BILATERAL ESTIMATES OF THE MAXIMUM LYAPUNOV INDICATOR

Abstract

Many systems encountered in problems of mechanics, control theory, and other fields of science and engineering are described by nonlinear differential equations with time delay. Most studies consider systems with constant delay; however, information about the delay function is often unavailable, with only its upper bound known; furthermore, the system may contain distributed delay. Known methods, in most cases, allow one to obtain only sufficient conditions for stability or upper bounds on the maximum Lyapunov exponent. A drawback of such results is that the degree of their conservativeness remains unknown. In this regard, the task of localizing the maximum Lyapunov exponent is relevant; that is, in addition to an upper bound, it is necessary to find its lower bound. The proximity of these bounds guarantees the accuracy of the obtained upper bound and, consequently, of the sufficient stability conditions. The article provides a detailed examination of a class of systems of nonlinear differential equations with a given linear part and a norm-bounded nonlinear term containing a variable delay. Particular attention is paid to the effect of the delay on the system’s dynamics and the estimation of its stability characteristics. Was obtained two-sided estimates of the maximum Lyapunov exponent, expressed in terms of the norm of the nonlinear term as well as in terms of the maximum values of the delay functions. This allows us to establish quantitative bounds on the behavior of the solutions and to estimate the rate of their convergence or divergence. For certain classes of systems, it was possible to determine exact values of the maximum Lyapunov exponent, which is an important result for stability theory. Based on the obtained estimates, sufficient, and in certain cases necessary, conditions for the exponential stability of the studied systems have been formulated. A characteristic feature of these conditions is their invariance with respect to delay, which significantly expands their scope of application. A simple and effective method for verifying exponential stability is also proposed, which does not require complex calculations and has a computational complexity that is practically independent of the system’s dimension (order). This makes the approach convenient for practical use, particularly for high-dimensional systems. Finally, a series of examples is provided that illustrate the application of the developed method, demonstrate its effectiveness, and confirm the theoretical results.