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On asymptotic stability of semi-linear distributed parameter dissipative systems
Automatica 46 (2010) 1042–1046
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
On asymptotic stability of semi-linear distributed parameter dissipative systemsI Ilyasse Aksikas ∗ , J. Fraser Forbes Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada, T6G 2G6
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Article history: Received 26 May 2009 Received in revised form 16 February 2010 Accepted 23 February 2010 Available online 31 March 2010 Keywords: Asymptotic stability Semi-linear infinite-dimensional systems Dissipativity Nonlinear contraction semigroup
abstract The concept of asymptotic stability is studied for a class of semi-linear distributed parameter dissipative systems with nonlinearity defined on a convex subset of the state space. This is achieved by using infinite-dimensional Banach state space description. Stability criteria are established, which are based on a weaker technical condition of the m-dissipativity. These theoretical results are applied to a class of transport–reaction processes. Different types of nonlinearities are studied by adapting the criteria given in the early portions of the paper. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Stability is one of the most important aspects of system theory. The fundamental theory of stability is extensively developed for finite-dimensional systems. Many results on the asymptotic behavior of nonlinear infinite-dimensional systems are known, for which the dissipativity property plays an important role, see e.g. Belleni-Morante and Mc Bride (1998), Brezis (1973), Crandall and Pazy (1969), Dafermos and Slemrod (1973), Laskshmikantham and Leela (1981), Luo, Guo, and Morgül (1999), Martin (1976), Miyadera (1992) and Pazy (1983). In Aksikas, Winkin, and Dochain (2007), asymptotic stability was studied for a class of semi-linear infinite-dimensional systems. With respect to the domain of definition of the nonlinearity in the system, two scenarios were treated. The first one deals with semilinear systems with a nonlinear term defined everywhere on the state space. In this case, some stability criteria were established on the basis of the m-dissipativity concept (see Aksikas et al., 2007, Theorem 12 and Corollary 13). The second case is when the nonlinear term is not necessarily defined everywhere, but only defined on a closed convex subset of the state space. This case is more important from application point of view due to the fact that some physical limitations are imposed. Aksikas et al. (2007,
I The material in this paper was partially presented at IFAC Workshop on Control of Distributed Parameter Systems, Toulouse, France, July 20–24, 2009. This paper was recommended for publication in revised form by Associate Editor Nicolas Petit under the direction of Editor Miroslav Krstic. ∗ Corresponding author. Tel.: +1 780 492 6238; fax: +1 780 492 2881. E-mail addresses: [email protected] (I. Aksikas), [email protected] (J.F. Forbes).
0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.02.030
Theorem 16) prove the asymptotic stability under some technical conditions, which will not be easy to check for practicing control engineers. The objective of this paper is twofold. The first goal is to study the asymptotic stability for different nonlinearities and at the same time replace the technical conditions with conditions that are more amenable to use by control engineers. The second goal is to apply this theory to first-order PDE systems in one spatial dimension. More specifically, this paper focuses on the asymptotic stability of a class of dissipative semi-linear distributed parameter systems, that admit at least one solution. Some additional conditions, viz compactness and density, are assumed in order to complete the set of stability criteria. The paper is organized as follows. Section 2 contains some basic results on nonlinear contraction semigroup theory. Notably, an asymptotic stability criterion in Banach space is reported, which is based on a technical and weaker condition than the m-dissipativity concept. In Section 3, asymptotic stability criteria are established for a class of semi-linear infinite-dimensional systems by applying the result stated in the previous section. Section 4 deals with the asymptotic stability of an infinite-dimensional description of a transport–reaction process model. 2. Basic results This section presents: (a) some properties of nonlinear contraction semigroup theory (Brezis, 1973; Dafermos & Slemrod, 1973; Luo et al., 1999); and (b) a paramount result related to LaSalle’s Invariance Principle (see Theorem 4). Let us consider a real reflexive Banach space X equipped with the norm k · k. Definition 1. Let A be a nonlinear operator with (possibly nonconvex) domain D(A).
I. Aksikas, J.F. Forbes / Automatica 46 (2010) 1042–1046
(i) A is said to be dissipative if, ∀ x, x0 ∈ D(A), there exists a bounded linear functional f on X such that f (x − x0 ) = kx − x0 k2 = kf k2 and f (Ax − Ax0 ) ≤ 0 (Note that, since X is a reflexive Banach space, such f is unique (Luo et al., 1999, p. 32)). (ii) A is said to be strictly dissipative if the conditions above hold with strict inequalities, for all x, x0 ∈ D(A) such that x 6= x0 . Let A be a dissipative operator such that the following condition holds, conv(D(A)) ⊂
\
R(I − λA)
(1)
λ>0
where conv(S ) is the closure (with respect to the strong topology) of the convex hull of S. By Luo et al. (1999, Prop. 2.109, p. 100; Cor. 2.120, pp. 106–107) A is the generator of a unique nonlinear contraction semigroup Γ (t ) on D := D(A). Moreover, for any x0 ∈ D(A), x(t , x0 ) := Γ (t )x0 is the unique solution (in the sense of Luo et al., 1999, Definition 2.111, p. 101) of the following nonlinear abstract Cauchy problem: x˙ (t ) = Ax(t ), x(0) = x0 .
t >0
(2)
Definition 2. Let us consider system (2) and assume that A generates a nonlinear contraction semigroup Γ (t ). Let us consider an equilibrium point x of (2), i.e. x ∈ D(A) and Ax = 0. x is said to be an asymptotically stable equilibrium point of (2) on D if
∀x0 ∈ D
lim x(t , x0 ) := lim Γ (t )x0 = x.
