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 difierential 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 su–- 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 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 difierentiable with respect to their flrst 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 ]. Then for every t 2 → (t 0 ,t 1 ) there exists d > 0 and C > 0 such that inequalities |x 0 (t) ? x(t)|? C? for all t → [t 0 ,t 1 ] and |y 0 (t) ? y(t)|? C? for all t → [t 2 ,t 1 ] for all solutions of (10.1) with |x(t 0 ) ? x 0 (t 0 )|? ?, |y(t 0 ) ? y 0 (t 0 )|? d, and ? → (0,d). The theorem was originally proven by A. Tikhonov in 1930-s. It expresses a simple principle, which suggests that, for small ? > 0, x = x(t) can be considered a constant when predicting the behavior of y. From this viewpoint, for a given t ? → (t 0 ,t 1 ), one can expect that y(t ? + ??) … y 1 (?), where y 1 : [0,?) is the solution of the “fast motion” ODE y˙ 1 (?) = g(x 0 (t ? ),y 1 (?)), y 1 (0) = y(t ? ). Since y 0 (t ? ) is an equilibrium of the ODE, and the standard linearization around this equilibrium yields – ˙ (?) … A(t ? )–(?) where –(?) = y 1 (?) ? y 0 (t ? ), one can expect that y 1 (?) ? y 0 (t ? ) exponentially as ? ? ? whenever A(t ? ) is a Hurwitz matrix and |y(t ? ) ? y 0 (t ? )| is small enough. Hence, when ? > 0 is small enough, one can expect that y(t) … y 0 (t). 10.1.2 Proof of Theorem 10.1 First, let us show that the interval [t 0 ,t 1 ] can be subdivided into subintervals ￠ k = [? k?1 ,? k ], where k → {1,2,...,N} and t 0 = ? 0 < ? 1 < ￠￠￠ < ? N = t 1 in such a way that for every k there exists a symmetric matrix P k = P k ? > 0 for which P k A(t) + A(t) ? P k < ?I ? t → [? k?1 ,? k ]. Indeed, since A(t) is a Hurwitz matrix for every t → [t 0 ,t 1 ], there exists P(t) = P(t) ? > 0 such that P(t)A(t) + A(t) ? P(t) < ?I. Since A depends continuously on t, there exists an open interval ￠(t) such that t → ￠(t) and P(t)A(?) + A(?) ? P(t) < ?I ? ? → ￠(t). 3 Now the open intervals ￠(t) with t → [t 0 ,t 1 ] cover the whole closed bounded interval [t 0 ,t 1 ], and taking a flnite number of t ? k , k = 1,...,m such that [t 0 ,t 1 ] is completely covered by ￠(t ? k ) yields the desired partition subdivision of [t 0 ,t 1 ]. Second, note that, due to the continuous difierentiability of f,g, for every ? > 0 there exist C,r > 0 such that ? ? ? ? |f(x 0 (t) + – x ,y 0 (t) + – y ,t) ?f(x 0 (t),y 0 (t),t)|? C(|– x |+ |– y |) and ? ? ? ? ? |g(x 0 (t) + – x ,y 0 (t) + – y ,t) ?A(t)– y |? C|– x |+ ?|– y | ? for all t → R, – ? x → R n , – y → R m satisfying ? ? t → [t 0 ,t 1 ], |– x ?x 0 (t)|? r, |– y ?y 0 (t)|? r. For t → ￠ k let |– y | k = (– y ? P k – y ) 1/2 . Then, for – x (t) = x(t) ?x 0 (t), – y (t) = y(t) ?y 0 (t), we have |– ˙ x |? C 1 (|– x |+ |– y | k ), ?|– ˙ y | k ??q|– y | k + C 1 |– x |+ ?C 1 (10.2) as long as – x ,– y are su–ciently small, where C 1 ,q are positive constants which do not depend on k. Combining these two derivative bounds yields d (|– x |+ (?C 1 /q)|– y |) ? C 2 |– x |+ ?C 2 dt for some constant C 2 independent of k. Hence |– x (? k?1 + ?)|? e C 3 ? (|– x (? k?1 )|+ (?C 1 /q)|– y (? k?1 )|) + C 3 ? for ? → [0,? k ?? k?1 ]. With the aid of this bound for the growth of |– x |, inequality (10.2) yields a bound for |– y | k : |– y (? k?1 + ?)|? exp(?q?/?)|– y (? k?1 )|+ C 4 (|– x (? k?1 )|+ (?C 1 /q)|– y (? k?1 )|) + C 4 ?, which in turn yields the result of Theorem 10.1. 4 10.2 Averaging Another case of “potentially discontinuous” dependence on parameters is covered by the following “averaging” result. Theorem 10.2 Let f : R n ￡ R ￡ R ∞? R n be a continuous function which is ?-periodic with respect to its second argument t, and continuously difierentiable with respect to its flrst argument. Let ?x 0 → R n be such that f(?x 0 , t, ?) = 0 for all t, ?. For ?x → R n deflne ? ? ? f(?x,?) = f(?x, t, ?). 0 ? If df/dx| x=0,?=0 is a Hurwitz matrix, then, for su–ciently small ? > 0, the equilibrium x · 0 of the system x˙(t) = ?f(x, t, ?) (10.3) is exponentially stable. Though the parameter dependence in Theorem 10.2 is continuous, the question asked is about the behavior at t = ?, which makes system behavior for ? = 0 not a valid indicator of what will occur for ? > 0 being su–ciently small. (Indeed, for ? = 0 the equilibrium ?x 0 is not asymptotically stable.) To prove Theorem 10.2, consider the function S : R n ￡ R ∞? R n which maps x(0), ? to x(?) = S(x(0), ?), where x(￠) is a solution of (10.3). It is su–cient to show that the x,?) of S ˙ with respect to its flrst argument, evaluated at ? x 0 derivative (Jacobian) S ˙ (? x = ? and ? > 0 su–ciently small, is a Schur matrix. Note flrst that, according to the rules on difierentiating with respect to initial conditions, S ˙ (?x 0 , ?) = ￠(?,?), where d￠(t, ?) df = ? (0,t, ?)￠(t, ?), ￠(0,?) = I. dt dx ? Consider D(t, ?) deflned by ? d￠(t, ?) df ? ? = ? (0,t,0)￠(t,?), ￠(0, ?) = I. dt dx ? Let –(t) be the derivative of ￠(t,?) with respect to ? at ? = 0. According to the rule for difierentiating solutions of ODE with respect to parameters, ? t df –(t) = (0, t 1 ,0)dt 1 . dx 0 Hence ? –(?) = df/dx| x=0,?=0 5 is by assumption a Hurwitz matrix. On the other hand, ? ￠(?,?) ? ￠(?,?) = o(?). Combining this with ? ￠(?,?) = I + –(?)? + o(?) yields ￠(?,?) = I + –(?)? + o(?). Since –(?) is a Hurwitz matrix, this implies that all eigenvalues of ￠(?,?) have absolute value strictly less than one for all su–ciently small ? > 0.