745 research outputs found

    Feedback Nash Equilibria for Linear Quadratic Descriptor Differential Games

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    In this note we consider the non-cooperative linear feedback Nash quadratic differential game with an infinite planning horizon for descriptor systems of index one. The performance function is assumed to be indefinite. We derive both necessary and sufficient conditions under which this game has a Nash equilibrium.linear-quadratic games;linear feedback Nash equilibrium;affine systems;solvability conditions;Riccati equations

    The Optimal Linear Quadratic Feedback State Regulator Problem for Index One Descriptor Systems

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    In this note we present both necessary and sufficient conditions for the existence of a linear static state feedback controller if the system is described by an index one descriptor system. A priori no definiteness restrictions are made w.r.t. the quadratic performance criterium. It is shown that in general the set of solutions that solve the problem constitutes a manifold. This feedback formulation of the optimization problem is natural in the context of differential games and we provide a characterization of feedback Nash equilibria in a deterministic context.linear quadratic optimal control;descriptor systems;static stabilizing state feedback control

    Feedback saddle point equilibria for soft-constrained zero-sum linear quadratic descriptor differential game

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    In this paper the feedback saddle point equilibria of soft-constrained zero-sum linear quadratic differential games for descriptor systems that have index one will be studied for a finite and infinite planning horizon. Both necessary and sufficient conditions for the existence of a feedback saddle point equilibrium are considere

    The Optimal Linear Quadratic Feedback State Regulator Problem for Index One Descriptor Systems

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    In this note we present both necessary and sufficient conditions for the existence of a linear static state feedback controller if the system is described by an index one descriptor system. A priori no definiteness restrictions are made w.r.t. the quadratic performance criterium. It is shown that in general the set of solutions that solve the problem constitutes a manifold. This feedback formulation of the optimization problem is natural in the context of differential games and we provide a characterization of feedback Nash equilibria in a deterministic context.

    Necessary and Sufficient Conditions for Feedback Nash Equilibria for the Affine Quadratic Differential

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    In this note we consider the non-cooperative linear feedback Nash 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 Nash equilibrium.linear-quadratic games;linear feedback Nash equilibrium;affine systems;solvability conditions;Riccati equations

    Feedback Nash Equilibria for Descriptor Differential Games Using Matrix Projectors

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    In this article we address the problem of finding feedback Nash equilibria for linear quadratic differential games defined on descriptor systems. First, we decouple the dynamic and algebraic parts of a descriptor system using canonical projectors. We discuss the effects of feedback on the behavior of the descriptor system. We derive necessary and sufficient conditions for the existence of the feedback Nash equilibria for index 1 descriptor systems and show that there exist many informationally non-unique equilibria corresponding to a single solution of the game. Further, for descriptor systems with index greater than 1, we give a regularization based approach and discuss the associated drawbacks.

    Essays on dynamic games.

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    Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order Reduction

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    Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications. Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated. The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters. Some conclusions and an appendix complete the thesis
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