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)