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Additional info for Differential Equations: Stability, Oscillations, Time Lags
The transformation C ( t ; to)is linear. Proposition. Proof. (t; t o to)% = , + a2X2). 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.
En; let A = (aii) be the matrix attached to the transformation in this basis. Consider the matrix A' obtained through transposition from A ; let u be an eigenvector of this matrix. Hence we have Zj"=laiiuj = hui . Consider the relation EL, xiui = 0.
Let us set ~ /S(E)= /3, OL = -(l/T,) In E . T h e n ,i3 > 0, 01 > 0, E = era=, and the previous estimate becomes I C ( t ;to)l < I C(t; to + + + + + < < + < I C ( t ;to)\ < pe-a(t-to). Th us we have proved once again that for linear systems, uniform asymptotic stability is always exponential. We remark that in this demonstration we have not used the hypothesis that matrix A(t ) is bounded. We shall now prove with the help of this fundamental property of linear systems that if the trivial solution of a linear system is uniform asymptotically stable, then there exists a Lyapunov function which is a quadratic form.
Differential Equations: Stability, Oscillations, Time Lags by Halanay