Construction of algebraic and difference equations with a prescribed solution space

Moysis, L. ; Karampetakis, N. P.

International Journal of Applied Mathematics and Computer Science, Tome 27 (2017) no. 1, pp. 19-32.

Résumé

This paper studies the solution space of systems of algebraic and difference equations, given as auto-regressive (AR) representations A(σ) β (k) = 0, where σ denotes the shift forward operator and A(σ) is a regular polynomial matrix. The solution space of such systems consists of forward and backward propagating solutions, over a finite time horizon. This solution space can be constructed from knowledge of the finite and infinite elementary divisor structure of A(σ) . This work deals with the inverse problem of constructing a family of polynomial matrices A(σ) such that the system A(σ) β (k) = 0 satisfies some given forward and backward behavior. Initially, the connection between the backward behavior of an AR representation and the forward behavior of its dual system is showcased. This result is used to construct a system satisfying a certain backward behavior. By combining this result with the method provided by Gohberg et al. (2009) for constructing a system with a forward behavior, an algorithm is proposed for computing a system satisfying the prescribed forward and backward behavior.

Détail

Keywords: algebraic equation, difference equation, behavior, exact modeling, autoregressive representation, discrete time system, higher order system
Mots-clés : równanie algebraiczne, równanie różnicowe, system czasu dyskretnego, system wyższego rzędu

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