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

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.