On the partial stochastic realization problem

: Given a mean-square continuous stochastic vector process y with stationary increments and a rational spectral density Phi such that Phi(infinity) is finite an nonsingular, consider the problem of finding all minimal Gauss-Markov representations (stochastic realizations) of y. All such realizations are characterized and classified with respect to deterministic as well as probabilistic properties. It is shown that only certain realizations (internal Stochastic realizations) can be determined from the given output process y. All others (external stochastic realizations) require that the probability space be extended with an exogeneous random component. A complete characterization of the sets of internal and external stochastic realizations is provided. It is shown that the state process of any internal stochastic realization can be expressed in terms of two steady-state Kalman-Bucy filters, one evolving forward in time over the infinite past and one backward over the infinite future. An algorithm is presented which generates families of external realizations defined on the same probability space and totally ordered with respect to state covariances. (Author)

In this paper, the nonlinear matrix equation , where is a positive integer, , M is an nonsingular matrix, A is a positive definite matrix and B is a positive semidefinite matrix, is considered. The notation is the t-weighted geometric mean of the positive definite matrices A and B. Based on the properties of the Thompson metric, we prove that the nonlinear matrix equation always has a unique positive definite solution and we compare it with the unique positive definite solution of the equation , which has been studied in Jung et al. [On the solution of the nonlinear matrix equation . Linear Algebra Appl. 2009;430:2042–2052]; Meng and Kim [The positive definite solution of the nonlinear matrix equation . J Comput Appl Math. 2017;322:139–147]. A fixed-point iteration method for obtaining the unique positive definite solution and an elegant estimate of the solution are given. Perturbation analysis of the unique positive definite solution is presented.

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

A directional partial realization problem

We consider a problem modeling a porous medium with a random perturbation. This model occurs in many applications such as biology, medical sciences, oil exploitation, and chemical engineering. Many authors focused their study mostly on the deterministic case. The more classical one was due to Biot in the 50s, where he suggested to ignore everything that happens at the microscopic level, to apply the principles of the continuum mechanics at the macroscopic level. Here we consider a stochastic problem, that is, a problem with a random perturbation. First we prove a result on the existence and uniqueness of the solution, by making use of the weak formulation. Furthermore, we use a numerical scheme based on finite differences to present numerical results.

Minimal state-space realization in linear system theory: an overview

The nonlinear matrix equationX-A*X-pA=Qwithp>0is investigated. We consider two cases of this equation: the casep≥1and the case0<p<1.In the casep≥1, a new sufficient condition for the existence of a unique positive definite solution for the matrix equation is obtained. A perturbation estimate for the positive definite solution is derived. Explicit expressions of the condition number for the positive definite solution are given. In the case0<p<1, a new sharper perturbation bound for the unique positive definite solution is derived. A new backward error of an approximate solution to the unique positive definite solution is obtained. The theoretical results are illustrated by numerical examples.

In this paper it is shown that a natural representation of a state space is given by the predictor space, the linear space spanned by the predictors when the system is driven by a Gaussian white noise input with unit covariance matrix. A minimal realization corresponds to a selection of a basis of this predictor space. Based on this interpretation, a unifying view of hitherto proposed algorithmically defined minimal realizations is developed. A natural minimal partial realization is also obtained with the aid of this interpretation.

This paper uses noncommutative dilation theory to develop a theory of stochastic bilinear systems. A certain stochastic bilinear realization and partial realization problem is solved. A new Caratheodory interpolation problem is presented and solved.

In 1969, J. Massey published a now-famous paper showing, among other things, that an iterative algorithm introduced by Berlekamp for decoding BCH codes also solved the problem of finding a shortest-length feedback shift register circuit for generating a given finite sequence of digits. This nice physical interpretation opened the door to connections with many other problems, including the minimal partial realization problems of linear system theory, Pade approximations and continued fractions, the fast algorithms of Levinson and Schur for Toeplitz matrices, inverse scattering, VLSI implementations, etc. This paper is an informal account of some of the different contexts in which the Berlekamp-Massey algorithm have been encountered in the work of the author and his students.

The problem of finding minimal realizations of linear constant systems from finite order input-output Markov matrix sequences is considered. The paper identifies from the sequences sets of independent structural and numerical quantities which are invariants of equivalent state space representations and completely characterize any minimal realization of the sequence. These sets, termed bases of invariants, acquire a “nesting” property by which a subsequent basis of a higher order finite sequence is obtained from the previous basis by addition of some new invariants. Two canonical state space representations of special forms that reflect the input and output structural properties of the underlying systems are presented and readily derived from these bases by a simple algorithm which is provided. Necessary and sufficient conditions for the existence of a unique minimal partial realization to a given finite Markov sequence are given in terms of the invariants of its nested basis. The set of all minimal partial realizations $S_{n_r }^r $ that, in the case of existence of more than one solution, corresponds to many distinct systems, is thoroughly investigated. A minimal set of undetermined quantities that parametrize $S_{n_r }^r $ is obtained. These parameters are used to characterize $S_{n_r }^r $ either in the form of bases of invariants or in the form of the canonical representations, and it is also shown that an arbitrary assignment of values to these parameters leads to a minimal realization of the given finite sequence. Additional properties of these parameters that may be desirable in certain identification problems are also discussed.

Based on a generalization of the Markov parameters, a generalized method for the construction of minimal realizations is presented. The method can give a family of minimal realizations, which may be useful, for example, in the problem of minimal partial realization. In contrast to the Markov parameters, special forms of the generalized Markov parameters may readily be obtained from experimental data. In particular, it is shown that the Laguerre expansion coefficients of a system provide a special form of the generalized Markov parameters of an associated system, and by using the proposed method, an exact minimal state variable model of the system can be restored from its Laguerre truncation.

On minimality in the partial realization problem

The partial realization problem for 2D discrete linear shift-invariant systems is discussed. The authors attempt to establish a reasonable definition of minimality and propose a method for finding the degree and the system function of a minimal realization of a 2D linear system characterized by a given 2D impulse response array. The method used is based on the 2D Berlekamp-Massey algorithm which has a close connection with the 2D Hankel matrix derived from the 2D array. While it compares in efficiency with other methods for identification and approximation of 2D linear systems, it gives a novel approach to the problem in the sense that one does not need any knowledge or assumptions concerning the system degree. >

This paper deals with a semilinear parabolic system with reaction terms $${v^p, u^q}$$ and a free boundary $${x = s(t)}$$ in one space dimension, where $${s(t)}$$ evolves according to the free boundary condition $${s'(t) = -\mu(u_x + \rho v_x)}$$ . The main aim of this paper was to study the existence, uniqueness, regularity and long-time behavior of positive solution (maximal positive solution). Firstly, we prove that this problem has a unique positive solution when $${p, q \geq 1}$$ , and a (unique) maximal positive solution when $${p < 1}$$ or $${q < 1}$$ . Then, we study the regularity of $${(u,v)}$$ and $${s}$$ . At last, we discuss the global existence, finite-time blowup of the unique positive solution (maximal positive solution) and long-time behavior of bounded global solution.