Partial Stabilization and Control of Distributed Parameter Systems With Elastic Elements

This monograph provides a rigorous treatment of problems related to partial asymptotic stability and controllability for models of flexible structures described by coupled nonlinear ordinary and partial differential equations or equations in abstract spaces. The text is self-contained, beginning with some basic results from the theory of continuous semigroups of operators in Banach spaces. The problem of partial asymptotic stability with respect to a continuous functional is then considered for a class of abstract multivalued systems on a metric space. Next, the results of this study are applied to the study of a rotating body with elastic attachments. Professor Zuyev demonstrates that the equilibrium cannot be made strongly asymptotically stable in the general case, motivating consideration of the problem of partial stabilization with respect to the functional that represents “averaged” oscillations. The book’s focus moves on to spillover analysis for infinite-dimensional systems with finite-dimensional controls. It is shown that a family of L2-minimal controls, corresponding to low frequencies, can be used to obtain approximate solutions of the steering problem for the complete system.
The book turns from the examination of an abstract class of systems to particular physical examples. Timoshenko beam theory is exploited in studying a mathematical model of a flexible-link manipulator. Finally, a mechanical system consisting of a rigid body with the Kirchhoff plate is considered. Having established that such a system is not controllable in general, sufficient controllability conditions are proposed for the dynamics on an invariant manifold.
Academic researchers and graduate students interested in control theory and mechanical engineering will find Partial Stabilization and Control of Distributed-Parameter Systems with Elastic Elements a valuable and authoritative resource for investigations on the subject of partial stabilization.

Professor Zuyev’s research activity at the National Academy of Sciences of Ukraine and Donetsk National University is concentrated on the modeling and analysis of complex mechanical systems. The Department of Mechanics in Donetsk has been focusing on problems of multibody system dynamics and mathematical control theory for more than four decades. Our regular contacts with engineers from the State Space Agency of Ukraine have stimulated the development of new control design schemes for remote sensing satellites, in particular, for the “Sich-2” satellite. A particular outcome of this activity, published in a paper on the stabilization of a model with a pair of flexible beams, was awarded an honorable mention at the 15th IFAC World Congress in Barcelona, 2002. His collaboration with the group of Prof. Oliver Sawodny (currently at the University of Stuttgart), supported by the Alexander von Humboldt Foundation since 2004, has resulted in several publications on the modeling and control of flexible-link manipulators (such as IVECO turntable ladders).
An academic viewpoint on similar problems, presented in the monograph, should be useful for the reader interested in the theory of distributed parameter control systems.