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lec6_6243_2003

# lec6_6243_2003 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 6: Storage Functions And Stability Analysis 1 This lecture presents results describing the relation between existence of Lyapunov or storage functions and stability of dynamical systems. 6.1 Stability of an equilibria In this section we consider ODE models x ˙( t ) = a ( x ( t )) , (6.1) where a : X R n is a continuous function defined on an open subset X of R n . Remem- ber that a point ¯ x 0 ) = 0, i.e. if x ( t ) ¯ x 0 X is an equilibrium of (6.1) if a x 0 is a solution of (6.1). Depending on the behavior of other solutions of (6.1) (they may stay close to x 0 , or converge to ¯ ¯ x 0 as t , or satisfy some other specifications) the equilibrium may be called stable, asymptotically stable, etc. Various types of stability of equilibria can be derived using storage functions. On the other hand, in many cases existence of storage functions with certain properties is impled by stability of equilibria. 6.1.1 Locally stable equilibria Remember that a point ¯ x 0 X is called a (locally) stable equilibrium of ODE (6.1) if for every � > 0 there exists � > 0 such that all maximal solutions x = x ( t ) of (6.1) with x 0 | � are deinfed for all t 0, and satisfy | x ( t ) ¯ 0 | < � for all t 0. | x (0) ¯ x The statement below uses the notion of a lower semicontinuity : a function f : Y R , defined on a subset Y of R n , is called lower semicontinuous if lim inf f (¯) f x x x ) ¯ Y. r 0 ,r> 0 x Y : | ¯ x | <r ¯ x ¯ 1 Version of September 24, 2003

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2 Theorem 6.1 ¯ x 0 X is a locally stable equilibrium of (6.1) if and only if there exist c > 0 and a lower semicontinuous function V : B c x 0 ) R , defined on x 0 ) = { ¯ x 0 < c } B c x : x ¯ and continuous at x 0 , such that V ( x ( t )) is monotonically non-increasing along the solu- ¯ tions of (6.1), and V x ) ¯ x 0 ) / {
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