By Vangipuram Lakshmikantham, Srinivasa Leela, Anatoly A. Martynyuk
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Extra info for Stability Analysis of Nonlinear Systems
4) is true and the proof is complete. To avoid repetition, we have not so far considered the lower estimate for m(t) which can be obtained by reversing the inequalities. For later reference, we shall state the following result which yields a lower bound for m(t). 1) existing on [t0 , ∞). Suppose that m ∈ C[R+ , R+ ] and Dm(t) ≥ g(t, m(t)), t ≥ t0 , where D is any ﬁxed Dini derivative. Then m(t0 ) ≥ u0 implies m(t) ≥ ρ(t), t ≥ t0 . 2. 3), we now have to consider solutions v(t, ε) of v = g(t, v) − ε , v(t0 ) = u0 − ε , for suﬃciently small ε > 0 on [t0 , T ] and note that lim v(t, ε) = ρ(t) ε→0 uniformly on [t0 , T ].
If h is nondecreasing, we have, from the definition of p(t), g(m(t)) ≤ g(h(t0 ) + p(t)). Setting h(t0 ) + p(t) = w(t), we obtain w′ (t) = p′ (t) = v(t)g(m(t)) ≤ v(t)g(m(t)) , w(t0 ) = h(t0 ). 3). 2, g(u) is assumed to be nonincreasing and superadditive in u. 4, one can prove the following result. 3 Let m, h ∈ C[R+ , R+ ], g ∈ C[(0, ∞), (0, ∞)] and 3 , R ], K (t, s) g(u) be nondecreasing in u. Suppose that K ∈ C[R+ + t exists, is continuous, nonnegative and for t ≥ t0 , t m(t) ≤ h(t) + K(t, s)g(m(s)) ds .
2, suppose that h (t), Kt (t, s, u) exist, are continuous, nonnegative and Kt (t, s, u) is nondecreasing in u for each (t, s). Suppose that r(t) is the maximal solution of the diﬀerential equation t u (t) = h (t) + K(t, t, u(t)) + K(t, s, u(t)) ds, u(t0 ) = h(t0 ) t0 existing on [t0 , ∞). Then, m(t) ≤ r(t), t ≥ t0 . t Proof Set v(t) = h(t) + K(t, s, m(s)) ds so that t0 t v (t) = h (t) + K(t, t, m(t)) + Kt (t, s, m(s)) ds. t0 In view of the assumption, we note that v(t) is nondecreasing and hence t v (t) ≤ h (t) + K(t, t, v(t)) + Kt (t, s, v(t)) ds, v(t0 ) = h(t0 ).
Stability Analysis of Nonlinear Systems by Vangipuram Lakshmikantham, Srinivasa Leela, Anatoly A. Martynyuk