Abstract. In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ functions on the unit disc to problems concerning lifting onto H 2 of an operator T that is defined on K = H 2 ⊖φH 2 (φ is an inner function) and commutes with the (compressed) shift S. Inparticular,heshowed that interpolants (i.e., f ∈ H ∞ such that f(S)=T) having norm equal to �T � exist, and that in certain cases such an f is unique and can be expressed as a fraction f = b/a with a, b ∈ K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that �T � < 1, in which case they always exist). We parameterize the collection of all such pairs (a, b) ∈ K × K and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where φ is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint. 1.

Abstract—We introduce a Kullback–Leibler-type distance between spectral density functions of stationary stochastic processes and solve the problem of optimal approximation of a given spectral density 9 by one that is consistent with prescribed second-order statistics. In general, such statistics are expressed as the state covariance of a linear filter driven by a stochastic process whose spectral density is sought. In this context, we show i) that there is a unique spectral density 8 which minimizes this Kullback–Leibler distance, ii) that this optimal approximate is of the form 9 where the “correction term ” is a rational spectral density function, and iii) that the coefficients of can be obtained numerically by solving a suitable convex optimization problem. In the special case where 9=1, the convex functional becomes quadratic and the solution is then specified by linear equations. Index Terms—Approximation of power spectra, cross-entropy minimization, Kullback–Leibler distance, mutual information, optimization, spectral estimation. I.

Abstract—Over the last several years, a new theory of Nevanlinna–Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrix-valued case, also allowing for multiple interpolation points. We parameterize a class of interpolants consisting of “most interpolants ” of no higher degree than the central solution in terms of spectral zeros. This is a complete parameterization, and for each choice of interpolant we provide a convex optimization problem for determining it. This is derived in the context of duality theory of mathematical programming. To solve the convex optimization problem, we employ a homotopy continuation technique previously developed for the scalar case. These results can be applied to many classes of engineering problems, and, to illustrate this, we provide some examples. In particular, we apply our method to a benchmark problem in multivariate robust control. By constructing a controller satisfying all design specifications but having only half the McMillan degree of conventional controllers, we demonstrate the advantage of the proposed method. Index Terms—Complexity constraint, control, matrix-valued Nevanlinna–Pick interpolation, optimization, spectral

ABSTRACT In this paper we present a universal solution to the generalized moment problem, with a nonclassical complexity constraint. We show that this solution can be obtained by minimizing a strictly convex nonlinear functional. This optimization problem is derived in two different ways. We first derive this intrinsically, in a geometric way, by path integration of a one-form which defines the generalized moment problem. It is observed that this one-form is closed and defined on a convex set, and thus exact with, perhaps surprisingly, a strictly convex primitive function. We also derive this convex functional as the dual problem of a problem to maximize a cross entropy functional. In particular, these approaches give a constructive parameterization of all solutions to the Nevanlinna-Pick interpolation problem, with possible higher-order interpolation at certain points in the complex plane, with a degree constraint as well as all soutions to the rational covariance extension problem- two areas which have been advanced by the work of Hidenori Kimura. Illustrations of these results in system identifiaction and probablity are also mentioned. Key words. Moment problems, convex optimization, Nevanlinna-Pick interpolation, covariance extension, systems identification, Kullback-Leibler distance. 1

Abstract. Variational problems and the solvability of certain nonlinear equations have a long and rich history beginning with calculus and extending through the calculus of variations. In this paper, we are interested in “well-connected ” pairs of such problems which are not necessarily related by critical point considerations. We also study constrained problems of the kind which arise in mathematical programming as well as constraints of a geometric nature where a solution is sought on a leaf of a foliation. In these cases we are interested in interior minimizing points for the variational problem and in the well-posedness (in the sense of Hadamard) of solvability of the related systems of equations. We first prove a general result which implies the existence of interior points and which also leads to the development of certain generalization of the Hadamard-type global inverse function theorem, along the theme that uniqueness quite often implies existence. This result is illustrated by proving the non-existence of shock waves for certain initial data for the vector Burgers’ equation. The global inverse function theorem is also illustrated by a derivation of the existence of positive definite solutions of matrix Riccati equations without first

The minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint, and all solutions to this interpolation problem arise in this way by a suitable choice of weights. Selecting appropriate weights is pertinent to feedback control synthesis, where interpolants represent closed-loop transfer functions. In this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights.

Abstract — In this paper, we study the rational covariance extension problem when the chosen pseudopolynomial of degree at most n has zeros on the boundary of the unit circle. In particular, we derive a necessary and sufficient condition for a solution to be bounded (i.e. has no poles on the unit circle). Furthermore, we propose a new procedure for computing all bounded solutions for this special case of zeros of pseudopolynomials on the boundary and illustrate it by means of two examples. I.

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Abstract. Recently, a necessary and sufficient uniform log-integrability condition has been established for the canonical spectral factorization mapping to be sequentially continuous. However, this condition, along with several other equivalent conditions, is not straightforward to verify. In this paper, we first derive a new set of easily checkable sufficient conditions which guarantee uniform log-integrability. Based on the newly derived conditions, we establish the existence of certain convergent rational approximations for a class of matrix-valued spectral densities. We then propose a new spectral factorization algorithm and provide convergence results. Our approach does not require the spectral density to be coercive. Numerical examples are given to illustrate the effectiveness and convergence of the proposed algorithm. In particular, we compute approximate spectral factors of the noncoercive and nonrational Kolmogorov and von Karman power spectra which arise in the study of turbulence.

Dedicated to the memory of Mark Grigoryevich Krein on the occasion of the 100th anniversary of his birth Abstract. The moment problem matured from its various special forms in the late 19th and early 20th Centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception, where the applications have historically been to circuit theory, optimal control, robust control, signal processing, spectral estimation, stochastic realization theory and the use of the moments of a probability density. Many of these applications are also still works in progress. In this paper, we consider the generalized moment problem, expressed in terms of a basis of a finite-dimensional subspace P of the Banach space C[a, b] and a “positive ” sequences c, but with a new wrinkle inspired by the applications to systems and control. We seek to parameterize solutions which are positive “rational ” measures, in a suitably generalized sense. Our parameterization is given in terms of smooth objects. In particular, the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a Euclidean space and which is the domain of some canonically defined functions. The analysis of these functions, and related maps, yields interesting corollaries for the moment problems and its applications, which we compare to those in the recent literature and which play a crucial role in part of our proof. Our techniques are a combination of those drawn from the literature on the generalized moment problem, from the topology of smooth manifolds and maps, and from convex optimization. Key words. Moment problems, interpolation, rational positive measures