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

$$mathop smallint limits_0^t left| {X_A (t)P_1 X_A^{ - 1} (tau )} right|^p dtau + mathop smallint limits_t^{ + infty } left| {X_A (t)P_2 X_A^{ - 1} (tau )} right|^p dtau leqslant C_p (A) < + infty ,{mathbf{ }}p geqslant 1,{mathbf{ }}t geqslant 0,$$

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.