Stability of Dynamical Systems: Linear Approach

Abstract

Our understanding of the stability of a particular operating mode of a dynamical system is formed intuitively as we build up our experience and understanding of everyday life and nature. The first steps of a small child give him or her very real representations of the stability of walking, although these representations may not yet enter consciousness. Looking at the famous painting entitled Young Acrobat on a Ball by P. Picasso, we have a distinct feeling that the girl's equilibrium is not quite stable. As adults, we can already discuss the stability of a ship on a stormy sea, the stability of economic trends in relation to the actions of managers and politicians, the stability of our nervous system with regard to stressful perturbation, etc. In each case, we talk about different properties that are specific to the considered systems. However, if we think about it carefully, we can find something in common, inherent in any system. The common feature is that, when we talk about stability, we understand the way the dynamical system reacts to a small perturbation of its state. If arbitrarily small changes in the system state begin to grow in time, the system is unstable. Otherwise, small perturbations decay with time and the system is stable. It is extremely important from a practical point of view to be able to analyse the stability of the operating modes of dynamical systems. Stability of such systems as a car, an aircraft, or an ocean liner to perturbations is certainly a vital factor in the truest sense of the word, since such perturbations are always going to be present in one form or another. These arguments are qualitative and can be made precise only if we manage to translate them into the formal language of mathematics. The fundamentals of the rigorous mathematical theory of stability were laid down in the works of the prominent Russian mathematician A.M. Lyapunov 100 years ago, while the V.S.