Download PDF by Vangipuram Lakshmikantham, Srinivasa Leela, Anatoly A.: Stability Analysis of Nonlinear Systems

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By Vangipuram Lakshmikantham, Srinivasa Leela, Anatoly A. Martynyuk

ISBN-10: 3319271997

ISBN-13: 9783319271996

The booklet investigates balance idea when it comes to varied degree, displaying the benefit of applying households of Lyapunov capabilities and treats the idea of a number of inequalities, sincerely bringing out the underlying topic. It additionally demonstrates manifestations of the overall Lyapunov approach, exhibiting how this system could be tailored to numerous it seems that varied nonlinear difficulties. in addition it discusses the applying of theoretical effects to a number of diversified types selected from actual international phenomena, furnishing information that's rather suitable for practitioners.
Stability research of Nonlinear structures is a useful single-sourse reference for commercial and utilized mathematicians, statisticians, engineers, researchers within the technologies, and graduate scholars learning differential equations.

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Extra info for Stability Analysis of Nonlinear Systems

Example text

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 ).

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Stability Analysis of Nonlinear Systems by Vangipuram Lakshmikantham, Srinivasa Leela, Anatoly A. Martynyuk


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