Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point x e {displaystyle x_{e}} stay near x e {displaystyle x_{e}} forever, then x e {displaystyle x_{e}} is Lyapunov stable. More strongly, if x e {displaystyle x_{e}} is Lyapunov stable and all solutions that start out near x e {displaystyle x_{e}} converge to x e {displaystyle x_{e}} , then x e {displaystyle x_{e}} is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but 'nearby' solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs. Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point x e {displaystyle x_{e}} stay near x e {displaystyle x_{e}} forever, then x e {displaystyle x_{e}} is Lyapunov stable. More strongly, if x e {displaystyle x_{e}} is Lyapunov stable and all solutions that start out near x e {displaystyle x_{e}} converge to x e {displaystyle x_{e}} , then x e {displaystyle x_{e}} is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but 'nearby' solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs. Lyapunov stability is named after Aleksandr Mikhailovich Lyapunov, a Russian mathematician who defended the thesis The General Problem of Stability of Motion at Kharkov University in 1892. A. M. Lyapunov was a pioneer in successful endeavoring to develop the global approach to the analysis of the stability of nonlinear dynamical systems by comparison with the widely spread local method of linearizing them about points of equilibrium. His work, initially published in Russian and then translated to French, received little attention for many years. The mathematical theory of stability of motion, founded by A. M. Lyapunov, considerably anticipated the time for its implementation in science and technology. Moreover Lyapunov did not himself make application in this field, his own interest being in the stability of rotating fluid masses with astronomical application. He did not have doctoral students who followed the research in the field of stability and his own destiny was terribly tragic because of the Russian revolution of 1917. For several decades the theory of stability sank into complete oblivion. The Russian-Soviet mathematician and mechanician Nikolay Gur'yevich Chetaev working at the Kazan Aviation Institute in the 1930s was the first who realized the incredible magnitude of the discovery made by A. M. Lyapunov. Actually, his figure as a great scientist is comparable to the one of A. M. Lyapunov. The contribution to the theory made by N. G. Chetaev was so significant that many mathematicians, physicists and engineers consider him Lyapunov’s direct successor and the next-in-line scientific descendant in the creation and development of the mathematical theory of stability. The interest in it suddenly skyrocketed during the Cold War period when the so-called 'Second Method of Lyapunov' (see below) was found to be applicable to the stability of aerospace guidance systems which typically contain strong nonlinearities not treatable by other methods. A large number of publications appeared then and since in the control and systems literature.More recently the concept of the Lyapunov exponent (related to Lyapunov's First Method of discussing stability) has received wide interest in connection with chaos theory. Lyapunov stability methods have also been applied to finding equilibrium solutions in traffic assignment problems.