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Additional info for Differential Equations: Stability, Oscillations, Time Lags
2. ASYMPTOTIC STABILITY Then the solution x = 0 is asymptotically stable. Proof. 1, the solution is stable. By hypothesis, V(t, x ( t ; t o , xo) is monotone-decreasing; hence the limit VO lim V[t,x ( t ; t o ,xO)] - t+m Integrating, we have Thus V [ t ,x ( t ; t o , xo)] converges to the fact that V t , 4 t ; to x0)l I -GO as t -+ co, which contradicts 2 41x(t; t o , xo)l). It follows that V , = 0; from V [ t ,x ( t ; t o , xo)] 0, it follows that a(l x ( t ; t o , xo)l) -+ 0, and thus that 1 x(t; t o , xo)l -+ 0 when t + GO.
2) is a linear combination of solutions of system (3) and hence is a solution of the system. T h e proposition is thus proved. We shall write x(t; t o ,xo) = C(t;to)xo. If a basis of the space is fixed, every linear transformation is given by a matrix whose columns are the images, under the given transformation, of the vectors of the basis. 3. LINEAR SYSTEMS 41 unity matrix. Denoting the unit matrix by E , and making no distinction between the transformation C(t;to) and the corresponding matrix, we shall write C(t,; to) = E.
Us are linearly independent. Indeed, let us suppose that clul c2u2 c,u, = 0. c,Tu, = 0. Since Tu, = X,u, , we see c,$,u, = 0. * c,u, = -cIu1 - c,u2 - * * . - C ~ - ~ U , - ~I t. *. , - XSClUl - hsC2U2 - + (4- h,)c,u, + + *** ( L l * - + - XSC,-,U,-~ = 0; - X s ) C s - - l ~ s - l = 0. ,s - 1). ,s - I. But then C , ~ U ,= 0; hence c,? = 0. T h e linear independence of the system of s vectors can therefore be established by induction (for s = 1 it follows from the fact that uI # 0). 1. STABILITY THEORY 48 It follows from this that if the transformation T admits n distinct eigenvalues, then it admits n linearly independent eigenvectors.