By Halanay
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Extra resources for Differential Equations: Stability, Oscillations, Time Lags
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When V is defined for every x, 6, = limr+mS ( E ) = b - l [ u ( ~ ) ]T(r) , = b(S,)jc[S(~)]. Let us make now some remarks in connection with uniform asymptotic stability. 1. Let us prove that these functions can be chosen functions 8 ( ~ and continuous and monotone. 1. STABILITY THEORY 26 Proof. Let E, be a positive sequence, monotone, converging to zero, S1(e) = sup S(E,+~) for E < en; I xo I < implies 1 x(t; < < a,(€) t o , xo)i < E , + ~ E for t 3 to . Define S*(E) linear in E,] and 8*(%+1) = 81(%+2), a*(%) = %(%+l)*Since *1(%+1) d %(En) [%(En) having been defined as an upper bound of the set of numbers 6 with the property that I xo I < 6 implies I x ( t ; t o ,xo)l < E,], and in addition S1(e,) -+0 when n -+ CO, it follows that S1(~,+l) = &(en) only on finite parts of the sequence, which we can eliminate, with the result that S1(cn), and hence S*(E) be made strictly monotone-increasing.
Follows - I x(t; t o 7 x0)l2 < 1 V[t, P(1) - 7x(t; t o , xo)], and thus d 1 q t , x ( t ; t o , xo)) < - __ q t , s(t;t o 2P(T) 1 xo)). , 4. From what has just been proved, it follows that if f(t, x) is homogeneous in x of first degree, then uniform asymptotic stability implies exponential stability. Indeed, if f ( t , x) is homogeneous in x of first degree, it follows that x ( t ; t o ,kx,) = k x ( t ; t o , xo). We have solutions of the equation for both members of the equality, and these solutions coincide for t = t o .
STABILITY FOR LINEAR SYSTEMS 43 From dC(t; to)/& = A(t)C(t;to) it follows that C(t; to)(dy/dt) = f(t) and thus that dy-- dt [C(t;tO)]-'f(t)= C(to;t)f(t). From here From the relation which connects x(t) to y ( t ) it follows that %(to) = y(to), since C(to;to)= E . 4. Stability for Linear Systems After this introductory part concerning linear systems we may proceed to the study of stability problems for these systems. , there exist constants B and such that we shall have I x ( t ; t o , xo)l < Be-a(t-to)I xo 1.