Interpolation problems for some non-classical domains

Design problems in control theory sometimes lead to variants of classical problems in complex analysis. One such variant is the spectral Caratheodory-Fejer problem: <br/>Given k x k complex matrices V_0,..., V_n, determine whether there exists an analytic k x k function F in the open unit disc D such that F^{(j)}(0) = V_j for j=0,..., n and all eigenvalues of F(lambda) lie in D for every lambda in D. <br/>I shall present various results on this problem, obtained jointly with J. Agler, H-N. Huang and S. A. M. Marcantognini.