《非线性动力学》（英文版） Lecture 10 Singular Perturbations and Averaging

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 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 i(t)=f(ar(t),y(t),t), g(x(t),y(t),t), 10.1 where e E[0, Eo] is a small positive parameter. When e>0,(10.1)is an ODE model For e=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- "classicall"approach to dealing with uncertainty, complexity, and nonlineari ons is the quently studied in physics and mechanics. One can say that singular perturbatie 10.1.1 The Tikhonov's heorem A typical question asked about the singularly perturbed system(10. 1)is whether its solutions with e >0 converge to the solutions of(10.1)with e =0 as E-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 o: to, t1]H+ R", yo: [to, til brm be continuous functions tisfying equations io(t)=f(aro(t), yo(t), t),0=g(ao(t), yo(t), t) Version of October 15. 2003

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 Averaging1 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 x0 : [t0, t1] ∞� Rn, y0 : [t0, t1] ∞� Rm be continuous functions satisfying equations x˙ 0(t) = f(x0(t), y0(t), t), 0 = g(x0(t), y0(t), t), 1Version of October 15, 2003

where f:R"×Rn×R→ R" and g:R"×R"×R→ R are continuous functions. Assume that f, g are continuously differentiable with respect to their first two arguments in a neigborhood of the trajectory co(t), yo(t), and that the derivative ()=92(xo(t),o(t) is a Hurwitz matrin for all t e to, t1. Then for every t2 E(to, ti) there erists d>0 and C>0 such that inequalities ao(t)-c(t)l s Ce for all e [ to, t, and lyo(t)-y(t)lsce for all tE [t2, ti] for all solutions of (10. 1)with lr(to)-ro(to)l<e, ly(to)-yo(tol<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 e>0,a= a(t) can be considered a constant when predicting the behavior of y. From this viewpoint, for a given t E(to, t1), one can expect that y(t +e where y1: 0, oo) is the solution of the "fast motion"ODE i1(r)=9(x0(),y(7),1(0)=y(D) Since yo(t) is an equilibrium of the ODE, and the standard linearization around this equilibrium yields 6(7)≈A(E)6() where 8(1)=y1(r)-yo(t), one can expect that y1(T)- yo() exponentially as T-+oo whenever A(t)is a Hurwitz matrix and ly(b-yo (b)l is small enough. Hence, when E>0 is small enough, one can expect that y(t) a yo(t) 10.1.2 Proof of Theorem 10.1 First, let us show that the interval [to, til can be subdivided into subintervals Ak 7k-1,n], where k∈{1,2 and to= To <TI<..<TN=ti in such a way that for every k there exists a symmetric matrix Pk= Pk>0 for which Pk A(t)+A(t)Pk<-I VtEITk-1, Tk Indeed, since A(t)is a Hurwitz matrix for every tE to, ti, there exists P(t)=P(t)>0 such that P(t)A(t)+A(tP(t) Since A depends continuously on t, there exists an open interval A(t) such that tE A( P(t)A(7)+A()P(t)<-Ir∈△(t)

