7 research outputs found

    On the stabilization of bilinear systems via constant feedback

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    We study the problem of stabilization of a bilinear system via a constant feedback. The question reduces to an eigenvalue problem on the pencil A+α0B of two matrices. Using the idea of simultaneous triangularization of the matrices involved, some easily checkable conditions for the solvability of this question are obtained. Algorithms for checking these conditions are given and illustrated by a few examples

    Relations between the likehood ratios for two-dimensional continuous and discrete stochastic processes

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    The author considers the likelihood ratio for 2D processes. In order to detect this ratio, it is necessary to compute the determinant of the covariance operator of the signal-plus-noise observation process. In the continuous case, this is in general a difficult problem. For cyclic processes, using Fourier transforms it is possible to compute the determinant for continuous and discrete processes. For the 2D Poisson equation and its discretization, it is shown that the discretized determinant converges to the continuous one if the stepsize tends to zer

    Krein factorization of coverance operators of 2-parameter random fields and application to the likelihood ratio

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    A factorization of the covariance operator (I+R) is derived for the observation process of a 2-parameter random field. This result can be applied to express the determinant term appearing in L.A. Shepp's (1966) expression for the likelihood ratio in terms of the system parameters. This means that, in practice, one of the problems in computing the likelihood ratio for random fields is solved. Extensions to the multiparameter case are straightforward. The expression of the determinant of (I+R) in terms of the system parameters may also be used to reexpress the Wong-Zakai correction term (1977

    Approximations for the likelihood ratio for continuous multi-parameter stochastic processes

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    Based on finitely additive white noise theory, one may derive the likelihood ratio for random variables with values in any Hilbert space. This includes stochastic processes, defined on a one- or multi-dimensional continuous-parameter bounded domain. In certain circumstances, the likelihood ratio for continuous processes may be computed directly. In general however, one will have to approximate the likelihood ratio. In this paper approximations for the likelihood ratios for continuous-parameter processes are studied. Starting from a sequence of finite dimensional projection operators in the Hilbert space, strongly converging to identity, the authors show that the likelihood ratios for the projected processes converge to the likelihood ratio for the original process. Discretization of the stochastic process turns out to be one of the possibilities for such approximations. The discretization method is expected to give good results for signals satisfying elliptic PDEs, because discretization of these processes leads to nearest neighbor models, for which the likelihood ratio has been obtained in Luesink (1992)