RESEARCH OUTPUTS / RÉSULTATS DE RECHERCHE
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Asymptotic behaviour and stability for solutions of a biochemical reactor distributed
parameter model
Dramé, Abdou; Dochain, Denis; Winkin, Joseph
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IEEE Transactions on Automatic Control
Publication date:
2008
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Citation for pulished version (HARVARD):
Dramé, A, Dochain, D & Winkin, J 2008, 'Asymptotic behaviour and stability for solutions of a biochemical
reactor distributed parameter model', IEEE Transactions on Automatic Control, vol. 53, no. 1, pp. 412-416.
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Asymptotic Behavior and Stability for Solutions of a Biochemical Reactor Distributed Parameter Model
A. K. Dram´e, D. Dochain, and J. J. Winkin
Abstract—This paper deals with the dynamical analysis of a tubular
biochemical reactor. The existence of nonnegative state trajectories and the invariance of the set of all physically feasible state values under the dynamical equation as well as the convergence of the state trajectories to equilibrium profiles are proved. In addition, the existence of multiple equi-librium profiles is analyzed. It is proved that, under physically meaningful conditions, the system has two stable and one unstable equilibrium profiles.
Index Terms—Asymptotic behavior, biochemical reactor, multiple
equilibrium profiles, stability, state trajectories.
I. INTRODUCTION
The dynamical analysis and control of tubular biochemical reactors have motivated many research activities over the last decades [3], [4], [8], [9]. The dynamics of these reactors are described by distributed parameter systems and typically by nonlinear partial differential equa-tions with, e.g., Danckwerts-type boundary condiequa-tions [3] and [4]. The aim of this paper is to analyze the dynamical nonlinear model of a fixed bed tubular biochemical reactor with axial dispersion. The non-linearity in the model arises from the substrate inhibition term in the model equations and is a specific rational function of the state com-ponents. The model is based on experimental research performed on anaerobic digestion in the pilot fixed-bed reactor of the Laboratoire de Biotechnologie de l’Environnement–Institut National de la Recherche Agronomique (LBE-INRA), Narbonne, France, and is mainly inspired from the dynamical models built and validated on the process [2], [14]. This study follows the preliminary one performed in [9].
The existence and uniqueness of the state trajectories (limiting sub-strate and limiting biomass) as well as their asymptotic behavior are analyzed for this model. First of all, it is proved that the trajectories exist on the whole (nonnegative real) time axis, and the set of all physically feasible state values is invariant under the dynamical equation. This set takes into account the positivity of the state variables as well as a saturation condition on the substrate. Second, the asymptotic behavior of the trajectories is investigated, and it is proved that the trajectories converge to equilibrium solutions of the system. In addition, the exis-tence of multiple equilibrium profiles is analyzed. The multiplicity is established together with the stability of equilibrium profiles by using phase plane analysis of ordinary differential equations.
The paper is organized as follows. In Section II, the dynamical model is presented and some preliminary analysis is performed. In Section III, the state trajectories of the tubular biochemical reactor dynamical model are analyzed. Section IV and Section V deal with the existence of multiple equilibrium profiles and their stability. Finally, some concluding remarks are given in Section VI. Due to the lack of space and details, several proofs have been omitted. The interested reader is refered to the internal report [7] for more detail.
Manuscript received March 1, 2006; revised March 3, 2007. Recommended by Associate Editor M. Demetriou.
A. K. Dram´e and D. Dochain are with the Center for Systems Engineering and Applied Mechanics (CESAME), Universit´e Catholique de Louvain, Louvain-La-Neuve B-1348, Belgium (e-mail: drame@inma.ucl.ac.be; dochain@csam. ucl.ac.be).
J. J. Winkin is with the Department of Mathematics, University of Namur (FUNDP), B-5000, Namur, Belgium (e-mail: joseph.winkin@fundp.ac.be).
