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