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Dynamical analysis of distributed parameter tubular reactors. (English) Zbl 0979.93077
The authors describe dynamics of tubular reactors corresponding to two models: the so-called plug model, and the model which considers axial dispersion. The kinetics of reactions depend only on concentrations of the reactants. The reactions of the form $$C_1\to bC_2$$ are governed by mass balance equations $\partial x_1/\partial t= D_a\partial^2 x_1/\partial z^2- v\partial x_1/\partial z- r(x_1, x_2), \partial x_2/\partial t= D_a\partial^2 x_2/\partial z^2- v\partial x_2/\partial z+ br(x_1, x_2),$ with appropriate boundary conditions. $$z\in [0,L]$$ is the axial variable. Everything is happening in the sum of $$L^2$$ spaces. All terms on the right-hand side of these equations are chemical concentrations, and so obviously must be terms like $$\partial x_i/\partial t$$. Assuming that $$r= k_0 x_1$$, the authors present this partial differential equation dynamic model of the reaction. They define the operators $A_2= \begin{pmatrix} D_a & 0\\ 0 & D_a\end{pmatrix},\quad A_1= \begin{pmatrix} -v & 0\\ 0 & -v\end{pmatrix},\quad A_0= \begin{pmatrix} -k_0 & 0\\ k_0 & 0\end{pmatrix}$ and the operator $$A= A_1+ A_2+ A_0$$. The operator $$A$$ is a generator of a continuous semigroup. Bounded control and observation operators are introduced. The physical domain is shown to be reachable, thus proving controllability in this sense, while any dominant set of modes is shown to be observable, under certain conditions. Some finite-dimensional approximations are discussed and the authors note that lumped parameter approximations produce results that are similar to the distributed parameter case when only the dominant mode is considered.

##### MSC:
 93C95 Application models in control theory 93C20 Control/observation systems governed by partial differential equations 80A30 Chemical kinetics in thermodynamics and heat transfer 93B07 Observability 93B05 Controllability
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