Digital Object Identifier 10.1109/TAC.2007.914948 0018-9286/$25.00 © 2008 IEEE
II. DYNAMICALMODEL
Applying the mass balance principles to the limiting substrate con-centration S(τ, ζ) and the living biomass concon-centration X (τ, ζ) leads to the nonlinear system of partial differential equations
∂S ∂τ = D ∂2S ∂ζ2 − ν ∂S ∂ζ − kµ(S, X)X (2.1) ∂X ∂τ =−kdX + µ(S, X )X (2.2)
with the following boundary conditions: for all τ ≥ 0,
D∂S
∂ζ(τ, 0)− νS(τ, 0) + νSin= 0, ∂S
∂ζ(τ, L) = 0. (2.3)
The substrate inhibition is expressed via the following law:
µ(S, X ) = µ0
S
KSX + S + (1/Ki)S2
. (2.4) In the aforementioned equations, D, k, kd, KS, Ki, Sin, ν, and µ0
are positive parameters. In particular, D, ν, and kd denote the axial
dispersion coefficient, the superficial fluid velocity, and the kinetic constant, respectively. The parameters k and KS are dimensionless,
whereas Ki has the dimension of a concentration. In addition, the
specific growth rate µ(S, X ) has the dimension of the inverse of a time. Finally, ζ∈ [0, L] and τ ≥ 0 denote the spatial and time variables, respectively, and L denotes the length of the reactor. The considered reaction is autocatalytic (the biomass is both a product and a catalyst): this is modeled by the last term on the right-hand side (RHS) of (2.1) and (2.2). The selection of the aforementioned model is largely motivated by its analogy with the single microbial growth model with Haldane kinetics in perfectly mixed reactors, for which the existence of multiple (stable and unstable) equilibria have been emphasized [1].
From a physical point of view, it is expected that the saturation condition
0≤ S ≤ Sin (2.5)
holds, where Sin is the inlet limiting substrate concentration. One of
the contributions of this paper is to prove that the physiscal set defined by inequalities (2.5) is invariant under the dynamic equation of the biochemical reactor model introduced earlier. As usual, the dynamical analysis of this model will be performed on an equivalent dimensionless infinite dimensional system description. Let us introduce the new state components ˜x1 := (Sin− S)S−1in and ˜x2 := X Sin−1, and the new
spa-tial and time variables z := ζL−1 and t := ντ L−1, respectively; the
model (2.1), (2.2) can be rewritten as the partial differential equations (PDEs) ∂ ˜x1 ∂t = 1 Pe ∂2˜x 1 ∂z2 − ∂ ˜x1 ∂z + k ˜µ(˜x1, ˜x2)˜x2 (2.6) ∂ ˜x2 ∂t =−γ ˜x2+ ˜µ(˜x1, ˜x2)˜x2 (2.7)
with the following boundary conditions: for all t≥ 0, 1 Pe ∂ ˜x1 ∂z (t, 0)− ˜x1(t, 0) = 0 and ∂ ˜x1 ∂z (t, 1) = 0 (2.8)
where the modified substrate inhibition law given by ˜µ(˜x1, ˜x2) is
˜
µ = β (1− ˜x1)
KSx˜2+ (1− ˜x1) + α(1− ˜x1)2
(2.9)
and where the constants α, β, γ, and Pe (Peclet number) are given,
respectively, by
α := Si nKi−1, β := Lµ0ν−1,
γ := Lkdν−1, and Pe := νLD−1.
We also introduce the following change of variables in order to trans-form (2.6)–(2.8) into a reaction-diffusion equation with a bounded per-turbation. This is a one-to-one transformation; so, the two systems are equivalent. The new variables x1(z) := e−Pe/ 2 zx˜1(z) and x2(z) :=
e−Pe/ 2 zx˜
2(z) satisfy the equations
∂x1 ∂t = 1 Pe ∂2x 1 ∂z2 − Pe 4x1+ k ˜µ(x˜ 1, x2)x2 (2.10) ∂x2 ∂t =−γx2 + ˜µ(x˜ 1, x2)x2 (2.11)
with the following boundary conditions: for all t≥ 0,
∂x1 ∂z (t, 0) = Pe 2 x1(t, 0), ∂x1 ∂z (t, 1) =− Pe 2x1(t, 1) (2.12) where ˜µ(z, x˜ 1, x2) = ˜µ(e( Pe/ 2 ) zx1, e( Pe/ 2 ) zx2).
Now, the dimensionless model (2.10)–(2.12) can be given an infinite-dimensional state–space description as follows. Consider the Banach space C[0, 1] (equipped with the usual norm · C0) on which we
define the unbounded linear operator A1 by
D(A1) = # x1 ∈ C2[0, 1] : dx1 dz (0)− Pe 2x1(0) = 0 dx1 dz (1) + Pe 2 x1(1) = 0 $ A1x1 = 1 Pe d2x 1 dz2 − Pe 4 x1 ∀ x1 ∈ D(A1).
