On the stabilization of a delayed system (Q664158)

Linear continuous-time control systems with constant coefficients and two constant and variable delays in the state variable are considered. It is generally assumed that the rate of growth of the varying delays tends to unity as time tends to infinity. First, conditions for asymptotic stability are given. Next, stabilization by state feedback is defined and solved for systems with two constant delays and only one time-varying delay. In the second part of the paper, the stabilization problem for systems with two constant and two variable delays is solved. In the proofs of the stabilizing conditions, it is assumed that the systems are completely controllable [\textit{J. Klamka}, Controllability of dynamical systems. Dordrecht etc.: Kluwer Academic Publishers; Warszawa: PWN- Polish Scientific Publishers (1991; Zbl 0732.93008)]. Sufficient conditions for stabilizability are derived using reduced, simpler difference equations, obtained from original continuous-time systems. Finally, it should be pointed out that similar stabilization problems have been considered in the paper [\textit{B. G. Grebenshchikov} and \textit{A. B. Lozhnikov}, Differ. Equ. 40, No. 12, 1667--1675 (2004); translation from Differ. Uravn. 40, No. 12, 1587--1595 (2004; Zbl 1088.34068)].