A Lyapunov Functional for a Retarded Differential Equation
J. C. F. De Oliveira, L. Α. V. Carvalho · SIAM Journal on Mathematical Analysis · 1985
Consider the autonomous linear retarded differential equation \[\dot x(t) = A_0 x(t) + A_1 x(t - r) + \int_{ - r}^0 {A_{01} (\theta )x(t + \theta )d\theta .} \] A positive definite quadratic functional is given that yields an equivalent norm in the phase space, which measures the exact asymptotic behavior of its solutions, providing the best estimate for the rate of growth or decay of the said solutions.