We also consider the bounded linear operator A2 =−γI, where I is the
identity operator on C[0, 1]. Therefore, (2.10)–(2.12) can be rewritten as follows: for all t≥ 0,
˙x(t) = Ax(t) + N (x(t))
where x := (x1, x2)T is a vector of the Banach space Z := C[0, 1]⊕
C[0, 1] endowed with the natural norm · Z = · C0 + · C0,
A := diag(A1, A2) is defined on its domain D(A) = D(A1)⊕
C[0, 1], and N is the nonlinear operator defined on Z by N (x) :=
(N1(x), N2(x)), where N1 = kN2 and N2(x1, x2) = ˜µ(x˜ 1, x2)x2.
By the arguments given in [5], the operator A1 is the infinitesimal
generator of an exponentially stable C0-semigroup of bounded linear
operators T1(t) on C[0, 1]. It is easy to see that A2is the infinitesimal
generator of an exponentially stable C0-semigroup of bounded linear
operators T2(t) = e−γ tI on C[0, 1]. Therefore, A is the infinitesimal
generator of an exponentially stable C0-semigroup of bounded linear
operators T (t) on Z given by T (t) = diag(T1(t), T2(t)). Moreover,
the semigroups T1(t) and T2(t) are analytic in C[0, 1] and T1(t) is
compact in C1[0, 1]. Therefore, T (t) is analytic in Z.
III. STATETRAJECTORIES
A. Global Existence
The well-posedness and invariance properties of the state trajecto-ries of the biochemical reactor dimensionless model (2.10)–(2.12) are
now studied by using the description of an abstract semilinear Cauchy problem of the form
˙x(t) = Ax(t) + N (x(t)), x(0) = x0 ∈ Ω (3.1)
where the operators A and N are defined in the previous section and Ω, motivated by (2.5), is the physically admissible set given by Ω ={(x1, x2)∈ Z : 0 ≤ x1(z)≤ e−(Pe/ 2 ) z,
0≤ x2(z) for all z∈ [0, 1]}. (3.2)
The main tool used in the analysis of this problem is [11, Th. 5.1, p. 355]. This theorem gives sufficient conditions for the existence and the uniqueness of the mild solution of equations of type (3.1) on the whole interval [0, +∞) and for the invariance of the set Ω under the dynamical equations. The following lemma holds.
Lemma 3.1: The semigroup T (t) is positive and for all t≥ 0, T (t)Ω⊂ Ω.
Proof: Let x0 = (x0 , 1, x0 , 2)∈ Ω and x(t) = T (t)x0 = (T1(t)
x0 , 1, T2(t)x0 , 2)T, for t≥ 0. Obviously, x2(t) = T2(t)x0 , 2 ≥ 0 for
all t≥ 0. We have to prove that 0 ≤ x1(t, z) = (T1(t)x0 , 1) (z)≤
e−(Pe/ 2 ) z, for all t≥ 0 and z ∈ [0, 1]. Let us introduce y
1(t) :=
−x1(t) for all t≥ 0. By using the analyticity of the semigroup T1(t)
in C[0, 1] and the comparison theorem [13, Th. 8, Ch. 3], we show that for all (z, t)∈ [0, 1] × (0, +∞),
x1(t, z)≤ x1(t)C0 ≤ x0 , 1C0 ≤ e−(Pe/ 2 ) z.
It follows that T (t)x0 ∈ Ω, that is, the semigroup T (t) is positive and
Ω is invariant under T (t).
Remark 3.1: It is easy to see that the function N : Ω−→ Z is Lipschitz continuous and its Lipschitz constant,lN, can be obtained by rather simple calculations.
Now, let Λ be a closed interval ofR and consider the set given as
K (Λ, C[0, 1]) :={φ ∈ C[0, 1] : φ(z) ∈ Λ ∀ z ∈ [0, 1]}.
The proof of the following lemma is similar to the one of [8, Proposition 3.1], and is therefore, omitted.
Lemma: 3.2: Assume thatΛ = [a, b], fc : Λ−→ R is a continuous function andfp : [0, 1]× Λ −→ R is a nonnegative bounded measur-able function. Iffc(a)≥ 0 and fc(b)≤ 0, then
lim
h→0+
1
hd(φ + hB(φ), K(Λ, C[0, 1])) = 0
where the substitution operator B is defined on K(Λ, C[0, 1]) by
[B(φ)](z) := fp(z, φ(z)).fc(φ(z)) for all z∈ [0, 1] and for all φ ∈ K (Λ, C[0, 1]).
Remark 3.2: By replacing[a, b] by [a,∞) in Lemma 3.2, the
con-clusion holds if only the conditionfc(a)≥ 0 is satisfied.
