Fundamental theorem of linear state feedback for singular systems

The problem is solved of simultaneously assigning the finite and infinite eigenstructure of the controllable singular system Ex´(t)=Ax(t)+Bu(t) by the proportional state feedback law u(t)=Fx(t). Given m monic polynomials d₁(s), . . .,d_m(s) of degrees d₁,. . ., d_m satisfying d_i+1(s)|d_i(s), and η (η=nullity of E) nonnegative integers p₁, . . .,pη d₁+. . .+d _m+p₁+p_η=rank E, necessary and sufficient conditions are established for the existence of a real feedback map F so that the d_i(s)´s are the invariant polynomials and the p_i´s are the infinite pole orders of the closed-loop system Ex´(t)=(A+BF)x(t)