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 depend 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.LQ theory;Algebraic Riccati equations;Differential games

On the relationship between the open-loop Nash equilibrium in LQ-games and the inertia of a matrix

In this paper we consider the location of the eigenvalues of the composite matrix ( -A S1 S2 ) ( Q1 At 0 ) ( Q2 0 At ) , where the matrices Si and Qi are assumed to be semi-positive definite. Two interesting observations, which are not or only partially mentioned in literature before, challenge this study. The first observation is that this matrix appears naturally in a both necessary and sufficient condition for the existence of a unique open-loop Nash solution in the 2-player linear-quadratic dynamic game and, more in particular, its inertia play an important role in the analysis of the convergence of the associated state in this game. The second observation is that from the eigenvalue and eigenstructure of this matrix all solutions for the algebraic Riccati equations corresponding with the above mentioned dynamic game can be directly calculated and, moreover, also the eigenvalues of the associated closed-loop system. Simulation experiments suggest that the composite matrix will have at least n eigenvalues (here n is the state dimension of the system) with a positive real part. Unfortunately, it turns out that this property of the inertia of this matrix in general does not hold. Some specific cases for which the property does hold are discussed.Game Theory;Nash Equilibrium;game theory

Asymptotic analysis of Nash equilibria in nonzero-sum linear-quadratic differential games: The two player case

Game Theory;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 depend 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.

The open-loop Nash equilibrium in LQ-games revisited

In this paper we reconsider the conditions under which the finite-planning-horizon linear- quadratic differential game has an open-loop Nash equilibrium solution. Both necessary and sufficient conditions are presented for the existence of a unique solution in terms of an invertibility condition on a matrix. Moreover, we show that the often encountered solvability conditions stated in terms of Riccati equations are in general not correct. In an example we show that existence of a solution of the associated Riccati-type differential equations may fail to exist whereas an open-loop Nash equilibrium solution exists. The scalar case is studied in more detail, and we show that solvability of the associated Riccati equations is in that case both necessary and sufficient. Furthermore we consider convergence properties of the open-loop Nash equilibrium solution if the planning horizon is extended to infinity. To study this aspect we consider the existence of real solutions of the associated algebraic Riccati equation in detail and show how all solutions can be easily calculated from the eigenstructure of a matrix.

The open-loop Nash equilibrium in LQ-games revisited

In this paper we reconsider the conditions under which the finite-planning-horizon linear- quadratic differential game has an open-loop Nash equilibrium solution. Both necessary and sufficient conditions are presented for the existence of a unique solution in terms of an invertibility condition on a matrix. Moreover, we show that the often encountered solvability conditions stated in terms of Riccati equations are in general not correct. In an example we show that existence of a solution of the associated Riccati-type differential equations may fail to exist whereas an open-loop Nash equilibrium solution exists. The scalar case is studied in more detail, and we show that solvability of the associated Riccati equations is in that case both necessary and sufficient. Furthermore we consider convergence properties of the open-loop Nash equilibrium solution if the planning horizon is extended to infinity. To study this aspect we consider the existence of real solutions of the associated algebraic Riccati equation in detail and show how all solutions can be easily calculated from the eigenstructure of a matrix.Nash Equilibrium;Game Theory;game theory