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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 suﬃcient 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 suﬃcient 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 suﬃciently...
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 Spring '09
 AlexandreMegretski
 Dynamics, Topology, Continuous function, initial conditions, Lipschitz continuity

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