Hamiltonian Feedback Design for Mass Action Law Chemical Reaction Networks

Lipták, György; Szederkényi, Gábor; Hangos, Katalin M.

doi:10.1016/j.ifacol.2015.10.232

Cited by 1 publication

(1 citation statement)

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“…They showed that there are two simple ways to modify the transition from chaotic behaviour, by varying the strength of the feedback and the delay response. Lipták et al [19] considered a general open CSTR system obeying the mass action law. They proposed a class of polynomial feedback that stabilizes the system, which can then be described by a generalized Hamiltonian form.…”

Section: Introductionmentioning

confidence: 99%

Non-smooth feedback control for Belousov–Zhabotinskii reaction–diffusion equations: semi-analytical solutions

2016

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The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reaction-diffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full nonsmooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail. Abstract The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reactiondiffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full non-smooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail.

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Section: Introductionmentioning

confidence: 99%