STABILITY OF SYSTEMS WITH VARIABLE AND DISTRIBUTED DELAYS

Abstract

Most modern studies on the stability of dynamic systems focus on investigating stability criteria using Lyapunov functions and functionals. A significantly smaller number of studies are devoted to estimates of the maximum Lyapunov exponent, which characterizes the rate at which solutions decrease. For example, in the works of D.Y. Khusainov, such estimates were obtained using the method of Lyapunov functions. Another approach, based on estimates of the system’s evolution operator, allows one to obtain stability criteria and an upper bound on the maximum Lyapunov exponent, expressed directly in terms of the system’s parameters, without using Lyapunov functions or functionals. For systems without delay, such results were presented in early works on this topic from the 1970s. In the works of Professor O.A. Zevin, this approach was applied to systems containing delay. Most works consider systems with constant delay; however, in most cases, information about the delay function is missing, and only its upper bound is known; furthermore, the system may contain distributed delay. The stability of systems with variable and distributed delay has been studied to a much lesser extent. This paper examines a class of systems that contain variable delay in the linear part and also include elements with distributed delay. This factor complicates the analysis of the behavior and stability of the systems under consideration. The main focus is on investigating the influence of delay parameters. In particular, two-sided estimates of the maximum Lyapunov exponent are obtained, which are expressed in terms of the norm of the system’s nonlinear term and the maximum values of the delay functions. This allows for the determination of both upper and lower bounds for the convergence or divergence of solutions. For certain classes of systems, exact values of this index have been determined, which is an important result for practical stability analysis. For all the systems mentioned, the effectiveness of the proposed approach has been verified using model examples; the results obtained significantly expand the existing body of knowledge.