A number-theoretic part of control theory

Abstract

The importance of Q is primarily due to the fact that it defines a common Lyapunov function for the two stable matrices M = −diag(1, 2, . . . , N) and A + BC, where Aei = −iei+1 for i = 1, . . . , N , B = e1, and C = −(1/2)B∗Q. Here the ei, i = 1, . . . , N , form the standard basis of R and eN+1 = 0. In [6] it was shown that Q is an even integer matrix, that is, Qij ∈ 2Z, and it was conjectured that all elements of the matrix are divisible by N(N + 1). The proofs in [6] were based on considering orthogonal polynomials. Here we prove this conjecture using similar methods.