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Different models of chemotherapy taking into account drug resistance stemming from gene amplification

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Śmieja J., Świerniak A.
International Journal of Applied Mathematics and Computer Science
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2003
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tom 13
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nr 3
297-305
EN
This paper presents an analysis of some class of bilinear systems that can be applied to biomedical modelling. It combines models that have been studied separately so far, taking into account both the phenomenon of gene amplification and multidrug chemotherapy in their different aspects. The mathematical description is given by an infinite dimensional state equation with a system matrix whose form allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the other is infinite dimensional and tridiagonal. A methodology of the analysis of such models, based on system decomposition, is presented. An optimal control problem is defined in the l^1 space. In order to derive necessary conditions for optimal control, the model description is transformed into an integro-differential form. Finally, biomedical implications of the obtained results are discussed.
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Adaptive compensators for perturbed positive real infinite-dimensional systems

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Curtain R. F., Demetriou M. A., Ito K.
International Journal of Applied Mathematics and Computer Science
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2003
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tom 13
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nr 4
441-452
EN
The aim of this investigation is to construct an adaptive observer and an adaptive compensator for a class of infinite-dimensional plants having a known exogenous input and a structured perturbation with an unknown constant parameter, such as the case of static output feedback with an unknown gain. The adaptive observer uses the nominal dynamics of the unperturbed plant and an adaptation law based on the Lyapunov redesign method. We obtain conditions on the system to ensure uniform boundedness of the estimator dynamics and the parameter estimates, and the convergence of the estimator error. For the case of a known periodic exogenous input we design an adaptive compensator which forces the system to converge to a unique periodic solution. We illustrate our approach with a delay example and a diffusion example for which we obtain convincing numerical results.
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Modeling heat distribution with the use of a non-integer order, state space model

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Oprzędkiewicz K., Gawin E., Mitkowski W.
International Journal of Applied Mathematics and Computer Science
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2016
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tom 26
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nr 4
749-756
EN
A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function.
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