2 where f : Rn × Rm × R ∞� Rn and g : Rn × Rm × R ∞� Rm are continuous functions. Assume that f, g are continuously differentiable with respect to their first two arguments in a neigborhood of the trajectory x0(t), y0(t), and that the derivative A(t) = g2 � (x0(t), y0(t), t) is a Hurwitz matrix for all t → [t0, t1]. Then for every t2 → (t0, t1) there exists d > 0 and C > 0 such that inequalities |x0(t) − x(t)| ≈ C� for all t → [t0, t1] and |y0(t) − y(t)| ≈ C� for all t → [t2, t1] for all solutions of (10.1) with |x(t0) − x0(t0)| ≈ �, |y(t0) − y0(t0)| ≈ 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 ¯ → (t0, t1), one can expect that y(t ¯+ �� ) � y1(� ), where y1 : [0,∀) is the solution of the “fast motion” ODE y˙1(� ) = g(x0(t ¯), y1(� )), y1(0) = y(t ¯). Since y0(t ¯) is an equilibrium of the ODE, and the standard linearization around this equilibrium yields � ˙(� ) � A(t ¯)�(� ) where �(� ) = y1(� ) − y0(t ¯), one can expect that y1(� ) � y0(t ¯) exponentially as � � ∀ whenever A(t ¯) is a Hurwitz matrix and |y(t ¯) − y0(t ¯)| is small enough. Hence, when � > 0 is small enough, one can expect that y(t) � y0(t). 10.1.2 Proof of Theorem 10.1 First, let us show that the interval [t0, t1] can be subdivided into subintervals �k = [�k−1, �k], where k → {1, 2, . . . , N} and t0 = �0 < �1 < · · · < �N = t1 in such a way that for every k there exists a symmetric matrix Pk = Pk � > 0 for which PkA(t) + A(t) � Pk < −I � t → [�k−1, �k]. Indeed, since A(t) is a Hurwitz matrix for every t → [t0, t1], 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)

Now the open intervals A(t)with t E [to, t, cover the whole closed bounded interval to, ti, and taking a finite number of tk, k=1, . m such that [to, ti is completely covered by A(tk) yields the desired partition subdivision of to, til exist cod, o t h th due to the coutinuous differentiablity of g, for every u> o therd Jf(xo0(t)+6-,3o(t)+y,1)-f(o(t),3(1,t)≤C(|62|+|) and lg(xo(1)+6x,y0(t)+6y,t)-A(t)by≤C|bx+列 for all t∈R,bz∈Rn,by∈ R satisfying t∈{to,t1],|1x-x0()≤r,1,-3(t)≤r Fort∈△klet lk=(2P5)/2 Then. for 6(t)=x(t)-x0(t),y(t)=y(t)-o(t), we have x|≤C1(1x|+|6y|k) e|6k≤-ql6yk+C1|6+C1 (10.2) as long as 8, du are sufficiently small, where C1, g are positive constants which do not depend on k. Combining these two derivative bounds yield 18=|+(EC1/q)15, D)<C218-+EC2 for some constant C2 independent of k. Hence 162(7k-1+r)|≤er(62(7k-1)+(eC1/q)(7k-1))+C3 for T E0, Tk-Tk-1. With the aid of this bound for the growth of 8=I, inequality(10. 2) yields a bound for 8,li 6(7k-1+7)≤exp(-qr/e)|b(xk-1)+C4(|62(k-1)+(eC1/q)b(k-1))+C46, which in turn yields the result of Theorem 10.1

3 Now the open intervals �(t) with t → [t0, t1] cover the whole closed bounded interval [t0, t1], and taking a finite number of t ¯k, k = 1, . . . , m such that [t0, t1] is completely covered by �(t ¯k) yields the desired partition subdivision of [t0, t1]. Second, note that, due to the continuous differentiability of f, g, for every µ > 0 there exist C, r > 0 such that ¯ ¯ ¯ ¯ |f(x0(t) + �x, y0(t) + �y, t) − f(x0(t), y0(t), t)| ≈ C(|�x| + |�y|) and ¯ ¯ ¯ ¯ ¯ |g(x0(t) + �x, y0(t) + �y, t) − A(t)�y| ≈ C|�x| + µ|�y| ¯ for all t → R, � ¯ x → Rn, �y → Rm satisfying ¯ ¯ t → [t0, t1], |�x − x0(t)| ≈ r, |�y − y0(t)| ≈ r. For t → �k let |�y|k = (�y � Pk�y) 1/2 . Then, for �x(t) = x(t) − x0(t), �y(t) = y(t) − y0(t), we have |� ˙ x| ≈ C1(|�x| + |�y|k), �|� ˙ y|k ≈ −q|�y|k + C1|�x| + �C1 (10.2) as long as �x, �y are sufficiently small, where C1, q are positive constants which do not depend on k. Combining these two derivative bounds yields d (|�x| + (�C1/q)|�y|) ≈ C2|�x| + �C2 dt for some constant C2 independent of k. Hence |�x(�k−1 + � )| ≈ eC3� (|�x(�k−1)| + (�C1/q)|�y(�k−1)|) + C3� 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)| + C4(|�x(�k−1)| + (�C1/q)|�y(�k−1)|) + C4�, which in turn yields the result of Theorem 10.1

