Stabilizability, controllability and optimal strategies of linear and nonlinear dynamical games

摘要

In this work we investigate besides the optimal control theory, also dynamic game theory in case of linear and nonlinear differential systems. In Chapter 1 the most important results of the linear control theory are presented, whereby also for some classical statements modified proof and resuming consequences are indicated. In this section the tools, which are used later, are described. Chapter 2 deals with the theory of linear-quadratic Nash games. Apart from the well-known results on the optimal strategies, which are completed here with further conditions on the existence and uniqueness, we concern here the questions of the controllability and stabilizability of games. In the third chapter, Riccati equations are introduced briefly, and we also discuss approximation methods. Afterwards, in Chapter 4, an investigation of disturbed games takes place. This means that - apart from the optimal controls of the players - the system is influenced by a noise-like signal. Here strategies fur maximal noise reduction are calculated. Also the existence and uniqueness of these strategies are examined. Finally, in the fifth chapter the question is answered, under which assumptions a game on finite time horizon possesses the same stabilizing characteristics, as the one on infinite time horizon. Chapters 6,7,8 of this work are dedicated to systems on manifolds. After a short introduction in Chapter 6 control systems are examined, in particular invariant control systems over Lie groups on controllability and quite briefly also on stabilizability. Afterwards a method is shown, how one explicitely finds an optimal trajectory for controllable nonlinear control systems. Such curves are called ' splines '. Also a numerical algorithm for the constuction of such curves on different Lie groups is presented and examined. Finally, in Chapter 8, nonlinear differential games on Lie groups are presented and the existence of Nash strategies with and without boundary value problems are examined.