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A dynamical system is a mathematical model described by a high dimensional ordinary differential equation for a wide variety of real world phenomena, which can be as simple as a clock pendulum or as complex as a chaotic Lorenz system. Stability is an important topic in the studies of the dynamical system. A major challenge is that the analytical solution of a time-varying nonlinear dynamical system is in general not known. Lyapunov's direct method is a classical approach used for many decades to study stability without explicitly solving the dynamical system, and has been successfully employed in numerous applications ranging from aerospace guidance systems, chaos theory, to traffic assignment. Roughly speaking, an equilibrium is stable if an energy function monotonically decreases along the trajectory of the dynamical system. This paper extends Lyapunov's direct method by allowing the energy function to follow a rich set of dynamics. More precisely, the paper proves two theorems, one on globally uniformly asymptotic stability and the other on stability in the sense of Lyapunov, where stability is guaranteed provided that the evolution of the energy function satisfies an inequality of a non-negative Hurwitz polynomial differential operator, which uses not only the first-order but also high-order time derivatives of the energy function. The classical Lyapunov theorems are special cases of the extended theorems. the paper provides an example in which the new theorem successfully determines stability while the classical Lyapunov's direct method fails.
Proceedings of the Tenth Power Systems Computation Conference
Bridging the gap between elementary courses and the research literature in this field, the book covers the basic concepts necessary to study differential equations. Stability theory is developed, starting with linearisation methods going back to Lyapunov and Poincaré, before moving on to the global direct method. The Poincaré-Lindstedt method is introduced to approximate periodic solutions, while at the same time proving existence by the implicit function theorem. The final part covers relaxation oscillations, bifurcation theory, centre manifolds, chaos in mappings and differential equations, and Hamiltonian systems. The subject material is presented from both the qualitative and the quantitative point of view, with many examples to illustrate the theory, enabling the reader to begin research after studying this book.
Submanifold stabilization is the problem of steering a quantity towards a desired submanifold of the space in which it evolves. This is done by controlling the system producing the quantity in an appropriate fashion. In this thesis, methods for explicitly constructing controllers which solve submanifold stabilization problems are proposed. To this end, three distinct approaches are pursued: For control systems modeled by input-affne differential equations, a construction for turning the submanifold into an asymptotically stable invariant set is presented. For controllers which shall stabilize the submanifold with minimal energy consumption, the structure of such optimal controls is investigated. For control systems modeled by input-output relationships, a framework for bounding the integral deviation of the output from the submanifold is proposed.
Mathematical Methods in Nuclear Reactor Dynamics covers the practical and theoretical aspects of point-reactor kinetics and linear and nonlinear reactor dynamics. The book, which is a result of the lectures given at the University of Michigan, is composed of seven chapters. The opening chapter of the book describes various physical phenomena influencing the temporal behavior of neutrons to provide insights into the physics of reactor dynamics and the interrelationships between various diverse phenomena. The text then presents a set of equations, called point kinetic equation, which describes the time behavior of the total power generated in the medium. The book also provides a short discussion on Gyftopoulos modification and Becker's formulation. The next chapters explore the exact methods for solving the feedback-free point kinetic equations for a number of reactivity insertions and the validity of the various approximate methods of solution. The book also examines the derivation of models for a certain reactor type and briefly discusses the validity of these models in certain cases against experimental data. A chapter focuses on a concise presentation of the stability theory of linear systems with feedback. Lastly, the concepts of stability in nonlinear reactor systems and the criteria for asymptotic stability in the large as well as in a finite domain of initial disturbances are covered in the concluding chapter. The text is an ideal source for nuclear engineers and for those who have adequate background in reactor physics and operational and applied mathematics.