LMI approach to spectral stabilizability of linear delay systems and stabilizability of linear systems with complex parameter

The present paper is focused on the issue of pointwise [11], or (strong) delay-independent [16], stabilization of linear delay systems with commensurate delays. A criterion using LMI formulation is proposed, which is sufcient to have this property. Necessity is conjectured. 1 The problem under study and its background In a previous paper [1, 2] was provided a LMI condition necessary and su cient for (strong) delay-independent stability of linear delay systems with commensurate delays. The present paper proposes some progresses towards a LMI criterion for delay-independent stabilizability of these systems. Recall that the nite-dimensional system _ x = Ax+Bu, A 2 C n n ; B 2 C n p , is stabilizable if and only if [14, Theorem 4.5.6] 8s 2 C ;Re s 0 ) rank sIn A B = n ; (1) or equivalently [4, x7.2.1] if the LMI P = P > 0; PA +AP < BB is feasible. In this case, one veri es easily that any solution P generates a class of stabilizing proportional feedbacks, given by u = Kx;K def = B P ; 1=2 : (2) 14 The following system is examined here: _ x =A0x(t) + A1x(t h) + +Amx(t mh) +B0u(t) +B1u(t h) + +Bmu(t mh) ; (3) where m 2 N and, for some integers n; p, A0; A1; : : : ; Am 2 C n n , B0; B1; : : : ; Bm 2 C n p .