Lyapunov functionals for Volterra Integro-differential equations

Abstract

The aim of this thesis is to study the qualitative behavior of a specific non-linear Volterra integro-differential equation with finite delays by using Lyapunov's second method. The non-linear Volterra integro-differential equation is: $x'(t)=b(t)x(t-r_1)-\int_{t-r_2}^{t}a(t,s)g(x(s))ds,$ where $r_1$, $r_2$ are positive constants representing 2 finite delays, $t \geq 0$ and

$a : \ [0,\infty) \times [-\tau, \infty) \rightarrow \R, \qquad \text{and} \qquad b : \ [0, \infty ) \rightarrow \R$ are two continuous functions.

In the first part, we study the qualitative behavior of the constant delay equation which is a specific case of the given integro-differential equation where $r_1 \neq 0$ and $r_2=0$. In the second part, we study the qualitative behavior of the integro-differential equation with one finite delay which is another specific case of the given integro-differential equation where $r_1=0$ and $r_2 \neq 0$. Three main steps are to be applied to each case separately. The first step is to construct a suitable, positive definite and non-decreasing, Lyapunov functional that yields the exponential stability of the zero solution of the given integro-differential equation. The second step is to derive inequalities and assumptions that guarantee the exponential stability of the zero solution of the given integro-differential equation. Finally, the third step is to derive inequalities and assumptions that guarantee the instability of the zero solution of the given integro-differential equation. Our theoretical results are extensions of many results found in the study of qualitative behavior of the zero solution of integro-differential equations with finite delay.