lec3_6243_2003

lec3_6243_2003 - Massachusetts Institute of Technology...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 3: Continuous Dependence On Parameters 1 Arguments based on continuity of functions are common in dynamical system analysis. They rarely apply to quantitative statements, instead being used mostly for proofs of existence of certain objects (equilibria, open or closed invariant set, etc.) Alternatively, continuity arguments can be used to show that certain qualitative conditions cannot be satisfied for a class of systems. 3.1 Uniqueness Of Solutions In this section our main objective is to establish sucient conditions under which solutions of ODE with given initial conditions are unique. 3.1.1 A counterexample Continuity of the function a : R n R n on the right side of ODE x ( t ) = a ( x ( t )) , x ( t ) = x (3.1) does not guarantee uniqueness of solutions. Example 3.1 The ODE x ( t ) = 3 | x ( t ) | 2 / 3 , x (0) = has solutions x ( t ) and x ( t ) t 3 (actually, there are infinitely many solutions in this case). 1 Version of September 12, 2003 2 3.1.2 A general uniqueness theorem The key issue for uniqueness of solutions turns out to be the maximal slope of a = a ( x ): to guarantee uniqueness on time interval T = [ t , t f ], it is sucient to require existence of a constant M such that | a ( x 2 ) | M | x 1 x 1 ) a ( x 2 | x 1 , for all x 2 from a neigborhood of a solution x : [ t , t f ] R n of (3.1). The proof of both existence and uniqueness is so simple in this case that we will formulate the statement for a much more general class of integral equations. Theorem 3.1 Let X be a subset of R n containing a ball x ) = { x B r ( x R n : | x | r } of radius r > , and let t 1 > t be real numbers. Assume that function a : X [ t , t 1 ] [ t , t 1 ] R n is such that there exist constants M, K satisfying | a ( x 1 , , t ) a ( x 2 , , t ) | K | x 1 x 2 | x 1 , x 2 B r ( x ) , t t t 1 , (3.2) and | a ( x, , t ) | M x B r ( x ) , t t t 1 . (3.3) Then, for a suciently...
View Full Document

Page1 / 6

lec3_6243_2003 - Massachusetts Institute of Technology...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online