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The LQ-optimal state feedback of a finite-dimensional linear time-invariant sys- tem determines a coprime factorization NM¡1 of the transfer function. We show that the same is true also for infinite-dimensional systems over arbitrary Hilbert spaces, in the sense that the factorization is weakly coprime, i.e., Nf;Mf 2 H2 =) f 2 H2 for every function f. The factorization need not be Bézout coprime. We prove that every proper quotient of two bounded holomorphic operator-valued func- tions can be presented as the quotient of two bounded holomorphic weakly coprime func- tions. This result was already known for matrix-valued functions with the classical definition gcd(N;M) = I, which we prove equivalent to our definition. We give necessary and sucient conditions and further results for weak coprimeness and for Bézout coprimeness. We then establish a variant of the inner-outer factorization with the inner factor being "weakly left-invertible". Most of our results hold also for continuous-time systems and many are new also in the scalar-valued case. AMS class: 49N10, 93D15, 93B52; 47A68, 47A56, 47B35.

Weakly coprime factorization and state-feedback stabilization of discrete-time systems