The following lemma can be deduced from Lemma 3.2 and Remark 3.2.
Lemma 3.3: For all (x1, x2)∈ Ω, limh→0+d((x1, x2) + hN ((x1,
x2)); Ω)h−1 = 0.
Finally, we can deduce from Remark 3.1 that the operator N− lN
is dissipative on Ω, where lN is the Lipschitz constant of N . In the
following theorem, the global existence of state trajectories is reported. It follows from the aforementioned lemmas and Remark 3.1, by using [11, Th. 5.1, p. 355].
Theorem 3.4: For everyx0 ∈ Ω, (3.1) has a unique mild solution
x(t, x0) on the interval [0, +∞[. Moreover, if one sets T(t)x :=
x(t, x0), then (T(t))t≥0 is a strongly continuous nonlinear semigroup
onΩ, generated by the operator A + N .
Hence, the state trajectories of the tubular biochemical reactor non-linear model given by (3.1) [or equivalently, (2.10)–(2.12)] are well defined on the whole time interval [0, +∞). Moreover, the physically feasible set Ω is invariant under this model dynamics, i.e., for all t≥ 0,
T(t)Ω⊂ Ω.
B. Asymptotic Behavior
The main result here is the convergence of the solutions of (3.1) to equilibrium profiles. We will successively give some regularity and relative compactness result of the solutions before dealing with their asymptotic behavior. Due to lack of space, the detailed proof of lem-mas will be omitted here. Also, for the sake of simplicity, we shall denote by x(t) the solution x(t, x0) of (3.1). The proof of the
follow-ing lemma is similar to the one of [5, Lemma 3.1], and is therefore, omitted.
Lemma 3.5: (i) For anyx0 ∈ Ω, the mild solution x(t) of (3.1) is a
classical solution, i.e.,x∈ C([0, +∞[; Z) ∩ C1(]0, +∞[; Z), x(t) ∈
D(A) for all t > 0 and x(t) satisfies (3.1) in the usual sense. (ii)For any t0 > 0, the subsets {Ax(t), t ≥ t0} and {(∂x(t))/∂t, t ≥ 0}
are bounded inZ. Moreover, the norms of C[0, 1] and C1[0, 1] are
equivalent on the subset{x1(t), t≥ t0}.
In view of Lemma 3.5, we shall understand by solution of (3.1) a global classical solution.
Let us denote by K (τ ), for any τ≥ 0, the subset of Z given by
K (τ ) ={x(t), t ≥ τ }.
The following lemma states a relative compactness of the solutions of (3.1). Note that N2is Lipschitz continuous in C[0, 1] with constant
lN2. Note that the invariance of the set Ω implies the boundedness of
solutions of (3.1) if lN2 < γ.
Lemma 3.6: Assume thatlN2 < γ and let x0 = (x0 , 1, x0 , 2)∈ Ω
andx(t) be the solution of (3.1), then K (0) is relatively compact in Z. Moreover, for anyt0 > 0, K (t0) is relatively compact in C1[0, 1]⊕
C[0, 1].
Lemma 3.7: Letx(t) = (x1(t), x2(t)) be a solution of (3.1). Then,
theω-limit set of x(t) with respect to the topology of Z coincides with itsω-limit set with respect to the topology of C1[0, 1]⊕ C[0, 1]; i.e.,
ω(x0/Z) = ω(x0/C1[0, 1]⊕ C[0, 1]).
In view of this lemma, the ω-limit set of a solution x(t) will simply be denoted by ω(x0). The following theorem is one of the main results
related to the asymptotic behavior of solutions of (3.1) established here.
Theorem 3.8: Assume thatlN2 < γ, and let x0 = (x0 , 1, x0 , 2)∈ Ω
andx(t) be the solution of (3.1) with x(0) = x0 andω(x0) its ω-limit
set. Then,ω(x0) is nonempty and is contained in D(A). Moreover,
ω(x0) consists of equilibrium profiles of (3.1).
The convergence of solutions of (3.1) to its equilibrium profiles is reported now. The proof is based on Theorem 3.8 and [10, Th. A].
Theorem 3.9: Asssume thatlN2 < γ, and let x0 = (x0 , 1, x0 , 2)∈ Ω
andx(t) be the solution of (3.1) with x(0) = x0. Then, there exists an
equilibrium profilex of (3.1) such that lim¯ t→∞x(t) = ¯x in C1[0, 1]⊕ C[0, 1].