t →∞
t →∞
Remark 3. Note that the asymptotic stability definition captures only the attractivity part of this property since stability is guaranteed by the fact that this definition is given for a system that generates a nonlinear contraction semigroup. Indeed, ∀x0 , y0 ∈ D,
kΓ (t )x0 − Γ (t )y0 k ≤ kx0 − y0 k, and by using this inequality for y0 = x and the fact that x is a fixed point of the semigroup Γ (t ), we have
kx(t , x0 ) − xk ≤ kx0 − xk. In the rest of the paper we shall also need the important result stated in Theorem 4, whose proof can be found in Aksikas et al. (2007). This result is strongly related to the well-known LaSalle’s invariance principle, see Aksikas et al. (2007, Theorems 5 and 6). In order to state this result, the following concepts and notation are needed. If Γ (t ) is a nonlinear semigroup of contractions on D, for any x0 ∈ D, the orbit γ (x0 ) through x0 is defined by
γ (x0 ) := {Γ (t )x0 : t ≥ 0}, and the (possibly empty) ω-limit set, ω(x0 ) of x0 , is defined (with respect to the strong topology) by
n o ω(x0 ) := x ∈ D : ∃tn → ∞ such that x = lim Γ (tn )x0 . n→∞
Observe that ω(x0 ) is Γ (t )-invariant, i.e. for all t ≥ 0, Γ (t )ω(x0 ) ⊂ ω(x0 ). Theorem 4. Consider the system (2) with A as a nonlinear dissipative operator such that (1) holds, and let Γ (t ) be the nonlinear contraction semigroup on D = D(A), generated by A. Assume that x is the unique equilibrium point of (2) and (I − λA)−1 is compact for some λ > 0. Then for any x0 ∈ D, x(t , x0 ) := Γ (t )x0 converges, as t → ∞, to ω(x0 ) ⊂ {z : kz − xk = r }, for some positive r such that r ≤ kx0 − xk, i.e. lim d(x(t , x0 ), ω(x0 )) = 0.
t →∞
If in addition A is strictly dissipative, then x(t , x0 ) → x as t → ∞ i.e. x is an asymptotically stable equilibrium point of (2) on D.
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3. Asymptotic stability Let us consider a real reflexive Banach space X equipped with the norm k · k. Let us consider the following class of semi-linear systems:
x˙ (t ) = A0 x(t ) + N0 (x(t )) x(0) = x0 ∈ D(A0 ) ∩ F ,
(3)
where A0 is a linear operator defined on its domain D(A0 ) and N0 is a nonlinear operator defined on a closed convex subset F of X such that the following assumptions hold: (A1) A0 generates a C0 -semigroup of contractions on X . (A2) there exists a positive constant λ such that
(I − λA0 )−1 is compact in X .
(4)
(A3) N0 is a Lipschitz continuous dissipative nonlinear operator on F , with Lipschitz constant l0 . (A4) for all x0 ∈ D(A0 ) ∩ F , (3) has at least one solution, in the sense of (Luo et al., 1999, Definition 2.111, p. 101), denoted by x(t , x0 ). Remark 5. (a) The existence of the trajectories of this class of models has already been studied by several authors. For instance, Pazy (1983) contains an investigation of the general abstract model (3), where A0 is the generator of a strongly continuous semigroup S0 (t ) on an abstract Banach space X , and the nonlinear operator N0 is locally Lipschitz continuous. It is shown in Pazy (1983, pp. 185–186) that Eq. (3) has a unique local mild solution on some interval [0, T ], T ∈ [0, ∞) given by x(t ) = S0 (t )x0 +
t
Z
S0 (t − s)N0 (x(s))ds,
0 ≤ t ≤ T.
0
Moreover, if T < ∞ then, limt →T kx(t )k = ∞. Martin (1976, Theorem 5.1, p. 355) gives sufficient conditions for the existence and the uniqueness of the mild solution of system (3) on the whole interval [0, ∞). An equivalent version of this theorem is given by Laabissi, Achhab, Winkin, and Dochain (2001, Theorem 2.1). (b) Since A0 and N0 are dissipative, then A := A0 + N0 is dissipative. Therefore, if (A1), (A3) and (A4) are satisfied, then the solution x(t , x0 ) is unique. Moreover, the operator A generates a nonlinear contraction semigroup Γ (t ) on D := D(A0 ) ∩ F (see Luo et al., 1999, Theorem 2.112, p. 101). In order to study the asymptotic stability of system (3), the following lemmas are useful. Lemma 6. Consider a semi-linear system given by (3) and satisfying (A1)–(A3) . Then the operator (I − λA)−1 is compact in X , where λ > 0 is a constant such that (4) holds. Proof. Let λ > 0 be such that (4) holds. In order to prove the compactness of the operator (I − λA)−1 , let us consider any bounded sequence (vn ) in X and prove that the sequence (un ) := (I − λA)−1 vn , defined in D, has a converging subsequence. Observe that vn = un − λAun and consider a bounded linear functional fn such that fn (un ) = kun k2 = kfn k2 . Then fn (vn ) = kun k2 − λfn (Aun ). Using the fact that A is dissipative and assume without loss of the generality that N0 (0) = 0, we have
kun k2 ≤ kun k2 − λfn (Aun ) = fn (vn ) ≤ kun k kvn k. It follows that the sequence (un ) is bounded. Now consider the sequence (wn ) defined by
wn := (I − λA0 )un = (I − λA + λN0 )un = vn + λN0 (un ).
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I. Aksikas, J.F. Forbes / Automatica 46 (2010) 1042–1046
Since N0 is a Lipschitz operator and the sequence (un ) is bounded, then the sequence (N0 (un )) is bounded. Thus, using the boundedness of (vn ), one can conclude that (wn ) is bounded. Consequently, by using the compactness of the operator (I − λA0 )−1 in X , the sequence (un ) = ((I −λA0 )−1 wn ) has a converging subsequence. Lemma 7. Consider a semi-linear system given by (3) and satisfying conditions (A1), (A3) and (A4). Then the following condition holds: D⊂
\
R(I − λA).
