Class of controllable systems of differential equations for infinite time

In the article necessary conditions for a controllability of systems of nonlinear differential equations in an infinite time are obtained without assuming the existence of an asymptotic equilibrium for the system of linear approximation. Thus, a new class of controlled systems of differential equations is presented. The problem of controllability for an infinite time (i.e. the transfer of an arbitrary point into an arbitrary small domain of another point) comes down to choosing an operator depending on the selected control, which in turn depends on the point being transferred. Then one is to prove the existence of a fixed point for this operator. It is known that the theorems on controllability require existence of an asymptotic equilibrium for system of the first approximation. It is shown in the paper that in general case the condition of asymptotic equilibrium’s existence is not necessary for controllability of systems in an infinite time. An example on the theorem on controllability for an infinite time is given. The theorem generalizing Vazhevsky inequality is proved by implementation of Cauchy-Bunyakovsky inequality. A remark is made about the theorem’s validity for the case when the matrix and vector from the right-hand side of nonlinear differential equation are complex and $x$ is vector with complex components. Basing on the left-hand side of the inequality in the theorem generalizing Vazhevsky inequality, the necessary conditions for controllability in an infinite time are obtained. These conditions are verified on the same example of a scalar equation that was mentioned before.