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Unformatted text preview: March 27, 2011 20:29 World Scientific Book - 9in x 6in ws-book9x6 Chapter 5 Elements of Stability This chapter is entirely dedicated to the study of the stability of solutions to certain systems of differential equations. In the first section we introduce and illustrate the main concepts referring to stability. The second one is concerned with several necessary and sufficient conditions for various types of stability in the particular case of first-order systems of linear differential equations. In the third section we present some sufficient conditions under which the asymptotic stability of the null solution of a first-order differential system is inherited by the null solution of a certain perturbed system, provided the perturbation is small enough. In the fourth section we prove several sufficient conditions for stability expressed by means of some functions decreasing along the trajectories, while in the fifth section we include several results regarding the stability of solutions of dissipative systems. In the sixth section we analyze the stability problem referring to automatic control systems, while the seventh section is dedicated to some considerations concerning instability and chaos. As each chapter of this book, this one also ends with an Exercises and Problems section. 5.1 Types of Stability In its usual meaning, stability is that property of a particular state of a given system of preserving the features of its evolution, as long as the perturbations of the initial data are sufficiently small. This meaning comes from Mechanics, where it describes that property of the equilibrium state of a conservative system of being insensitive “ `a la longue ” to any kind of perturbations of “small intensity”. Mathematically speaking, this notion 155 March 27, 2011 20:29 World Scientific Book - 9in x 6in ws-book9x6 156 Elements of Stability has many other senses, all coming from the preceding one, and describing various kinds of continuity of a given global solution of a system as function of the initial data, senses which are more or less different from one another. The rigorous study of stability has its origins in the works of Celestial Mechanics of both Poincar´ e and Maxwell, and has culminated in 1892 with the doctoral thesis of the founder of this modern branch of differential equations, Lyapunov. As we have already shown in Theorem 2.5.2, under certain regularity conditions on the function f , the map η 7→ x ( · ,a,η ) — the unique saturated solution of the Cauchy problem ‰ x = f ( t,x ) x ( a ) = η CP ( D ) — is locally Lipschitz from Ω to C ([ a,b ]; R n ), for each b ∈ ( a,b ξ ), where [ a,b ξ ) is the domain of definition of the saturated solution x ( · ,a,ξ ). A much more delicate problem, and of great practical interest, is that of finding sufficient conditions on the function f such that, on one hand, x ( · ,a,ξ ) be defined on [ a, + ∞ ) and, on the other hand, the map η 7→ x ( · ,a,η ) be continuous from a neighborhood of...
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