(5)
λ>0
Proof. Let Ah : D → X be the operator defined by Ah x := h−1 [Γ (h)x − x],
h > 0.
The idea is to construct a contraction mapping and apply the contraction mapping theorem to prove that there is a unique solution yλ,h ∈ D to the equation (I − λAh )y = x given by yλ,h = (I − λAh )−1 x. Since Ah is dissipative, we know that (I − λAh ) exists and is nonexpansive on R(I − λAh ) for each λ and h positive. Let x ∈ D. The equation (I − λAh )y = x is equivalent to
λ Γ (h)y. h+λ h+λ Since x and Γ (h)y belong to D, then the mapping h
y=
y 7−→
x+
h h+λ
x+
(6)
λ Γ (h)y h+λ
maps D into itself, since D is convex. Moreover, the mapping is a contraction, since
h
λ h λ
x + Γ ( h ) y − x − Γ ( h ) y 1 2
h + λ h+λ h+λ h+λ λ ky1 − y2 k. ≤ h+λ By using contraction mapping theorem, there is a unique solution yλ,h ∈ D to (6) given by yλ,h = (I − λAh )−1 x.
In order to treat the case when the nonlinear operator N0 is not necessarily dissipative, we need these new assumptions: (A10 ) the linear operator A0 is the infinitesimal generator of an exponentially stable C0 -semigroup of bounded linear operators S0 (t ) on a Banach space X such that
kS0 (t )k ≤ Me−µt , for all t ≥ 0, for some µ > 0 and M ≥ 1. (A30 ) N0 is a Lipschitz continuous nonlinear operator defined on a closed convex subset F of X , with Lipschitz constant l0 . Now we are in position to state the following result: Theorem 10. Consider a semi-linear system given by (3) and 0 satisfying conditions (A10 ), (A2), (A3 ) and (A4). Let x be the unique equilibrium point of (3). Assume that µ ≥ Ml0 . Then for any x0 ∈ F , lim d(x(t , x0 ), ω(x0 )) = 0.
t →∞
If in addition, µ > Ml0 then x(t , x0 ) → x as t → ∞, i.e. x is an asymptotically stable equilibrium of (3) on F . Proof. Let us prove the dissipativity of the operator A. Consider the norm given by
|x| := sup {exp(µt )kS0 (t )xk : t ≥ 0} which is equivalent to the norm k · k (see e.g. Martin, 1976 p. 277 and Pazy, 1983 p. 12). It follows that the corresponding operator norm of the C0 -semigroup generated by A0 satisfies the condition
|S0 (t )| ≤ e−µt ,
∀t ≥ 0.
Moreover, N0 is a Lipschitz operator relative to the norm k · k with Lipschitz constant l0 . Hence, N0 is a Lipschitz operator with respect to |·|, with Lipschitz constant l00 ≤ l0 M. Note that in the case where M = 1, we do not need to change the norm. Now, the operator A can be written as follows.
A = A0 + l00 I + N0 − l00 I . Note that the fact that N0 is a Lipschitz operator implies that N0 − l00 I is a Lipschitz dissipative operator. On the other hand, since A0 is the generator of a C0 -semigroup S0 (t ) with |S0 (t )| ≤ e−µt , the operator A0 + l00 I is the generator of a C0 -semigroup Sl (t ) such that |Sl (t )| ≤
e(−µ+l0 )t (see e.g. Curtain & Zwart, 1995, Theorem 3.2.1, p. 110). It follows from the fact that µ ≥ Ml0 that Sl (t ) is a contraction semigroup, whence A0 + l00 I is dissipative and then A is dissipative. In order to show that the operator A is strictly dissipative when µ > Ml0 , it suffices to write it as A = A0 + lI + N0 − lI, where µ ≥ Ml > Ml0 , and to observe that the operator N0 − lI is strictly dissipative and A0 + lI is dissipative. The conclusion is a direct consequence of Theorem 9. 0
Consequently, D⊂
\ λ,h>0
R(I − λAh ) ⊂
\
R(I − λA).
λ>0
Remark 8. As a consequence of Lemma 7, if the conditions (A1), (A3) and (A4) hold, then the operator A generates a unique nonlinear contraction semigroup on D (see Luo et al., 1999, Corollary 2.120, p. 107).
Theorem 9. Consider a semi-linear system given by (3) and satisfying conditions (A1)–(A4). Assume that x is the unique equilibrium point of (3). Then for any x0 ∈ D,
Remark 11. Technically, the Lipschitz constant is the greatest lower bound of all constants satisfying the Lipschitz condition. In general, we do not need necessarily to determine the Lipschitz constant. Any constant that satisfies the Lipschitz condition will work. Finding the Lipschitz constant will enlarge the interval of satisfaction of the inequality condition Ml0 ≤ µ in Theorem 10.
lim d(x(t , x0 ), ω(x0 )) = 0.
4. Application
The following theorem follows directly from Lemmas 6 and 7, and Theorem 4.
t →∞
If in addition N0 is strictly dissipative, then x(t , x0 ) → x as t → ∞, i.e. x is an asymptotically stable equilibrium point of (3) on D. Proof. Observe that the compactness of (I − λA)−1 follows from Lemma 6. By Lemma 7, condition (1) holds. Then all the conditions of Theorem 4 are satisfied. Finally, the asymptotic behavior follows from Theorem 4 when N0 is (resp strictly) dissipative.