IV. MULTIPLEEQUILIBRIUMPROFILES
In this section, the equilibrium profiles analysis of the biochem-ical reactor dimensionless model (2.6)–(2.9) is performed by using a perturbation theory. The equilibrium profiles (¯x1, ¯x2) are solutions
of 1 Pe ¯ x1 − ¯x1+ k ˜µ(¯x1, ¯x2)¯x2 = 0 −γ ¯x2+ ˜µ(¯x1, ¯x2)¯x2 = 0 1 Pe ¯ x1(0)− ¯x1(0) = ¯x1(1) = 0. (4.1)
Obviously, the system (4.1) has a trivial solution (¯x1, ¯x2) = (0, 0) that
corresponds to the reactor washout ( ¯S, ¯X ) = (Sin, 0) when
consid-ering the initial state variables. In the following, we shall be inter-ested in the solution of (4.1) that satisfies ˜µ(¯x1, ¯x2) = γ. Recall that
Pe= νLD−1. We give here a result of the existence of multiple
equi-librium profiles whose proof is based on pertubation theory and is more simple. The following statement holds.
Proposition 4.1: There existsD∗> 0 sufficiently large and ν∗> 0 such that, for allD≥ D∗, the system (4.1) has: 1) at least two nontrivial solutions if the parameterν satisfies 0≤ ν < ν∗and 2) at least one nontrivial solution forν = ν∗.
Proof: From the second equation of (4.1), we have ˜µ(¯x1, ¯x2) =
γ, which implies that ¯x2 = (1− ¯x1)(M + αkdx¯1)(kdKS)−1, where M = µ0− kd− αkd< 0. Let us consider the real-valued function g
defined by g(¯x1) = kL(1− ¯x1)(M + αkdx¯1)KS−1. Let us introduce u(z) := ¯x1(1− z) and v(z) := ¯x
1(1− z) for all 0 ≤ z ≤ 1. Then,
we have u=−v, v=−1 D(νv− g(u)) u(0) = a, v(0) = 0 and v(1) = ν Du(1). (4.2)
Note that (4.2) can be solved by finding a parameter ν = ν(a, D) (depending on a and D), whenever a and D are given, such that the solution (u, v) of the Cauchy problem in (4.2) satisfies the final condition v(1) = νD−1u(1). Therefore, if there are a1 = a2and D >
0 such that ν(a1, D) = ν(a2, D), then (4.2) has at least two solutions.
So, the existence of multiple equilibrium profiles is equivalent to the existence of a1, a2, . . . , and D > 0 such that ν(ai, D) = ν(aj, D) for
all i and j.
Now, we assume that D is large enough, and we introduce ε =
D−1, uε = u, and vε = vε−1, and we regard ν as a function of ε
[ν = ν(a, ε)] instead of a function of D. This leads to the following regular perturbation problem:
uε=−εvε, vε =−(νεvε− g(uε))
uε(0) = a, vε(0) = 0 and vε(1) = νuε(1). (4.3)
Considering the nonperturbed problem, u0 ≡ a, v0(1) = g(u0),
whence νa = g(a). Then, for ε = 0, we have ν(a, 0) = (g(a))/a. From direct computations, one can check that
∂ν(a, 0) ∂a =− kL KSa2 M + αkda2 and ∂2ν(a, 0) ∂a2 = 2kLM KSa3 < 0.
It follows that the function a−→ ν(a, 0) is concave and has the fol-lowing form (Fig. 1).
From the theorem of dependence of the solutions of ordinary differ-ential equations on initial conditions,
lim
ε→0ν(a, ε) = ν(a, 0) in C
2
[0, 1]. And the result follows.
Fig. 1. Graph of ν (., 0).
V. STABILITY OFEQUILIBRIUMPROFILES
The stability of equilibrium profiles of the biochemical reactor di-mensionless model (2.6)–(2.9) is analyzed here. This analysis is based on the linearized equations at the equilibrium. As in Lemma 3.6, one can see that the semigroup associated with the linearized equations is compact and by [15, Ths. 11.20 and 11.22], the stability of equilibria is determined by the eigenvalue problem. We first prove that the trivial equilibrium (0, 0) is locally asymptotically stable. Among the multiple equilibrium profiles, it is shown that the middle one (¯x1, ¯x2) is unstable
whereas the third equilibrium profile (x∗1, x∗2) is locally asymptotically stable.
Let us consider an equilibrium solution (¯x1, ¯x2) of (2.6)–(2.9).
