Control Theory, Integral Matrices, and Orthogonal Polynomials

In control theory and approximation theory, there naturally arise matrices which are the inverses of the Gram matrices for the monomial basis in the space of square integrable functions with respect to a measure. For example, such a matrix arises in the problem of finite time feedback stabilization of a linear system, and in the Hilbert problem on the minimal $L_2$-norm of an integral polynomial. We show in a series of examples that the above inverse matrix is integral and has a large divisor. Our method is based on the arithmetic study of orthogonal polynomials naturally associated with the problem.