Many unit operations in chemical plants include transport processes that can best be described by partial differential equations (PDEs): see Dochain (1994), Christofides (2001) and Laabissi et al. (2001). When diffusive transport is negligible and convective transport is dominant, processes are commonly described by first-order hyperbolic PDEs (see Example 4.1).
I. Aksikas, J.F. Forbes / Automatica 46 (2010) 1042–1046
Our focus in this section is to apply the results of the previous section to the following semi-linear first-order PDEs system in one spatial dimension:
∂x ∂x = − + Mx(t ) + f (x(t )) ∂t ∂z
(7)
subject to the boundary and initial conditions given by: x(0, t ) = 0,
and
x(z , 0) = x0 (z )
(8)
where x(z , t ) = [x1 (z , t ) · · · xn (z , t )]T ∈ H := L2 (0, l)n denotes the vector of state variables, z ∈ [0, l] and t ∈ [0, ∞) denote position and time, respectively. f = [f1 · · · fn ] is a continuous vector function defined on a closed convex subset F ⊂ H. M is a diagonal matrix with nonpositive entries, i.e. M = diag(−αi ), i = 1, . . . , n, where αi ≥ 0 and x0 ∈ H. Without loss of the generality assume that l = 1. Actually, the equivalent state space description of the model (7)–(8) is given by (3), where A0 is the linear (unbounded) operator defined on its domain D(A0 ) :=
x ∈ H : x is a.c
dx dz
∈ H and x(0) = 0 ,
(9)
Note that if there exists i = 1, . . . , n such that αi = 0, then the constant α = 0. In this case, the inequality l0 < α cannot be satisfied because l0 is positive. The following theorem is stated to avoid the above situation. Theorem 15. Consider a semi-linear system given by (3), (9)–(11) and satisfying condition (H). Assume that f is Lipschitz continuous on F , with a Lipschitz constant l0 such that l0 < e−1 . Then for any x0 ∈ F , lim d(x(t , x0 ), x) = 0,
t →∞
i.e. the equilibrium point x is asymptotically stable on F . We shall prove Theorem 15 by using the following lemma (see Curtain & Zwart, 1995 Theorem 2.1.6 (c) and its proof for more details). Lemma 16. Let S0 (t ) be a C0 -semigroup and ω0 < 0 denotes its growth constant. For all ω ∈ (0, −ω0 ), there exists a constant Mω such that
∀t ≥ 0,
kS0 (t )k ≤ Mω e−ωt ,
where Mω is given by the following expression Mω = eωtω M0,ω
by d
+M ·I dx and the nonlinear operator N0 is defined on F by
(10)
N0 (x) := [f1 (x) · · · fn (x)]T .
(11)
A0 := −
1045
and the constants tω and M0,ω are given, respectively, by
tω := inf τ ≥ 0, kS0 (t )k ≤ e−ωt , ∀t ≥ τ
The following lemma focuses on the properties of the linear operator A0 (see e.g. Aksikas et al., 2007 and Winkin, Dochain, & Ligarius, 2000).
M0,ω = sup{kS0 (t )k, t ∈ [0, tω ]}. Proof of Theorem 15. Observe that S0 (t ) satisfies the following
kS0 (t )k =
1, 0,
0≤t <1 t ≥ 1.
Lemma 12. Let us consider the linear operator A0 given by (9)–(10). The following properties hold: (i) A0 is m-dissipative and generates an exponentially stable C0 semigroup S (t ). Moreover, for all t ≥ 0,
Then, (a) the growth constant is equal −∞, (b) the constant tω defined in Lemma 16 is equal 1 and (c) the constant M0,ω is equal 1. Therefore, the condition ω > Ml0 in Theorem 10 can be replaced by the weaker condition
kS (t )k ≤ e−αt ,
l0 < sup ωMω−1 = sup ωe−ω = e−1 .
with α = min αi . 1≤i≤n
(ii) There exists a positive constant λ such that the operator (I − λA0 )−1 is compact. In what follow, we need to assume the following condition: (H) For all x0 ∈ D(A0 ) ∩ F , system (7)–(8) has at least one solution denoted by x(t , x0 ). Now we are in a position to state the following corollaries about the asymptotic behavior of the system (3), (9)–(11). Two scenarios are presented. First we are interested in a dissipative nonlinearity (Corollary 13) and second we assume that the nonlinearity is not necessarily dissipative, but only Lipschitz continuous (Corollary 14). Corollary 13. Consider a semi-linear system given by (3), (9)–(11) and satisfying condition (H). Assume that f is Lipschitz continuous dissipative on F . Then for any x0 ∈ F , lim d(x(t , x0 ), ω(x0 )) = 0.
t →∞
If in addition f is strictly dissipative, then the equilibrium point x is asymptotically stable on F . Corollary 14. Consider a semi-linear system given by (3), (9)–(11) and satisfying condition (H). Assume that f is Lipschitz continuous on F , with a Lipschitz constant l0 such that l0 ≤ α . Then for any x0 ∈ F , lim d(x(t , x0 ), ω(x0 )) = 0.
t →∞
If in addition l0 < α , then the equilibrium point x is asymptotically stable on F .
ω>0
ω>0
Remark 17. The exponential stability of the linearized model of (7)–(8) is proved in Aksikas, Fuxman, Forbes, and Winkin (2009, Theorem 2) under some regularity conditions while our analysis imposes some dissipativity conditions in order to show the asymptotic stability of the nonlinear system. However, the linearization-based technique can be used only if the initial profile is close enough to the equilibrium profile, our approach guarantees the stability for any initial state on the domain of the system generator. Example 4.1. Plug flow reactor Let us consider a chemical plug flow reactor with the following two endothermic reactions: A −→ B and A −→ D. If the kinetics of the first reaction is first order with respect to the reactant concentration CA (mol/l) and has an Arrheniustype dependence on temperature T (K ), and if the kinetics of the second reaction is first-order kinetics with respect to the reactant concentration, the dynamics of the process are described by the following energy and mass balance PDEs:
α ∂T ∂T = −v + k1 1HCA exp − − β(T − Te ) ∂τ ∂z T α ∂ CA ∂ CA = −v − k1 CA exp − − k2 CA ∂τ ∂z T with the boundary conditions given for all τ ≥ 0, by T (0, τ ) = Tin and CA (0, τ ) = CA,in .
