Linear Quadratic Differential Games Research Articles

Linear-quadratic differential games: from finite to infinite dimension

Linear-quadratic differential games: from finite to infinite dimension

Game of Defending a Target: A Linear Quadratic Differential Game Approach

Pursuit-evasion (PE) differential games have recently received much attention in military applications involving adversaries. We extend the PE game problem to a problem of defending target, where the roles of the players are changed. The evader is to attack some fixed target, whereas the pursuer is to defend the target by intercepting the evader. We propose a practical strategy design approach based on the linear quadratic game theory with a receding horizon implementation. We prove the existence of solutions for the Riccati equations associated with games with simple dynamics. Simulation results justify the method.

The Open-Loop Linear Quadratic Differential Game for index one Descriptor Systems

In this paper we consider the linear quadratic differential game for descriptor systems that have index one. We derive both necessary and sufficient conditions for existence of an open-loop Nash equilibrium.

The Open-Loop Linear Quadratic Differential Game for Index One Descriptor Systems

In this paper we consider the linear quadratic differential game for descriptor systems that have index one. We derive both necessary and sufficient conditions for existence of an open-loop Nash equilibrium.

ROBUST OPTIMAL CONTROL IN NOT-COMPLETELY CONTROLLABLE LINEAR TIME-VARYING SYSTEMS

A linear-quadratic differential game related to robust control problems in linear time-varying systems is considered. We show that difficulties, which may arise in obtaining the solution of the game and the optimal control law, are related to the fact that the trajectories x(t) (where the state x = (x1, ..., xn)) of the controlled system may be exactly or approximately localized in the subspace of dimensionality less than n, under all possible bounded piecewise continuous control impacts acting on the inputs, provided that the initial vector x(t0) is fixed, x(t0) = x0. We demonstrate how to overcome these difficulties by using the solution D(t) of inverse matrix differential equation of Riccati type and the pseudo-inverse matrix D+(t). We establish new sufficient conditions for the existence of the saddle point. We show that the optimal control can be extended beyond conjugate points if D(t) ≥ 0. We also derive and prove new explicit formulae for robust optimal control law using the matrices D(t) and D+(t). With the use of the pseudo-inverse matrix D+(t) uncontrollable (or almost uncontrollable) components can be automatically eliminated in choosing robust optimal control. In this paper robust optimal control problems in linear time-varying systems are investigated in the finite-horizon case.

ROBUST SOLUTION OF A TIME-VARIABLE INTERCEPTION PROBLEM: A CHEAP CONTROL APPROACH

A planar interception problem of a maneuverable target is considered using the linearized kinematic model with variable velocities and first-order dynamics of the interceptor and target. The maneuverabilities of the interceptor and target are assumed to be variable. By using an auxiliary zero-sum linear-quadratic differential game with cheap controls of both players, an interception guidance law, linear with respect to the state vector, is derived. An analytical description of the set of initial positions (capture set) is obtained, from which this guidance law provides zero miss distance, subject to given maneuverabilities of the interceptor and target. A numerical example illustrating the analytical results is presented.

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.

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.

Capture Zones of Cheap Control Interception Strategies

The interception of a maneuverable target is considered using a linearized kinematical model with first-order acceleration dynamics. This study concentrates on control strategies, based on the solution of a linear-quadratic differential game with cheap controls, guaranteeing capture without exceeding some prescribed (physical) limits. The conditions for a nontrivial capture zone are derived. Detailed description of these capture zones is obtained in the space of the problem parameters.

Linear Quadratic Differential Games Guidance Law for Dual Controlled Missiles

A novel guidance law designed specifically for a missile having forward and aft control systems is presented. For the derivation a linear quadratic differential games formulation of the end-game interception scenario is used. Conditions for achieving perfect intercept are given and new game spaces are presented. It is also shown that increasing the direct lift associated with the canard control allows intercepting targets with higher maneuvering capability, while for the tail control the effect is opposite.

