带Poisson跳的线性二次随机微分博弈及其在鲁棒控制中的应用

研究了一类带Poisson跳扩散过程的线性二次随机微分博弈,包括非零和博弈的Nash均衡策略与零和博弈的鞍点均衡策略问题．利用微分博弈的最大值原理,得到Nash均衡策略的存在条件等价于两个交叉耦合的矩阵Riccati方程存在解,鞍点均衡策略的存在条件等价于一个矩阵Riccati方程存在解的结论,并给出了均衡策略的显式表达及最优性能泛函值．最后,将所得结果应用于现代鲁棒控制中的随机H₂/H_∞控制与随机H_∞控制问题,得到了鲁棒控制策略的存在条件及显式表达,并验证所得结果在金融市场投资组合优化问题中的应用．     

Analytical solution for a class of linear quadratic open-loop Nash game with multiple players

In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrix can be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.     

Analytical solution for a class of linear quadratic open-loop Nash game with multiple players

In this paper, the Nash equilibria for differential games with multiple players is studied. A method for solving the Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic game with multiple players is presented and analytical solution is given for a type of differential games in which the system matrix can be diagonalizable. As the special cases, the Nash equilibria for some type of differential games with particular structure is studied also, and some results in previous literatures are extended. Finally, a numerical example is given to illustrate the effectiveness of the solution procedure.     

Uniqueness conditions for the affine open-loop linear quadratic differential game

In this note we consider the open-loop Nash linear quadratic differential game with an infinite-planning horizon. The performance function is assumed to be indefinite and the underlying system affine. We derive both necessary and sufficient conditions under which this game has a unique Nash equilibrium.     

Analytic solution for a class of linear quadratic open-loop Nash games

A method for solving the asymmetric coupled Riccati-type matrix differential equations for open-loop Nash strategy in linear quadratic games is presented. The class of games studied here is one in which the state weighting matrices in player's cost functionals are proportional to each other. By writing in a special order the necessary conditions for open-loop Nash strategy, a matrix with specific properties is derived. These properties are then exploited to solve the two-point boundary-value problem. Some special cases are discussed and a simple example is given to illustrate the solution procedure.     

一类n×n矩阵博弈的Nash均衡的近似计算

给出了一种求解某类n×n矩阵博弈Nash均衡的近似解的算法。通过剖分单纯形,将混合策略空间离散化,利用初始的单纯形根据标号函数和替换规则求出此类矩阵博弈Nash均衡的近似解。并分析了其最优解与近似解的计算误差。     

Twelve monotonicity conditions arising from algorithms for equilibrium problems

In the last years many solution methods for equilibrium problems (EPs) have been developed. Several different monotonicity conditions have been exploited to prove convergence. The paper investigates all the relationships between them in the framework of the so-called abstract EP. The analysis is further detailed for variational inequalities and linear EPs, which include also Nash EPs with quadratic payoffs.     

A result on output feedback linear quadratic control

In this note we consider the static output feedback linear quadratic control problem. We present both necessary and sufficient conditions under which this problem has a solution in case the involved cost depends only on the output and control variables.This result is used to present both necessary and sufficient conditions under which the corresponding linear quadratic differential game has a Nash equilibrium in case the players use static output feedback control.Another consequence of this result is that the conditions also provide sufficient conditions for the static output stabilizability problem. Of course, in case these conditions are not met this does not mean that the system is not stabilizable via static output feedback.     

Properties of Nash solutions of a two-stage nonzero-sum game

This paper contains exact expressions for the complete class of uncountably many globally optimal affine Nasb equilibrium strategies for a two-stage two-person nonzero-sum game problem with quadratic objective functionals and with dynamic information for beth players. Existence conditions for each of these Nash equilibrium solutions are derived and it is shown that a recursive Nash solution is not necessarily globally optimal. Cost-uniqueness property of the derived Nash strategies is investigated and it is proven that the game problem under consideration admits a unique Nash cost pair if and only if it can be made equivalent to either a team problem or a zero-sum game. It is also shown that existence conditions of a globally optimal Nash solution will be independent of the parameters characterizing the nonuniques of the Nash strategies only if the game problem can be made equivalent to a team problem.     

Existence and uniqueness of open-loop Nash equilibria in linear-quadratic discrete time games

We present existence and uniqueness results for an equilibrium in an M-person Nash game with quadratic performance criteria and a linear difference equation as constraint, describing the system dynamics under an open-loop information pattern. The approach used is the construction of a value function which leads to existence assertions in terms of solvability of certain symmetric and nonsymmetric Riccati difference equations.     

Controllability of Fractional Order Stochastic Differential Inclusions with Fractional Brownian Motion in Finite Dimensional Space

In this paper, sufficient conditions are formulated for controllability of fractional order stochastic differential inclusions with fractional Brownian motion (fBm) via fixed point theorems, namely the Bohnenblust-Karlin fixed point theorem for the convex case and the Covitz-Nadler fixed point theorem for the nonconvex case. The controllability Grammian matrix is defined by using Mittag-Leffler matrix function. Finally, a numerical example is presented to illustrate the efficiency of the obtained theoretical results.      