(12) (13)
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I. Aksikas, J.F. Forbes / Automatica 46 (2010) 1042–1046
In the equations above, 1H is the heat of reaction and Te represents the equilibrium profile of jacket temperature. Let us consider the following new variables x1 =
T − Tin
and x2 =
Tin
CA − CA,in CA,in
and consider the new time variable t = τ v . Then the plug flow reactor PDE model can be written under the form (7) such that the matrix M is defined by
1 −β M = v 0
0 −k2
and the function f is defined on its domain F = (x1 , x2 ) ∈ L2 (0, 1)2 : x1 ≥ −1 and − 1 ≤ x2 ≤ 0
by
−λ a(1 + x2 ) exp + β xe 1 1+ x1 f (x1 , x2 ) := −λ v −k (1 + x ) exp 1 2 1 + x1 where a = k1 1H
CA,in Tin
,
λ=
α
and xe =
Tin
Te − Tin Tin
.
The Lipschitz constant of f can be computed by using similar calculations to Aksikas et al. (2007), and it is given by
l0 = max 1, −
1HCA,in
k1
1+
v
Tin
4
λe2
.
References Aksikas, I., Fuxman, A., Forbes, J. F., & Winkin, J. J. (2009). Lq-control for a class of hyperbolic pde systems: application to a fixed-bed reactor. Automatica, 45, 1542–1548. Aksikas, I., Winkin, J. J., & Dochain, D. (2007). Asymptotic stability of infinitedimensional semi-linear systems: application to a nonisothermal reactor. Systems & Control Letters, 56, 122–132. Belleni-Morante, A., & Mc Bride, A. (1998). Applied nonlinear semigroups: an introduction. Willey. Brezis, H. (1973). Opéateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. In Mathematics studies. North-Holland. Christofides, P. D. (2001). Nonlinear and robust control of partial differential equation systems: methods and application to transport-reaction processes. Birkhauser. Crandall, M. G., & Pazy, A. (1969). Semi-groups of nonlinear contractions and dissipative sets. Journal of Functional Analysis, 3, 376–418. Curtain, R. F., & Zwart, H. J. (1995). An introduction to infinite-dimensional linear systems theory. New York: Springer Verlag. Dafermos, C. M., & Slemrod, M. (1973). Asymptotic behavior of nonlinear contraction semigroups. Journal of Functional Analysis, 13, 97–106. Dochain, D. (1994). Contribution to the analysis and control of distributed parameter systems with application to (bio)chemical processes and robotics. Thèse d’Agrégation de l’Enseignement Supérieur, Université Catholique de Louvain, Belgium. Laabissi, M., Achhab, M. E., Winkin, J., & Dochain, D. (2001). Trajectory analysis of nonisothermal tubular reactor nonlinear models. Systems & Control Letters, 42, 169–184. Lakshmikantham, V., & Leela, S. (1981). Nonlinear differential equations in abstract spaces. Oxford: Pergamon. Luo, Z., Guo, B., & Morgül, O. (1999). Stability and stabilization of infinite dimensional systems with applications. London: Springer Verlag. Martin, R. H. (1976). Nonlinear operators and differential equations in Banach spaces. New York: John Wiley & Sons. Miyadera, I. (1992). Nonlinear semigroups. American Mathematical Society. Pazy, A. (1983). Semigroups of linear operators and application to partial differential equations. In Applied mathematical sciences: Vol. 44. New York: Springer Verlag. Winkin, J., Dochain, D., & Ligarius, P. (2000). Dynamical analysis of distributed parameter tubular reactors. Automatica, 36, 349–361.
The stability condition is
l0 = max 1, −
1HCA,in Tin
k1 1 +
4
λe2
< min(β, k2 ).
Remark 18. Note that if the jacket temperature is used as a state feedback, then the function f will change and then the Lipschitz condition must be checked for the new function. 5. Conclusion In this paper, we have studied the asymptotic stability property of a class of dissipative semi-linear infinite-dimensional systems with nonlinearity not defined everywhere. Different scenarios have been considered. First the stability property was studied for semi-linear systems with dissipative nonlinearity. Secondly, Lipschitz nonlinearity was considered and the asymptotic stability was proved by adding an exponential stability condition on the underlying linear semigroup. Finally, the previous results were applied to a class of transport–reaction systems that can be described by first-order hyperbolic PDEs.
