The principle of unbounded stability enhancement and finite transition time

In this paper, we consider an autonomous linear control system with one-dimensional control. We introduce a new concept of stability enhancement on a finite time interval. On the time interval [0,Θ], the control is chosen as a linear function of the state x, with coefficients that depend on time. These coefficients are selected such that the degree of stability, for each value of t, of the resulting linear system increases unboundedly as t→Θ. It is shown that the unbounded increase in the degree of stability over a finite time interval [0,Θ] allows the system to reach the origin in finite time Θ. Moreover, for a sufficiently large value of Θ, the absolute value of the control remains sufficiently small.