On the Nevanlinna-Pick interpolation problem: Analysis of the McMillan degree of the solutions

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On Observability of Chaotic Systems: An Example

The concept of observability of a special chaotic system, namely the dyadic map, is studied here in case the observation is not exact. The usual concept of observable subspace does not distinguish among the behavior of different models. It turns out that a suitable extension of this concept can be obtained using the idea of Hausdorff dimension. It is shown that this dimension increases as the observation error becomes smaller, and is equal to one only if the system is observable

Derivative pricing for a multi-curve extension of the Gaussian, exponentially quadratic short rate model

The recent financial crisis has led to so-called multi-curve models for the term structure. Here we study a multi-curve extension of short rate models where, in addition to the short rate itself, we introduce short rate spreads. In particular, we consider a Gaussian factor model where the short rate and the spreads are second order polynomials of Gaussian factor processes. This leads to an exponentially quadratic model class that is less well known than the exponentially affine class. In the latter class the factors enter linearly and for positivity one considers square root factor processes. While the square root factors in the affine class have more involved distributions, in the quadratic class the factors remain Gaussian and this leads to various advantages, in particular for derivative pricing. After some preliminaries on martingale modeling in the multi-curve setup, we concentrate on pricing of linear and optional derivatives. For linear derivatives, we exhibit an adjustment factor that allows one to pass from pre-crisis single curve values to the corresponding post-crisis multi-curve values

Minimal symmetric Darlington synthesis

We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue

An explicit state-space approach to the one-block super-optimal distance problem

An explicit state-space approach is presented for solving the super-optimal Nehari-extension problem. The approach is based on the all-pass dilation technique developed in (Jaimoukha and Limebeer in SIAM J Control Optim 31(5):1115–1134, 1993) which offers considerable advantages compared to traditional methods relying on a diagonalisation procedure via a Schmidt pair of the Hankel operator associated with the problem. As a result, all derivations presented in this work rely only on simple linear-algebraic arguments. Further, when the simple structure of the one-block problem is taken into account, this approach leads to a detailed and complete state-space analysis which clearly illustrates the structure of the optimal solution and allows for the removal of all technical assumptions (minimality, multiplicity of largest Hankel singular value, positive-definiteness of the solutions of certain Riccati equations) made in previous work (Halikias et al. in SIAM J Control Optim 31(4):960–982, 1993; Limebeer et al. in Int J Control 50(6):2431–2466, 1989). The advantages of the approach are illustrated with a numerical example. Finally, the paper presents a short survey of super-optimization, the various techniques developed for its solution and some of its applications in the area of modern robust control