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Optimal syntheses for control systems on 2-D manifolds. (English) Zbl 1137.49001
Mathématiques & Applications (Berlin) 43. Berlin: Springer (ISBN 3-540-20306-0/pbk). xiii, 261 p. (2004).
Publisher’s description: Control theory deals with systems that can be controlled, i.e. whose evolution can be influenced by some external agent. The birth of control theory can be a subject of discussion, however it was after the second world war that it had a great development due to engineering applications. Then it became a recognized mathematical research field.
About the same time, probably the most important tool in optimal control was proved, namely, the Pontryagin maximum principle. The goal of optimal control is to design trajectories that minimize some given cost. The respective theory can be viewed as a generalization of the calculus of variations.
Another typical problem is the one of prescribing the control automatically as a function of the state variables, i.e. in the form of a feedback, to avoid disturbances and ensure robustness of the system. For instance, one looks for a stabilizing feedback, that is a feedback guaranteeing Lyapunov stability of a given equilibrium. Among many applications, this can be used in aerospace engineering to stabilize communication satellites.
Starting from the late 1960s, the use of differential geometry for control problems gave birth to the so-called geometric control theory. The development of strong mathematical tools permitted one to attack problems of increasing difficulty and to suggest a systematic way towards the construction of optimal feedbacks. This ‘Holy Grail’ of control furnishes the solution to both kinds of problems: optimal trajectories implementation and feedback design. However, even simple optimal feedbacks are discontinuous, and the solution of the corresponding differential equation may not generate optimal trajectories. The optimal synthesis became an important solution concept for optimal control problems. Still, the optimal synthesis has been achieved only for some specific examples or classes of systems of low dimensions.
The aim of this book is to develop a complete synthesis theory for minimum time on two-dimensional manifolds. In addition to the construction of optimal synthesis for generic smooth single-input system, we are able to obtain a topological classification of the resulting nonsmooth flows, in the spirit of the work of Andronov-Pontryagin-Peixoto for two-dimensional dynamical systems. The research encompasses a comprehensive analysis of singularities, a detailed study of the minimum time function and a detailed description of the geometry underlying the Pontryagin maximum principle.
Contents: Introduction; 1. Geometric control; 2. Time optimal synthesis for 2-D systems; 3. Generic properties of the minimum time function; 4. Extremal synthesis; 5. Projection singularities; A. Some technical proofs of Chapter 2; B. Bidimensional sources; References; Index.
The volume is suitable for a graduate level one-semester course on optimal syntheses theory for mathematicians or engineers with a solid mathematical background.

##### MSC:
 49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control 49N35 Optimal feedback synthesis 93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory 93B29 Differential-geometric methods in systems theory (MSC2000) 93B50 Synthesis problems