Recall that ¯ x2 = (1− ¯x1)(M + αkdx¯1) kdKS ˜ µ(¯x1, ¯x2) = µ0 β (1− ¯x1) KSx¯2+ (1− ¯x1) + α(1− ¯x1)2 . (5.1) Let us introduce, for i = 1, 2, ϕi(z) = (β/µ0)(∂/∂ui)( ˜µ(u1,
u2)u2)|( u1, u2) = ( ¯x1( z ) , ¯x2( z ) ).
The stability of the equilibria is determined by the eigenvalue problem as 1 Pe x1− x1+ kµ0 β ϕ1(z)x1+ kµ0 β ϕ2(z)x2 =λx1 −γx2 + µ0 βϕ1(z)x1 + µ0 βϕ2(z)x2 =λx2 1 Pe x1(0) = x1(0) and x1(1) = 0. (5.2)
Then, from the second equation of (5.2), x2can be expressed as follows:
x2 = ((µ0/β)ϕ1)/λ+ γ− (µ0/β)ϕ2x1.
A. Stability of the Trivial Equilibrium(0, 0)
For the trivial equilibrium, the functions ϕ1and ϕ2 are constant and
given by ˜ ϕ1 = ϕ1(z) = M (αkd− M ) µ2 0KS < 0 ˜ ϕ2 = ϕ2(z) = k2 d(1 + α) µ2 0 > 0 for all z∈ [0, 1]. The parameters are assumed to satisfy the following condition.
Assumption A1: kd(1 + α) < µ20(
L ν∗)
2.
Hence, the following result holds.
Proposition 5.1: Assume that A1 holds. Then, the trivial equilibrium
(0, 0) is locally asymptotically stable.
Proof: Observe that, ifλ≤ −γ + (µ0/β) ˜ϕ2, then under
Assump-tion A1,λ< 0, and therefore, the equilibrium profile (0, 0) is locally
asymptotically stable. Now, assume that
λ>−γ +µ0
β ϕ˜2. (5.3)
From the expression of x2, we set x2 = ˜bx1, and also a = (kµ0/β)
˜
ϕ1 < 0 and b = (kµ0/β)˜b < 0. Let us introduce the following
vari-ables in (5.2):
u(z) := e−(Pe/ 2 ) zx
1(z), v(z) := u(z), ∀ z ∈ [0, 1].
Equation (5.2) is then equivalent to
u= v, v= Pe Pe 4 +λ− a − b u v(0) = Pe 2 u(0) and v(1) =− Pe 2u(1). (5.4) A necessary condition for (5.4) to have a solution is that (Pe/4) + λ− a − b < 0. It turns out thatλmust satisfyλ<−(Pe/4) + a + b < 0. It follows that the trivial equilibrium profile (0, 0) is locally
asymptotically stable.
B. Nontrivial Equilibria(x∗1, x∗2) and (¯x1, ¯x2)
Now, we shall consider that there exist two nontrivial equilibrium profiles (¯x1, ¯x2) and (x∗1, x∗2) as determined by Proposition 4.1. Note
that the equilibrium profiles satisfy 0≤ ¯x1(z)≤ x∗1(z)∀ z ∈ [0, 1].
Similarly to the previous case, we can prove that, under A1 and some appropriate conditions, we have the following.
Proposition 5.2: The nontrivial equilibrium profile(x∗1, x∗2) is locally
asymptotically stable.
Proposition 5.3: The nontrivial equilibrium profile (¯x1, ¯x2) is
unstable.
VI. CONCLUDINGREMARKS
This paper was devoted to the dynamical analysis of a tubular bio-chemical reactor nonlinear model. A system of nonlinear partial differ-ential equations was studied. The existence of the state trajectories and the invariance of the set of physically feasible state values under the dynamics of the reactor as well as the convergence of the state trajec-tories to equilibrium profiles of the dynamical model were proved. The multiple existence and stability of equilibrium profiles were analyzed under physically meaningful conditions. It was proved that, depending on the axial dispersion parameter, the dynamical model can have two stable and one unstable equilibrium profiles. And, as for completely mixed systems, the system studied here can have an unstable equilib-rium that can coincide with an interesting operating point from the practical viewpoint. So, as perspectives for further researches, we shall be interested in the control of this system that is nonlinear and infinite dimensional.
ACKNOWLEDGMENT
This paper presents research results of the Belgian Programme on Inter-University Poles of Attraction initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture. The scientific responsability rests with the authors.
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