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lec10_6243_2003

lec10_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 10: Singular Perturbations and Averaging 1 This lecture presents results which describe local behavior of parameter-dependent ODE models in cases when dependence on a parameter is not continuous in the usual sense. 10.1 Singularly perturbed ODE In this section we consider parameter-dependent systems of equations ± x ˙( t ) = f ( x ( t ) , y ( t ) , t ) , (10.1) �y ˙ = g ( x ( t ) , y ( t ) , t ) , where [0 , � 0 ] is a small positive parameter. When > 0, (10.1) is an ODE model. For = 0, (10.1) is a combination of algebraic and differential equations. Models such as (10.1), where y represents a set of less relevant, fast changing parameters, are fre- quently studied in physics and mechanics. One can say that singular perturbations is the “classical” approach to dealing with uncertainty, complexity, and nonlinearity. 10.1.1 The Tikhonov’s Theorem A typical question asked about the singularly perturbed system (10.1) is whether its solutions with > 0 converge to the solutions of (10.1) with = 0 as 0. A suffi- cient condition for such convergence is that the Jacobian of g with respect to its second argument should be a Hurwitz matrix in the region of interest. Theorem 10.1 Let x 0 : [ t 0 , t 1 ] ∞� R n , y 0 : [ t 0 , t 1 ] ∞� R m be continuous functions satisfying equations x ˙ 0 ( t ) = f ( x 0 ( t ) , y 0 ( t ) , t ) , 0 = g ( x 0 ( t ) , y 0 ( t ) , t ) , 1 Version of October 15, 2003
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2 where f : R n × R m × R ∞� R n and g : R n × R m × R ∞� R m are continuous functions. Assume that f, g are continuously differentiable with respect to their first two arguments in a neigborhood of the trajectory x 0 ( t ) , y 0 ( t ) , and that the derivative A ( t ) = g 2 ( x 0 ( t ) , y 0 ( t ) , t ) is a Hurwitz matrix for all t [ t 0 , t 1 ] .
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