Improper integrals for stability theory of multidimensional regulated systems

Abstract

A class of ordinary differential equations described the dynamics of nonlinear regulated systems
the right-hand part of which contains the nonlinear functions of the given set is considered. The
uncertainty of the right-hand side arises the non-uniqueness of the solution, that leads to the
necessity to study the group properties of solutions of the system. One such property is the absolute
stability of the trivial solution, i.e. properties at which all decisions coming from any starting point
for any non-linear functions of the given set tend over time to an equilibrium position. A completely
new method for the study of absolute stability of nonlinear regulated systems without involving
any Lyapunov functions and frequency theorems is proposed by evaluating improper integrals
along the solutions of the system. The motion equations of the system is led to a special form
by non-singular transformation, which allows to represent the integrand improper integrals as the
sum of two terms. The first term is a quadratic form reduced to the diagonal form, and the second
term is the total differential function on time. The representation of the integrand, ultimately,
leads to easily verifiable criteria for absolute stability.