Analysis of Boundary Value Problem for a System of Nonhogeneous Ode's : On the Direct Lefschetz Stability Criterion for a System of Nonhomogeneous Linear Ode's with Variable Coefficients

In most studies of stability, asymptotic stability appears to be the most common approach. This is because asymptotic stability implies stability; however, the reverse is not true. In most cases it is easier to confirm asymptotic stability than stability. The method whereby stability is studied without asymptotic stability is referred to as a direct stability method. We have chosen the Lefschetz direct stability method; modified it to suit our problem at hand. The direct method requires the construction of a Lyapunov function; not easy for a non-dynamic problem. For a dynamic problem the energy thereof is a suitable candidate for a Lyapunov function. For a non-dynamic problem it is harder to construct a Lyapunov function as there are no rules for the purpose. In this presentation we modify the Lefschetz system for the direct stability method and apply it to study the stability of a system of linear first order ODEs with variable coefficients.