Parametric Stabilization Is Easy*

A polynomially parametrized family of continuous-time controllable linear systems is always stabilizable by polynomially parametrized feedback. 1. RESULTS Theorem 1. Let (Aλ,Bλ) be a pair of matrices, all whose entries are real polynomial functions of the parameter λ∈Rr. Assume that Aλ is n×n , Bλ is n×m, and that the pair (Aλ,Bλ) is controllable for each λ∈Rr. Then, there exists an m×n matrix Kλ whose entries are also real polynomials in λ, such that, for every λ, each eigenvalue of the matrix A+BK has a negative real part. This result will be a consequence of the following much more general fact. Theorem 2. Let n,m be integers, and A=(aij), B=(bij) two matrices of distinct indeterminates, of sizes n×n and n×m respectively. There exist then: • an m×n matrix K(A,B,γ) of real polynomials in the aij, bij, and another variable γ, and • (scalar) polynomials p(A,B,γ) and s(A,B,γ) in the variables aij,bij, and γ, such that: (a) when the variables aij, bij take values making (A,B) controllable, p(A,B,γ) is nonzero, for every real γ, and (b) for any such values of the aij, bij, and for each γ, the matrix A+B(qK) has all eigenvalues with real part less than -γ whenever q is a (real) number such that pq>s. *