10.2 Averaging Another case of "potentially discontinuous"dependence on parameters is covered by the following"averaging" result Theorem 10.2 Let f: R"XRxRHR be a continuous function which is T-periodic with respect to its second argument t, and continuously differentiable with respect to its first argument. Let toE r be such that f(Co, t, e)=0 for all t, E. ForiEr define ∫(z,∈) ∫(正,t,e) If df /d xl2=0 e-o is a Hurwitz matric, then, for sufficiently small e>0, the equilibrium x≡0 of the system i(t)=∈f(x,t,) is exponentially stable Though the parameter dependence in Theorem 10.2 is continuous, the question asked about the behavior at t= oo, which makes system behavior for e =0 not a valid indicator of what will occur for e>0 being sufficiently small.(Indeed, for e=0 the quilibrium io is not asymptotically stable. To prove Theorem 10.2, consider the function S: R"XRH R which maps z(0), to r(o)=s((0), e), where a( ) is a solution of (10.3). It is sufficient to show that the derivative (Jacobian)S(I, e) of S with respect to its first argument, evaluated at i=To and e>0 sufficiently small, is a Schur matrix. Note first that, according to the rules on differentiating with respect to initial conditions, S(0, e)=A(T, e), where d△(t,e)df (0,t,∈)△(t,e),△(0,∈)=I Consider D(t, e) defined by d△(t,e)df (0,t,0)(t,e),△(0,)=1 Let S(t)be the derivative of A(t, e) with respect to e at e=0. According to the rule for differentiating solutions of ODE with respect to parameters 6(t) ,1,O)d1 Hence d(r=df/ d

4 10.2 Averaging Another case of “potentially discontinuous” dependence on parameters is covered by the following “averaging” result. Theorem 10.2 Let f : Rn × R × R ∞� Rn be a continuous function which is � -periodic with respect to its second argument t, and continuously differentiable with respect to its first argument. Let x¯0 → Rn be such that f(¯x0,t,�) = 0 for all t,�. For x¯ → Rn define � � ¯f(¯x, �) = f(¯x,t,�). 0 ¯ If df/dx|x=0,�=0 is a Hurwitz matrix, then, for sufficiently 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 sufficiently small. (Indeed, for � = 0 the equilibrium x¯0 is not asymptotically stable.) To prove Theorem 10.2, consider the function S : Rn × R ∞� Rn which maps x(0),� to x(� ) = S(x(0),�), where x(·) is a solution of (10.3). It is sufficient to show that the derivative (Jacobian) S x, �) of S˙ with respect to its first argument, evaluated at ¯ x0 ˙(¯ x = ¯ and � > 0 sufficiently small, is a Schur matrix. Note first that, according to the rules on differentiating with respect to initial conditions, S˙(¯x0,�) = �(�, �), where d�(t,�) df = � (0, t,�)�(t,�), �(0, �) = I. dt dx Consider D¯(t,�) defined 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 differentiating solutions of ODE with respect to parameters, � t df �(t) = (0,t1, 0)dt1. 0 dx Hence ¯ �(� ) = df/dx|x=0,�=0

is by assumption a hurwitz matrix. On the other hand △(r,)-△(7,e)=o(e Combining this with △(r,e)=I+6(7)e+o(∈) yields △(r,e)=I+6(7)e+o() Since S()is a Hurwitz matrix, this implies that all eigenvalues of A(T, e) have absolute value strictly less than one for all sufficiently small E>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 sufficiently small � > 0