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1
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Polynomial systems theory applied to the analysis and design of multidimensional systems

100%
Hatonen J., Ylinen R.
International Journal of Applied Mathematics and Computer Science
|
2003
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tom 13
|
nr 1
15-27
EN
The use of a principal ideal domain structure for the analysis and design of multidimensional systems is discussed. As a first step it is shown that a lattice structure can be introduced for IO-relations generated by polynomial matrices in a signal space X (an Abelian group). It is assumed that the matrices take values in a polynomial ring F[p] where F is a field such that F[p] is a commutative subring of the ring of endomorphisms of X. After that it is analysed when a given F[p] acting on X can be extended to its field of fractions F(p). The conditions on the pair (F[p],X) are quite restrictive, i.e. each non-zero a(p)∊F[p] has to be an automorphism on X before the extension is possible. However, when this condition is met, say for operators {p1, p2, . . . , pn−1}, a polynomial ring F[p1, p2, . . . , pn] acting on X can be extended to F(p1, p2, . . . , pn−1)[pn], resulting in a principal ideal domain structure. Hence in this case all the rigorous principles of ‘ordinary’ polynomial systems theory for the analysis and design of systems is applicable. As an example, both an observer for estimating non-measurable outputs and a stabilizing controller for a distributed parameter system are designed.
2
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A modified filter SQP method as a tool for optimal control of nonlinear systems with spatio-temporal dynamics

100%
Rafajłowicz E., Styczeń K., Rafajłowicz W.
International Journal of Applied Mathematics and Computer Science
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2012
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tom 22
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nr 2
313-326
EN
Our aim is to adapt Fletcher's filter approach to solve optimal control problems for systems described by nonlinear Partial Differential Equations (PDEs) with state constraints. To this end, we propose a number of modifications of the filter approach, which are well suited for our purposes. Then, we discuss possible ways of cooperation between the filter method and a PDE solver, and one of them is selected and tested.
3
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Towards a framework for continuous and discrete multidimensional systems

100%
Rabenstein R., Trautmann L.
International Journal of Applied Mathematics and Computer Science
|
2003
|
tom 13
|
nr 1
73-85
EN
Continuous multidimensional systems described by partial differential equations can be represented by discrete systems in a number of ways. However, the relations between the various forms of continuous, semi-continuous, and discrete multidimensional systems do not fit into an established framework like in the case of one-dimensional systems. This paper contributes to the development of such a framework in the case of multidimensional systems. First, different forms of partial differential equations of physics-based systems are presented. Secondly, it is shown how the different forms of continuous multidimensional systems lead to certain discrete models in current use (finite-difference models, multidimensional wave digital filters, transfer function models). The links between these discrete models are established on the basis of the respective continuous descriptions. The presentation is based on three examples of physical systems (heat flow, transmission of electrical signals, acoustic wave propagation).
4
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$L^p$-$L^q$-Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity

88%
Gawinecki J.
Annales Polonici Mathematici
|
1991
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tom 54
|
nr 2
135-145
EN
We prove the $L^p$-$L^q$-time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman's approach [12], [5] to the $L^p$-$L^q$-time decay estimates.
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