Feedback control on Nash equilibrium for discrete-time stochastic systems with Markovian jumps: Finite-horizon case

In this paper, we consider the feedback control on nonzero-sum linear quadratic (LQ) differential games in finite horizon for discrete-time stochastic systems with Markovian jump parameters and multiplicative noise. Four-coupled generalized difference Riccati equations (GDREs) are obtained, which are essential to find the optimal Nash equilibrium strategies and the optimal cost values of the LQ differential games. Furthermore, an iterative algorithm is given to solve the four-coupled GDREs. Finally, a suboptimal solution of the LQ differential games is proposed based on a convex optimization approach and a simplification of the suboptimal solution is given. Simulation examples are presented to illustrate the effectiveness of the iterative algorithm and the suboptimal solution.     

H∞控制理论在纳什均衡动态对策问题中的应用研究

在l2范数有界不确定性扰动影响情况下,把N人线性二次纳什对策问题归结为由非合作、非零和线性二次动态对策及二人零和动态对策组成的复合对策问题;基于纳什均衡对策理论的H∞控制理论方法,给出并证明了不确定环境下纳什对策问题状态反馈解存在的条件及其求解方法;通过一个政府债务稳定问题简例,说明了算法的有效性。     

Controllability, stabilizability, and continuous-time Markovianjump linear quadratic control

Consideration is given to the control of continuous-time linear systems that possess randomly jumping parameters which can be described by finite-state Markov processes. The relationship between appropriately defined controllability, stabilizability properties, and the solution of the infinite time jump linear quadratic (JLQ) optimal control problems is also examined. Although the solution of the continuous-time Markov JLQ problem with finite or infinite time horizons is known, only sufficient conditions for the existence of finite cost, constant, stabilizing controls for the infinite time problem appear in the literature. In this paper necessary and sufficient conditions are established. These conditions are based on new definitions of controllability, observability, stabilizability, and detectability that are appropriate for continuous-time Markovian jump linear systems. These definitions play the same role for the JLQ problem as the deterministic properties do for the linear quadratic regulator (LQR) problem     

Linear−quadratic optimal control and nonzero‐sum differential game of forward−backward stochastic system

An existence and uniqueness result for one kind of forward–backward stochastic differential equations with double dimensions was obtained under some monotonicity conditions. Then this result was applied to the linear‐quadratic stochastic optimal control and nonzero‐sum differential game of forward–backward stochastic system. The explicit forms of the optimal control and the Nash equilibrium point are obtained respectively. We note that our method is effective in studying the uniqueness of Nash equilibrium point. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Characterization of feedback Nash equilibria for multi-channel systems via a set of non-fragile stabilizing state-feedback solutions and dissipativity inequalities

We consider the problem of state-feedback stabilization for a multi-channel system in a game-theoretic framework, where the class of admissible strategies for the players is induced from a solution set of the individual objective functions that are associated with certain dissipativity properties of the system. In such a framework, we characterize the feedback Nash equilibria via a set of non-fragile stabilizing state-feedback solutions corresponding to a family of perturbed multi-channel systems. Moreover, we show that the existence of a weak-optimal solution to a set of constrained dissipativity problems is a sufficient condition for the existence of a feedback Nash equilibrium, whereas the set of non-fragile stabilizing state-feedbacks solutions is described in terms of a set of dilated linear matrix inequalities.     

State controllability and optimal regulator control of time-delayed systems

An important recent advance in the solution of the optimal regulator control problem for time-delayed systems is extended here to multivariable systems and to systems which exhibit multiple time delays. The state equations are partitioned into discrete and continuous portions through a state transformation such that the solution of the optimal regulator problem reduces to finding a steady-state controller gain based on both a discrete and continuous Riccati matrix. The discrete Ricatti matrix is found independently of the continuous solution due to the partitioning of the state equations, and it is not necessary to solve the system of partial differential Riccati equations which arise in the traditional solution of the linear quadratic regulator (LQR) problem for time-delayed systems. In addition, through this state transformation it becomes possible to extend the standard state controllability tests to time-delayed systems. It is shown that the controllability of the transformed state space is necessary for a feasible solution to the optimal regulator problem for time-delayed systems. This is an important test to determine the practicality of various time-delayed system realizations. Numerical examples illustrate the application of the technique to systems exhibiting multiple time delays, multivariable systems and time-series models. It is shown that the classic Wood-Berry distillation model realization does not possess state controllability properties which explains why this system has been historically difficult to control using feedback techniques.     

On relaxation algorithms in computation of noncooperativeequilibria

This paper considers a special class of numerical algorithms, the so-called relaxation algorithm, for Nash equilibrium points in noncooperative games. The relaxation algorithms have been studied by various authors for the deterministic case. Convergence conditions of this algorithm are based on fixed point theorems. For example, Basar (1987) and Li and Basar (1987) have proved its convergence for a two-player game via the contraction mapping theorem. For the quadratic case these conditions can be easily checked. For other nonlinear payoff functions it is sometimes difficult to check these convergence conditions. In this paper, the authors propose an alternative approach using the residual terms of the Nikaido-Isoda function. The convergence theorem is proved for nonsmooth weakly convex-concave Nikaido-Isoda functions. The family of weakly convex-concave functions is broad enough for applications, since if includes the family of smooth functions. When the payoff functions are twice continuously differentiable, the condition for the residual terms is reduced to strict positiveness of a matrix representing the difference of the Hessians of the Nikaido-Isoda function with respect to the first and second groups of variables. An analogous condition was used by Uryas'ev (1988) to prove convergence of the gradient-type algorithm for the Nash equilibrium problem. In this paper the authors discuss only the deterministic case; nevertheless this approach can be generalized for the stochastic Nash equilibrium problems with uncertainties in parameters