Ilyasse Aksikas was born on September 17, 1977, in Azemmour (El Jadida), Morocco. He received a Bachelor degree from the University of Chouaib Doukkali (Morocco) in 2000. He joined the Université catholique de Louvain (Belgium), where he obtained a Master degree (DEA) and a doctorate degree in Applied Mathematics in 2002 and 2005, respectively. He is currently a Research Associate at the Department of Chemical and Materials Engineering, University of Alberta, Canada. His main research interest is in the area of system and control theory, especially infinite-dimensional (distributed parameter) system theory, linear-quadratic optimal control, spectral factorization techniques and dynamical analysis and control of tubular reactors models. J. Fraser Forbes is a Professor and Chair of the Department of Chemical and Materials Engineering at the University of Alberta, Canada. He has over 25 years of experience in control/systems engineering as a researcher, educator and practitioner, and has worked in the steel, food products, forest products, and petrochemical industries. His current research interests include the use of the control of fixedbed reactor systems and optimization techniques for the design and synthesis of industrial automation systems.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
On asymptotic stability of semi-linear distributed parameter dissipative systemsI Ilyasse Aksikas ∗ , J. Fraser Forbes Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, Canada, T6G 2G6
article
info
Article history: Received 26 May 2009 Received in revised form 16 February 2010 Accepted 23 February 2010 Available online 31 March 2010 Keywords: Asymptotic stability Semi-linear infinite-dimensional systems Dissipativity Nonlinear contraction semigroup
abstract The concept of asymptotic stability is studied for a class of semi-linear distributed parameter dissipative systems with nonlinearity defined on a convex subset of the state space. This is achieved by using infinite-dimensional Banach state space description. Stability criteria are established, which are based on a weaker technical condition of the m-dissipativity. These theoretical results are applied to a class of transport–reaction processes. Different types of nonlinearities are studied by adapting the criteria given in the early portions of the paper. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Stability is one of the most important aspects of system theory. The fundamental theory of stability is extensively developed for finite-dimensional systems. Many results on the asymptotic behavior of nonlinear infinite-dimensional systems are known, for which the dissipativity property plays an important role, see e.g. Belleni-Morante and Mc Bride (1998), Brezis (1973), Crandall and Pazy (1969), Dafermos and Slemrod (1973), Laskshmikantham and Leela (1981), Luo, Guo, and Morgül (1999), Martin (1976), Miyadera (1992) and Pazy (1983). In Aksikas, Winkin, and Dochain (2007), asymptotic stability was studied for a class of semi-linear infinite-dimensional systems. With respect to the domain of definition of the nonlinearity in the system, two scenarios were treated. The first one deals with semilinear systems with a nonlinear term defined everywhere on the state space. In this case, some stability criteria were established on the basis of the m-dissipativity concept (see Aksikas et al., 2007, Theorem 12 and Corollary 13). The second case is when the nonlinear term is not necessarily defined everywhere, but only defined on a closed convex subset of the state space. This case is more important from application point of view due to the fact that some physical limitations are imposed. Aksikas et al. (2007,
I The material in this paper was partially presented at IFAC Workshop on Control of Distributed Parameter Systems, Toulouse, France, July 20–24, 2009. This paper was recommended for publication in revised form by Associate Editor Nicolas Petit under the direction of Editor Miroslav Krstic. ∗ Corresponding author. Tel.: +1 780 492 6238; fax: +1 780 492 2881. E-mail addresses: [email protected] (I. Aksikas), [email protected] (J.F. Forbes).
0005-1098/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2010.02.030
Theorem 16) prove the asymptotic stability under some technical conditions, which will not be easy to check for practicing control engineers. The objective of this paper is twofold. The first goal is to study the asymptotic stability for different nonlinearities and at the same time replace the technical conditions with conditions that are more amenable to use by control engineers. The second goal is to apply this theory to first-order PDE systems in one spatial dimension. More specifically, this paper focuses on the asymptotic stability of a class of dissipative semi-linear distributed parameter systems, that admit at least one solution. Some additional conditions, viz compactness and density, are assumed in order to complete the set of stability criteria. The paper is organized as follows. Section 2 contains some basic results on nonlinear contraction semigroup theory. Notably, an asymptotic stability criterion in Banach space is reported, which is based on a technical and weaker condition than the m-dissipativity concept. In Section 3, asymptotic stability criteria are established for a class of semi-linear infinite-dimensional systems by applying the result stated in the previous section. Section 4 deals with the asymptotic stability of an infinite-dimensional description of a transport–reaction process model. 2. Basic results This section presents: (a) some properties of nonlinear contraction semigroup theory (Brezis, 1973; Dafermos & Slemrod, 1973; Luo et al., 1999); and (b) a paramount result related to LaSalle’s Invariance Principle (see Theorem 4). Let us consider a real reflexive Banach space X equipped with the norm k · k. Definition 1. Let A be a nonlinear operator with (possibly nonconvex) domain D(A).
I. Aksikas, J.F. Forbes / Automatica 46 (2010) 1042–1046
(i) A is said to be dissipative if, ∀ x, x0 ∈ D(A), there exists a bounded linear functional f on X such that f (x − x0 ) = kx − x0 k2 = kf k2 and f (Ax − Ax0 ) ≤ 0 (Note that, since X is a reflexive Banach space, such f is unique (Luo et al., 1999, p. 32)). (ii) A is said to be strictly dissipative if the conditions above hold with strict inequalities, for all x, x0 ∈ D(A) such that x 6= x0 . Let A be a dissipative operator such that the following condition holds, conv(D(A)) ⊂
\
R(I − λA)
(1)
λ>0
where conv(S ) is the closure (with respect to the strong topology) of the convex hull of S. By Luo et al. (1999, Prop. 2.109, p. 100; Cor. 2.120, pp. 106–107) A is the generator of a unique nonlinear contraction semigroup Γ (t ) on D := D(A). Moreover, for any x0 ∈ D(A), x(t , x0 ) := Γ (t )x0 is the unique solution (in the sense of Luo et al., 1999, Definition 2.111, p. 101) of the following nonlinear abstract Cauchy problem: x˙ (t ) = Ax(t ), x(0) = x0 .
t >0
(2)
Definition 2. Let us consider system (2) and assume that A generates a nonlinear contraction semigroup Γ (t ). Let us consider an equilibrium point x of (2), i.e. x ∈ D(A) and Ax = 0. x is said to be an asymptotically stable equilibrium point of (2) on D if
∀x0 ∈ D
lim x(t , x0 ) := lim Γ (t )x0 = x.
