$\mathcal{L}$-stability of nonlinear systems represented by state models

Stability theory plays a key role in systems theory and engineering. The stability of equilibrium points is usually considered within the framework of the stability theory developed by the Russian mathematician and mechanic A.M. Lyapunov (1857–1918), who laid its foundations and gave it its name. Nowadays, the point of view on stability has become very widespread, as stability in relation to disturbance of the input signal. The research is based on the space-state approach for modelling nonlinear dynamic systems and an alternative “input-output” approach. The input-output model is implemented without explicit knowledge of the internal structure determined by the equation of state. The system is considered as a “black box” , which is accessed only through the input and output terminals ports. The concept of stability in terms of “input-output” is based on the definition of $\mathcal{L}$-stability of a nonlinear system, the method of Lyapunov functions and its generalization to the case of nonlinear dynamical systems. The interpretation of the problem on accumulation of perturbations is reduced to the problem on finding the norm of an operator, which makes it possible to expand the range of models under study, depending on the space in which the input and output signals act.