Lyapunov Functions in Nonlinear Unsteady Dynamics and Control

This monograph is a little gem. It is a must read for everyone who deals with just as the problems of stability of dynamic systems and control design, so too mathematical modeling and simulation in all areas of modern natural philosophy and engineering. Why? First, because to a certain degree current science and technology are all about mathematical models. Second, after you have successfully constructed the adequate mathematical model of a physical process the next step you have to make is checking its stability out. This sine qua non is dictated by thousand years of human experience implying that any manifestations of instability around you usually are a bad sign. Third, the knowledge the book provides its reader with is unique and original. You will not find it in other sources because the pivotal ideas used in the research, the view of the problem and the methods of its study lie far away from the mainstream of the today's approach to Lyapunov functions. But perhaps to the big surprise of the readers this is not the highest value the book possesses. The methodology and the techniques developed and used in the research return us to the fountainheads of the theory of stability and topology. This is what cuts the diamond in the brilliant. The reader will dip into the vision of the theory and the branch of mathematics their founders, Henri Poincar and Aleksandr Lyapunov, respectively had. The text is composed in such a way that the readers are gradually guided up the Ladder of Cognition where climbing on the next step requires them to unlock the door hiding the corresponding knowledge by constructivism method or in other words to rediscover it per say as the author did it in his time. On the first stage they will learn about the geometric-topological structure of parametric differential inclusions and free dynamic systems by means of the techniques of foliations and fiber bundles illustrated geometrically with figures to create visual associations. The second stage reveals the mysterious ubiquity of the universal law of canonicity that in our case evinces itself as the cascades of flattening diffeomorphisms converting the initial objects to the ones of canonical forms. The third stage tells us about the second universal law, namely the principle of equivalency stating the conditions, under which the cascade of flattening diffeomorphisms conserves the fundamental property of stability. In physics its twin acts as the principle of symmetry or topological invariants. The fourth stage familiarizes the readers with the structural stability of the elementary constructional "bricks" of the extended phase spaces of free dynamic systems within a given parametric differential inclusion. These structural elements are made of the leaves of the invariant foliations of codinensionality one and divided into the classes of attractors, repellers, left and right shunts. The further analysis is founded on the introduced concept of the classificational stability of the covering maps of the coverings with stratified total and base spaces. The classificational stability is a transitional bridge from the classic Lyapunov stability of free dynamic systems through Andronov-Pontryagin structural stability of rough systems ("syst mes grossiers") to the robustness of dynamic system with control in the most general nonlinear, non-autononous case. The final fifth stage deals with the design of a new class of robust controls, namely wide-sense ones, for the parametric van der Pol equation with an external force as a control. The wide-sense robust control is defined to ensure the stability of a given system with control and the topological equivalence of its (extended) phase spaces not only at the unpredictable variations of system parameters but also at the sudden partial loss of control resources. The book is richly illustrated with the figures explaining the theory and the graphs of computer simulations.