t →∞
t →∞
Remark 3. Note that the asymptotic stability definition captures only the attractivity part of this property since stability is guaranteed by the fact that this definition is given for a system that generates a nonlinear contraction semigroup. Indeed, ∀x0 , y0 ∈ D,
kΓ (t )x0 − Γ (t )y0 k ≤ kx0 − y0 k, and by using this inequality for y0 = x and the fact that x is a fixed point of the semigroup Γ (t ), we have
kx(t , x0 ) − xk ≤ kx0 − xk. In the rest of the paper we shall also need the important result stated in Theorem 4, whose proof can be found in Aksikas et al. (2007). This result is strongly related to the well-known LaSalle’s invariance principle, see Aksikas et al. (2007, Theorems 5 and 6). In order to state this result, the following concepts and notation are needed. If Γ (t ) is a nonlinear semigroup of contractions on D, for any x0 ∈ D, the orbit γ (x0 ) through x0 is defined by
γ (x0 ) := {Γ (t )x0 : t ≥ 0}, and the (possibly empty) ω-limit set, ω(x0 ) of x0 , is defined (with respect to the strong topology) by
n o ω(x0 ) := x ∈ D : ∃tn → ∞ such that x = lim Γ (tn )x0 . n→∞
Observe that ω(x0 ) is Γ (t )-invariant, i.e. for all t ≥ 0, Γ (t )ω(x0 ) ⊂ ω(x0 ). Theorem 4. Consider the system (2) with A as a nonlinear dissipative operator such that (1) holds, and let Γ (t ) be the nonlinear contraction semigroup on D = D(A), generated by A. Assume that x is the unique equilibrium point of (2) and (I − λA)−1 is compact for some λ > 0. Then for any x0 ∈ D, x(t , x0 ) := Γ (t )x0 converges, as t → ∞, to ω(x0 ) ⊂ {z : kz − xk = r }, for some positive r such that r ≤ kx0 − xk, i.e. lim d(x(t , x0 ), ω(x0 )) = 0.
t →∞
If in addition A is strictly dissipative, then x(t , x0 ) → x as t → ∞ i.e. x is an asymptotically stable equilibrium point of (2) on D.
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3. Asymptotic stability Let us consider a real reflexive Banach space X equipped with the norm k · k. Let us consider the following class of semi-linear systems:
x˙ (t ) = A0 x(t ) + N0 (x(t )) x(0) = x0 ∈ D(A0 ) ∩ F ,
(3)
where A0 is a linear operator defined on its domain D(A0 ) and N0 is a nonlinear operator defined on a closed convex subset F of X such that the following assumptions hold: (A1) A0 generates a C0 -semigroup of contractions on X . (A2) there exists a positive constant λ such that
(I − λA0 )−1 is compact in X .
(4)
(A3) N0 is a Lipschitz continuous dissipative nonlinear operator on F , with Lipschitz constant l0 . (A4) for all x0 ∈ D(A0 ) ∩ F , (3) has at least one solution, in the sense of (Luo et al., 1999, Definition 2.111, p. 101), denoted by x(t , x0 ). Remark 5. (a) The existence of the trajectories of this class of models has already been studied by several authors. For instance, Pazy (1983) contains an investigation of the general abstract model (3), where A0 is the generator of a strongly continuous semigroup S0 (t ) on an abstract Banach space X , and the nonlinear operator N0 is locally Lipschitz continuous. It is shown in Pazy (1983, pp. 185–186) that Eq. (3) has a unique local mild solution on some interval [0, T ], T ∈ [0, ∞) given by x(t ) = S0 (t )x0 +
t
Z
S0 (t − s)N0 (x(s))ds,
0 ≤ t ≤ T.
0
Moreover, if T < ∞ then, limt →T kx(t )k = ∞. Martin (1976, Theorem 5.1, p. 355) gives sufficient conditions for the existence and the uniqueness of the mild solution of system (3) on the whole interval [0, ∞). An equivalent version of this theorem is given by Laabissi, Achhab, Winkin, and Dochain (2001, Theorem 2.1). (b) Since A0 and N0 are dissipative, then A := A0 + N0 is dissipative. Therefore, if (A1), (A3) and (A4) are satisfied, then the solution x(t , x0 ) is unique. Moreover, the operator A generates a nonlinear contraction semigroup Γ (t ) on D := D(A0 ) ∩ F (see Luo et al., 1999, Theorem 2.112, p. 101). In order to study the asymptotic stability of system (3), the following lemmas are useful. Lemma 6. Consider a semi-linear system given by (3) and satisfying (A1)–(A3) . Then the operator (I − λA)−1 is compact in X , where λ > 0 is a constant such that (4) holds. Proof. Let λ > 0 be such that (4) holds. In order to prove the compactness of the operator (I − λA)−1 , let us consider any bounded sequence (vn ) in X and prove that the sequence (un ) := (I − λA)−1 vn , defined in D, has a converging subsequence. Observe that vn = un − λAun and consider a bounded linear functional fn such that fn (un ) = kun k2 = kfn k2 . Then fn (vn ) = kun k2 − λfn (Aun ). Using the fact that A is dissipative and assume without loss of the generality that N0 (0) = 0, we have
kun k2 ≤ kun k2 − λfn (Aun ) = fn (vn ) ≤ kun k kvn k. It follows that the sequence (un ) is bounded. Now consider the sequence (wn ) defined by
wn := (I − λA0 )un = (I − λA + λN0 )un = vn + λN0 (un ).
