Stability Analysis of Ships’ Movement Along Optimal Routes

Abstract

In this report, we consider the system of difference-differential equations of neutral type with homogeneous, of the larger unit order, right-hand sides. The following fact is well known. If a system of retarded-type difference-differential equations with homogeneous, of the larger unit order, righthand sides is asymptotically Lyapunov stable at zero delays, then the zero solution of the initial system is also asymptotically Lyapunov stable for any continuous and bounded delays. For this case, the Lyapunov-Krasovskii functional is constructed to estimate the asymptotic stability domain of the zero solution. For a linear system of neutral type, the concept of the Lyapunov matrix is introduced and the Lyapunov-Krasovskii functional is constructed. This functional was then used to analyze exponential stability. This paper presents sufficient conditions for asymptotic Lyapunov stability and Lyapunov instability of the zero solution for a class of homogeneous difference-differential systems of neutral type. In addition, a constructive algorithm for checking the stability and instability of the zero solution is formulated. Another result is the development of a method for constructing a complete type Lyapunov-Krasovskii functional, previously used for the analysis of homogeneous differencedifferential systems of retarded type.

Lyapunov–Krasovskii functionals for homogeneous time delay systems of neutral type

“…The basic assumptions we make are that (i) the matrix characterizing the difference part has all eigenvalues inside the unit circle, (ii) the corresponding system with all delays equal to zero is asymptotically stable. The work 28 presents some preliminary results in this direction for a single delay case. However, the constructions of the functionals in those work are substantially different (and, in particular, much more awkward) from that presented here.…”