Quasi-equilibrium in LQ differential games with bounded uncertain disturbances: robust and adaptive strategies with pre-identification via sliding mode technique

A finite time multi-persons linear-quadratic differential game with bounded disturbances and uncertainties is considered. When players cannot measure these disturbances, it is demonstrated that the standard feedback Nash strategies bear to a quasi Nash-equilibrium depending on an uncertainty upper bound that confirms the robustness property of such standard strategies. In the case of periodic disturbances, another concept, namely adaptive concept, containing three different versions is suggested. They are sliding modes, second order sliding modes and window integration. All of them realize the identification of unknown periodic disturbances during an “identification period” when all participants apply the standard feedback Nash strategies with the, so-called, “shifting signal” generated only by a known external exciting signal. After that period the complete standard strategies with “pre-identification” including the recalculated shifting signal are activated. A numerical example dealing with a two participants game shows that the cost functional for each player achieves better values when the adaptive approach is applied.

A new class of nonsymmetric algebraic Riccati equations

Motivated by the study of linear quadratic differential games, we introduce a new class of nonsymmetric algebraic Riccati equations. It is shown that every equation in this class has a unique stabilizing solution, which is the solution required to find the open-loop Nash equilibrium for the differential game. We show that the doubling algorithm can be used to find this solution efficiently. The solution may also be found by the Schur method, and under further assumptions by Newton’s method and a basic fixed-point iteration.

Numerical computation of sign-indefinite linear quadratic differential games for weakly coupled large-scale systems

In this paper, N-player linear quadratic differential games that are sign-indefinite for infinite horizon weakly coupled large-scale systems are discussed. After establishing the asymptotic structure and local uniqueness of the solution for cross-coupled sign-indefinite algebraic Riccati equations (CSARE), a new algorithm for solving CSARE is provided. It is shown that the proposed algorithm attains linear convergence. Moreover, in order to reduce the computational workspace, the recursive algorithm is combined. Finally, a high-order approximation strategy based on the proposed iterative solutions is described. As a result, it was recently proved that the numerical strategy achieves a high-order approximation of the equilibrium value. As another important feature, when the small parameters are unknown, a parameter-independent strategy is developed.

Linear Quadratic Differential Games: Saddle Point and Riccati Differential Equation

Zhang [SIAM J. Control Optim., 43 (2005), pp. 2157–2165] recently established the equivalence between the finiteness of the open loop value of a two-player zero-sum linear quadratic (LQ) game and the finiteness of its open loop lower and upper values. In this paper we complete and sharpen the results of Zhang for the finiteness of the lower value of the game by providing a set of necessary and sufficient conditions that emphasizes the feasibility condition: $(0,0)$ is a solution of the open loop lower value of the game for the zero initial state. Then we show that, under the assumption of an open loop saddle point in the time horizon $[0,T]$ for all initial states, there is an open loop saddle point in the time horizon $[s,T]$ for all initial times s, $0\le s<T$, and all initial states at time s. From this we get an optimality principle and adapt the invariant embedding approach to construct the decoupling symmetrical matrix function $P(s)$ and show that it is an $H^1(0,T)$ solution of the matrix Riccati differential equation. Thence an open loop saddle point in $[0,T]$ yields closed loop optimal strategies for both players. Furthermore, a necessary and sufficient set of conditions for the existence of an open loop saddle point in $[0,T]$ for all initial states is the convexity-concavity of the utility function and the existence of an $H^1(0,T)$ symmetrical solution to the matrix Riccati differential equation. As an illustration of the cases where the open loop lower/upper value of the game is $-\infty$/$+\infty$, we work out two informative examples of solutions to the Riccati differential equation with and without blow-up time.

Necessary and Sufficient Conditions for Solving Cooperative Differential Games

In this note we present as well necessary as sufficient conditions for existence of a Pareto optimum for general non-convex differential games. The obtained results are used to analyze the non-convex regular indefinite linear quadratic differential game. For the scalar case an algorithm is devised to find all Pareto efficient solutions.

