Stabilization of two-dimensional linear systems by time-delay feedback controls (Q1907741)

The author considers the action of a delayed feedback control (1) \(u(t)= Kx (t- \tau)\) on finite-dimensional, controllable, single input linear systems (2) \(\dot x= Ax+ Bu\). The stability analysis is reduced to the study of the roots of a transcendental polynomial. In the two-dimensional case, the author shows that (2) is stabilized by (1) if and only if \(\tau\) is not larger than some upper bound. Such a bound depends on the signs of the coefficients of the characteristic polynomial of \(A\) and they are explicitly computed.