Extremal Problems of Interpolation Theory

We consider problems where one seeks m×m matrix valued H∞ functions w(ξ) which satisfy interpolation constraints and a bound (0.1) w∗(ξ)w(ξ) ≤ ρmin, |ξ| < 1, where the m×m positive semi-definite matrix ρmin is minimal (no smaller than) any other matrix ρ producing such a bound. That is, if (0.2) w∗(ξ)w(ξ) ≤ ρ, |ξ| < 1, and if ρmin − ρ is positive semi-definite, then ρmin = ρ. This is an example of what we shall call a “minimal interpolation problem.” Such problems are studied extensively in the book [13, Chapter 7]. When the bounding matrices ρ are restricted to be scalar multiples of the identity, then the problem where we extremize over them is just the classical matrix valued interpolation problem containing those of Schur and NevalinnaPick (which in typical cases has highly nonunique solutions). Our minimal interpolation forces tighter conditions. In this paper we actually study a framework more general than that of Nevanlinna-Pick and Schur, and in this general context we show under some assumptions that our minimal interpolation problem, with ρmin defined formally by a minimal rank condition in Definition 3.3, has a unique solution ρmin and wmin(ξ). It is important both from applied and theoretical view points that the solution wmin(ξ) turns out to be a rational matrix function, indeed for the matrix NevanlinnaPick and Schur problems we obtain an explicit formulas generalizing those known classically. Also in this paper we compare minimal interpolation problems to superoptimal interpolation problem, cf. [14] and [11], and see that they have very different answers. Whether one chooses super-optimal criteria or our minimal criteria in a particular situation depends on which issues are important in that situation. The case m = 1 was investigated by many people with a formulation close to the one we use being found in Akhiezer Received by the editors on October 11, 2001, and in revised form on July 28, 2003. This work was partially supported by NSF, ONR, DARPA and the Ford Motor Company. Copyright c ©2005 Rocky Mountain Mathematics Consortium 819 820 J.W. HELTON AND L.A. SAKHNOVICH [1]. Interpolation with matrix valued analytic functions has found great application in control theory, cf. the books [2, 3, 6, 7, 15]. 1. Outline. The main consequences for analytic function theory of the general results of this paper are presented in Sections 4 and 5. Theorem 4.1 and the corollaries which follow it thoroughly describe minimal solutions to a class of matrix valued Nevannlinna-Pick interpolation problems. Also these corollaries connect the definition of minimal interpolation given in the abstract, see inequality (0.1), with the more general minimal rank Definition 3.3. Section 5 parallels Section 4 with Theorem 5.1 and its corollaries solving a class of matrix valued Schur problems as a consequence of the theory of a general interpolation problem. The general interpolation problem and some consequences of it appear in Section 3. It is a problem, about matrices with consequences for analytic function theory. This matrix, or more generally operator theoretic approach, comes from the book [13]. There are different matrix theoretic approaches to analytic function theory, which correspond to state space linear systems theory, cf. [2, 6, 15]; linear systems theory. Also there is the approach in [3]. While it would be interesting to know the connection between these ways of converting between linear algebra and analytic function theory, this has not been done. Possibly state space methods might be effective on our minimal interpolation problems, however, this has never to our knowledge been tried. In summary, this paper begins with some background on matrix inequalities, Section 2, moves to the general interpolation problem, Section 3, and then that goes to Nevanlinna-Pick interpolation and Schur interpolation applications, Sections 4 and 5. Finally in Section 6 we compare minimal interpolation to super-optimal interpolation. 2. Background on matrix equations. In the solution of extremal problem (0.1) an important role is played by the matrix nonlinear equation (2.1) X = R + C∗X−1C, R > 0 where matrices X,R,C are N ×N matrices. When studying equation EXTREMAL PROBLEMS OF INTERPOLATION THEORY 821 (2.1) we apply the method of successive approximations. We put (2.2) X0 = R, Xn+1 = R + C∗X−1 n C. It follows from (2.2) that (2.3) Xn ≥ X0, n ≥ 0. As the righthand side of (2.1) decreases with the growth of X, then in view of (2.2) and (2.3) the inequalities (2.4) Xn ≤ X1, n ≥ 1 are true. Similarly we obtain that (2.5) Xn ≥ X2, n ≥ 2. This leads to the following assertion (found in [4, 5]).