Turnpike property of nonzero-sum linear-quadratic differential games

Matrix quadratic equation solution derivation applied in finding steady state solutions of Riccati differential equations with constant coefficients

Riccati Equations in Nash and Stackelberg Differential and Dynamic Games

Rotational Motion Control by Feedback with Minimum L1-Norm

The purpose of this chapter is to develop a theory for stochastic linear-quadratic two-person differential games. Open-loop and closed-loop Nash equilibria are explored in the context of nonzero-sum and zero-sum differential games. The existence of an open-loop Nash equilibrium is characterized in terms of a system of constrained forward-backward stochastic differential equations, and the existence of a closed-loop Nash equilibrium is characterized by the solvability of a system of coupled symmetric differential Riccati equations. It is shown that in the nonzero-sum case, the closed-loop representation for open-loop Nash equilibria is different from the outcome of closed-loop Nash equilibria in general, whereas they coincide in the zero-sum case when both exist. Some results for infinite-horizon zero-sum differential games are also established in terms of algebraic Riccati equation.

This paper discusses the infinite time horizon nonzero-sum linear quadratic (LQ) differential games of stochastic systems governed by Ito’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.

Global existence is proved for the solution, of a Riccati differential equation connected with the synthesis of a boundary control problem governed by parabolic partial differential equations.

The key of solving the noncooperative linear quadratic (LQ) differential game is to solve the coupled matrix Riccati differential equation. The precise integration method based on the adaptive choosing of the two parameters is expanded from the traditional symmetric Riccati differential equation to the coupled asymmetric Riccati differential equation in this paper. The proposed expanded precise integration method can overcome the difficulty of the singularity point and the ill-conditioned matrix in the solving of coupled asymmetric Riccati differential equation. The numerical examples show that the expanded precise integration method gives more stable and accurate numerical results than the “direct integration method” and the “linear transformation method”.

In this paper, Linear-Quadratic (LQ) differential games are studied, focusing on the notion of solution provided by linear feedback Nash equilibria. It is well-known that such strategies are related to the solution of coupled algebraic Riccati equations, associated to each player. Herein, we propose an algorithm that, by borrowing techniques from algebraic geometry, allows to recast the problem of computing all stabilizing Nash strategies into that of finding the zeros of a single polynomial function in a scalar variable, regardless of the number of players and the dimension of the state variable. Moreover, we show that, in the case of a scalar two-player differential game, the proposed approach permits a comprehensive characterization - in terms of number and values - of the set of solutions to the associated game.

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.

The standard H/sub /spl infin// problem is solved for LTI systems with a single, pure input lag. The solution is based on state-space analysis, mixing a finite-dimensional and an abstract evolution model. Utilizing the relatively simple structure of these distributed systems, the associated operator Riccati equations are reduced to a combination of two algebraic Riccati equations and one differential Riccati equation over the delay interval. The results easily extend to finite time and time-varying problems where the algebraic Riccati equations are substituted by differential Riccati equations over the process time duration.

SummaryWe study a class of coupled nonlinear matrix differential equations arising in connection with the solution of a zero‐sum two‐player linear quadratic (LQ) differential game for a dynamical system modeled by an Itô differential equation subject to random switching of its coefficients. The system of differential equations under consideration contains as special cases the game‐theoretic Riccati differential equations arising in the solution of the H∞ control problem from the deterministic and stochastic cases. A set of sufficient conditions that guarantee the existence of the bounded and stabilizing solution of this kind of Riccati differential equations is provided. We show how such stabilizing solution is involved in the construction of the equilibrium strategy of a zero‐sum LQ stochastic differential game on an infinite‐time horizon and give as a byproduct the solution of such a control problem.

Policy iteration based cooperative linear quadratic differential games with unknown dynamics

This paper discusses the linear quadratic (LQ) differential games for stochastic systems with Markov jumps and multiplicative noise in infinite-time case. We introduce the definitions of exact detectability and stochastic detectability and the connection between them, which have close relation to Lyapunov equation. Based on Lyapunov equation, we obtain four-coupled algebraic Riccati equations (AREs), which are essential on finding the optimal strategies (Nash equilibrium strategies) and the optimal cost values for infinite stochastic differential games with Markov jumps. In addition, we also propose the PBH criterions of exact detectability for stochastic systems with Markov jumps.

Properties of Coupled Riccati Equations in Stackelberg Games with Time Preference Rate

We mainly consider the stability of discrete-time Markovian jump linear systems with state-dependent noise as well as its linear quadratic (LQ) differential games. A necessary and sufficient condition involved with the connection between stochasticTn-stability of Markovian jump linear systems with state-dependent noise and Lyapunov equation is proposed. And using the theory of stochasticTn-stability, we give the optimal strategies and the optimal cost values for infinite horizon LQ stochastic differential games. It is demonstrated that the solutions of infinite horizon LQ stochastic differential games are concerned with four coupled generalized algebraic Riccati equations (GAREs). Finally, an iterative algorithm is presented to solve the four coupled GAREs and a simulation example is given to illustrate the effectiveness of it.