Non-linear strategies in a linear quadratic differential game

We study non-linear Markov perfect equilibria in a two agent linear quadratic differential game. In contrast to the literature owing to Tsutsui and Mino [1990. Nonlinear strategies in dynamic duopolistic competition with sticky prices. Journal of Economic Theory 52, 136–161], we do not associate endogenous subsets of the state space with candidate solutions. Instead, we use the ‘catching up optimality’ criterion to address the possibility of infinitely valued value functions. Applying sufficiency conditions for existence based on those in Dockner et al. [2000. Differential Games in Economics and Management Science. Cambridge University Press, Cambridge] yields the familiar linear MPE and a condition under which a continuum of non-linear MPEs exists. These include, as their limit, a previously unreported piecewise linear MPE. The condition relaxes with increasing patience, allowing more efficient steady states, thus suggesting a Folk Theorem for differential games. As the lower state and control bounds go to - ∞ , the non-linear strategies are eliminated.

Linear Quadratic Games: An Overview

In this paper we review some basic results on linear quadratic differential games. We consider both the cooperative and non-cooperative case.For the non-cooperative game we consider the open-loop and (linear) feedback information structure. Furthermore the effect of adding uncertainty is considered. The overview is based on [9]. Readers interested in detailed proofs and additional results are referred to this book.

Special issue on dynamic games

This special issue of Computational Management Science covers a series of topics in dynamic games, with four of the six papers having a survey flavor. Ben Abdelaziz and Krichen review the optimal stopping time problem in the context where several decision makers are involved. They classify the literature in terms of the main variables which are usually considered, i.e., arrival of offers, number of offers to be selected, nature of the game, utility function of the decision maker, etc. Engwerda reviews some algorithms for the computation of Nash equilibria in the class of linear-quadratic differential games, a class widely used in economics and management science applications. The author covers both the open-loop and the feedback information structures, and finite and infinite planning horizon.Genc studies market outcome equivalence of two dynamic production-capital investment games under uncertainty. One game is played under complete information, and the other, feedback game, is played under incomplete information about the opponents’ costs and market demand, as in, e.g., industries in which a homogeneous good is exchanged via an auction mechanism. He shows that in that case, the feedback game may predict the complete information equilibrium market outcomes. Jorgensen and Zaccour take a general look at the (huge) literature applying differential games in economics and management science, and conclude that this literature has aimed to a large extent for analytical results. After suggesting some explanations to this state of affair, they give an account of a number of (recent and earlier) developments in numerical methods in differential games that, by and large, have not been used in these fields.Krawczyk deals with games where the agents face joint constraints as in, e.g., several electric utilities using a

MACROECONOMIC STABILIZATION POLICIES IN THE EMU: SPILLOVERS, ASYMMETRIES AND INSTITUTIONS

ABSTRACTThis paper studies the institutional design of the coordination of macroeconomic stabilization policies within a monetary union in the framework of linear quadratic differential games. A central role in the analysis plays the partitioned game approach of the endogenous coalition formation literature. The specific policy recommendations in the European Economic and Monetary Union (EMU) context depend on the particular characteristics of the shocks and the economic structure. In the case of a common shock, fiscal coordination or full policy coordination is desirable. When anti‐symmetric shocks are considered, fiscal coordination improves the performance but full policy coordination does not produce further gains in policymakers' welfare.

ε-Equilibrium in LQ differential games with bounded uncertain disturbances: robustness of standard strategies and new strategies with adaptation

A finite time multi-persons linear-quadratic differential game (LQDG) with bounded disturbances and uncertainties is considered. When players cannot measure these disturbances and uncertainties, the standard feedback Nash strategies are shown to yield to an ε-(or quasi) Nash equilibrium depending on an uncertainty upper bound that confirms the robustness property of such standard strategies. In the case of periodic disturbances, another concept, namely adaptive concept, is suggested. It is defined an “adaptation period” where all participants apply the standard feedback Nash strategies with the, so-called, “shifting signal” generated only by a known external exciting signal. Then, during the adaptation, the readjustment (or correction) of the control strategies is realized to estimate the effect of unknown periodic disturbances by the corresponding correction of the shifting vector. After that adaptation period, the complete standard strategies including the recalculated shifting signal are activated allowing the achievement of pure (ε = 0) Nash equilibrium for the rest of the game. A numerical example dealing with a two participants game shows that the cost functional for each player achieves better values when the adaptive approach is applied.