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Conditional exponential stability of a differential system with linear dichotomous Coppel-Conti approximation

Conditional exponential stability of a differential system with linear dichotomous Coppel-Conti... We prove the conditional exponential stability of the zero solution of the nonlinear differential system $$ \dot y = A(t)y + f(t,y),{\mathbf{ }}y \in R^n ,{\mathbf{ }}t \geqslant 0, $$ with L p -dichotomous linear Coppel-Conti approximation .x = A(t)x whose principal solution matrix X A (t), X A (0) = E, satisfies the condition $$ \mathop \smallint \limits_0^t \left\| {X_A (t)P_1 X_A^{ - 1} (\tau )} \right\|^p d\tau + \mathop \smallint \limits_t^{ + \infty } \left\| {X_A (t)P_2 X_A^{ - 1} (\tau )} \right\|^p d\tau \leqslant C_p (A) < + \infty ,{\mathbf{ }}p \geqslant 1,{\mathbf{ }}t \geqslant 0, $$ where P 1 and P 2 are complementary projections of rank k ∈ {1, …, n − 1} and rank n − k, respectively, and with a higher-order infinitesimal perturbation f:[0, ∞) × U → R n that is piecewise continuous in t ≥ 0 and continuous in y in some neighborhood U of the origin. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

Conditional exponential stability of a differential system with linear dichotomous Coppel-Conti approximation

Izobov, N.
Differential Equations , Volume 44 (5) – Jul 19, 2008
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References (3)

Publisher
Springer Journals
Copyright
Copyright © 2008 by MAIK Nauka
Subject
Mathematics; Difference and Functional Equations; Partial Differential Equations; Ordinary Differential Equations
ISSN
0012-2661
eISSN
1608-3083
DOI
10.1134/S0012266108050030
Publisher site
See Article on Publisher Site

Abstract

We prove the conditional exponential stability of the zero solution of the nonlinear differential system $$ \dot y = A(t)y + f(t,y),{\mathbf{ }}y \in R^n ,{\mathbf{ }}t \geqslant 0, $$ with L p -dichotomous linear Coppel-Conti approximation .x = A(t)x whose principal solution matrix X A (t), X A (0) = E, satisfies the condition $$ \mathop \smallint \limits_0^t \left\| {X_A (t)P_1 X_A^{ - 1} (\tau )} \right\|^p d\tau + \mathop \smallint \limits_t^{ + \infty } \left\| {X_A (t)P_2 X_A^{ - 1} (\tau )} \right\|^p d\tau \leqslant C_p (A) < + \infty ,{\mathbf{ }}p \geqslant 1,{\mathbf{ }}t \geqslant 0, $$ where P 1 and P 2 are complementary projections of rank k ∈ {1, …, n − 1} and rank n − k, respectively, and with a higher-order infinitesimal perturbation f:[0, ∞) × U → R n that is piecewise continuous in t ≥ 0 and continuous in y in some neighborhood U of the origin.

Journal

Differential Equations – Springer Journals

Published: Jul 19, 2008

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