemes of the saddle point condition is proven theoretically.Finally, nonlinear numerical examples are included to demonstrate the effectiveness and performance of the developed guidance approaches.Comparison of control performance between different guidance schemes are presented and analysis.
文摘Stackelberg differential game models have been used to study sequential decision making in noncooperative games in diverse fields. In this paper, we survey recent applications of Stackelberg differential game models to the supply chain management and marketing channels literatures. A common feature of these applications is the specification of the game structure: a decentralized channel composed of a manufacturer and independent retailers, and a sequential decision procedure with demand and supply dynamics and coordination issues. In supply chain management, Stackelberg differential games have been used to investigate inventory issues, wholesale and retail pricing strategies, and outsourcing in dynamic environments. The underlying demand typically has growth dynamics or seasonal variation. In marketing, Stackelberg differential games have been used to model cooperative advertising programs, store brand and national brand advertising strategies, shelf space allocation, and pricing and advertising decisions. The demand dynamics are usually extensions of the classical advertising capital models or sales-advertising response models. We begin by explaining the Stackelberg differential game solution methodology and then provide a description of the models and results reported in the literature.
基金supported by the National Science Foundation (No.ECCS-0801330)the Army Research Office (No.W91NF-05-1-0314)
文摘This paper will present an approximate/adaptive dynamic programming(ADP) algorithm,that uses the idea of integral reinforcement learning(IRL),to determine online the Nash equilibrium solution for the two-player zerosum differential game with linear dynamics and infinite horizon quadratic cost.The algorithm is built around an iterative method that has been developed in the control engineering community for solving the continuous-time game algebraic Riccati equation(CT-GARE),which underlies the game problem.We here show how the ADP techniques will enhance the capabilities of the offline method allowing an online solution without the requirement of complete knowledge of the system dynamics.The feasibility of the ADP scheme is demonstrated in simulation for a power system control application.The adaptation goal is the best control policy that will face in an optimal manner the highest load disturbance.
基金co-supported by the National Natural Science Foundation of China (Nos.61603115,91438202 and91638301)China Postdoctoral Science Foundation (No.2015M81455)+1 种基金the Open Fund of National Defense Key Discipline Laboratory of Micro-Spacecraft Technology of China (No.HIT.KLOF.MST.201601)the Heilongjiang Postdoctoral Fund of China (No.LBH-Z15085)
文摘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.
基金supported by the Science and Technology Innovation 2030-Key Project of “New Generation Artificial Intelligence”,China(No.2020AAA0108200)the National Natural Science Foundation of China(Nos.61873011,61922008,61973013 and 61803014)+3 种基金the Defense Industrial Technology Development Program,China(No.JCKY2019601C106)the Innovation Zone Project,China(No.18-163-00-TS-001-00134)the Foundation Strengthening Program Technology Field Fund,China(No.2019-JCJQ-JJ-243)the Fund from Key Laboratory of Dependable Service Computing in Cyber Physical Society,China(No.CPSDSC202001)。
文摘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.
基金National Natural Science Foundation of China (60874040)
文摘For intercepting modern high maneuverable targets, a novel adaptive weighted differential game guidance law based on the game theory of mixed strategy is proposed, combining two guidance laws which are derived from the perfect and imperfect in- formation pattern, respectively. The weights vary according to the estimated error of the target's acceleration, the guidance law is generated by directly using the estimation of target's acceleration when the estimated error is small, and a differential game guidance law with adaptive penalty coefficient is implemented when the estimated error is large. The adaptive penalty coeffi- cients are not constants and they can be adjusted with current target maneuverability. The superior homing performance of the new guidance law is verified by computer simulations.
基金国家自然科学基金,Outstanding Young Teachers of Ministry of Education of China,Special Fund for Ph.D.Program of Ministry of Education of China,Fok Ying Tung Education Foundation
文摘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.