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Since N0 is a Lipschitz operator and the sequence (un ) is bounded, then the sequence (N0 (un )) is bounded. Thus, using the boundedness of (vn ), one can conclude that (wn ) is bounded. Consequently, by using the compactness of the operator (I − λA0 )−1 in X , the sequence (un ) = ((I −λA0 )−1 wn ) has a converging subsequence. Lemma 7. Consider a semi-linear system given by (3) and satisfying conditions (A1), (A3) and (A4). Then the following condition holds: D⊂
\
R(I − λA).
(5)
λ>0
Proof. Let Ah : D → X be the operator defined by Ah x := h−1 [Γ (h)x − x],
h > 0.
The idea is to construct a contraction mapping and apply the contraction mapping theorem to prove that there is a unique solution yλ,h ∈ D to the equation (I − λAh )y = x given by yλ,h = (I − λAh )−1 x. Since Ah is dissipative, we know that (I − λAh ) exists and is nonexpansive on R(I − λAh ) for each λ and h positive. Let x ∈ D. The equation (I − λAh )y = x is equivalent to
λ Γ (h)y. h+λ h+λ Since x and Γ (h)y belong to D, then the mapping h
y=
y 7−→
x+
h h+λ
x+
(6)
λ Γ (h)y h+λ
maps D into itself, since D is convex. Moreover, the mapping is a contraction, since
h
λ h λ
x + Γ ( h ) y − x − Γ ( h ) y 1 2
h + λ h+λ h+λ h+λ λ ky1 − y2 k. ≤ h+λ By using contraction mapping theorem, there is a unique solution yλ,h ∈ D to (6) given by yλ,h = (I − λAh )−1 x.
In order to treat the case when the nonlinear operator N0 is not necessarily dissipative, we need these new assumptions: (A10 ) the linear operator A0 is the infinitesimal generator of an exponentially stable C0 -semigroup of bounded linear operators S0 (t ) on a Banach space X such that
kS0 (t )k ≤ Me−µt , for all t ≥ 0, for some µ > 0 and M ≥ 1. (A30 ) N0 is a Lipschitz continuous nonlinear operator defined on a closed convex subset F of X , with Lipschitz constant l0 . Now we are in position to state the following result: Theorem 10. Consider a semi-linear system given by (3) and 0 satisfying conditions (A10 ), (A2), (A3 ) and (A4). Let x be the unique equilibrium point of (3). Assume that µ ≥ Ml0 . Then for any x0 ∈ F , lim d(x(t , x0 ), ω(x0 )) = 0.
t →∞
If in addition, µ > Ml0 then x(t , x0 ) → x as t → ∞, i.e. x is an asymptotically stable equilibrium of (3) on F . Proof. Let us prove the dissipativity of the operator A. Consider the norm given by
|x| := sup {exp(µt )kS0 (t )xk : t ≥ 0} which is equivalent to the norm k · k (see e.g. Martin, 1976 p. 277 and Pazy, 1983 p. 12). It follows that the corresponding operator norm of the C0 -semigroup generated by A0 satisfies the condition
|S0 (t )| ≤ e−µt ,
∀t ≥ 0.
Moreover, N0 is a Lipschitz operator relative to the norm k · k with Lipschitz constant l0 . Hence, N0 is a Lipschitz operator with respect to |·|, with Lipschitz constant l00 ≤ l0 M. Note that in the case where M = 1, we do not need to change the norm. Now, the operator A can be written as follows.
A = A0 + l00 I + N0 − l00 I . Note that the fact that N0 is a Lipschitz operator implies that N0 − l00 I is a Lipschitz dissipative operator. On the other hand, since A0 is the generator of a C0 -semigroup S0 (t ) with |S0 (t )| ≤ e−µt , the operator A0 + l00 I is the generator of a C0 -semigroup Sl (t ) such that |Sl (t )| ≤
e(−µ+l0 )t (see e.g. Curtain & Zwart, 1995, Theorem 3.2.1, p. 110). It follows from the fact that µ ≥ Ml0 that Sl (t ) is a contraction semigroup, whence A0 + l00 I is dissipative and then A is dissipative. In order to show that the operator A is strictly dissipative when µ > Ml0 , it suffices to write it as A = A0 + lI + N0 − lI, where µ ≥ Ml > Ml0 , and to observe that the operator N0 − lI is strictly dissipative and A0 + lI is dissipative. The conclusion is a direct consequence of Theorem 9. 0
Consequently, D⊂
\ λ,h>0
R(I − λAh ) ⊂
\
R(I − λA).
λ>0
Remark 8. As a consequence of Lemma 7, if the conditions (A1), (A3) and (A4) hold, then the operator A generates a unique nonlinear contraction semigroup on D (see Luo et al., 1999, Corollary 2.120, p. 107).
Theorem 9. Consider a semi-linear system given by (3) and satisfying conditions (A1)–(A4). Assume that x is the unique equilibrium point of (3). Then for any x0 ∈ D,
Remark 11. Technically, the Lipschitz constant is the greatest lower bound of all constants satisfying the Lipschitz condition. In general, we do not need necessarily to determine the Lipschitz constant. Any constant that satisfies the Lipschitz condition will work. Finding the Lipschitz constant will enlarge the interval of satisfaction of the inequality condition Ml0 ≤ µ in Theorem 10.
lim d(x(t , x0 ), ω(x0 )) = 0.
4. Application
The following theorem follows directly from Lemmas 6 and 7, and Theorem 4.
t →∞
If in addition N0 is strictly dissipative, then x(t , x0 ) → x as t → ∞, i.e. x is an asymptotically stable equilibrium point of (3) on D. Proof. Observe that the compactness of (I − λA)−1 follows from Lemma 6. By Lemma 7, condition (1) holds. Then all the conditions of Theorem 4 are satisfied. Finally, the asymptotic behavior follows from Theorem 4 when N0 is (resp strictly) dissipative.
Many unit operations in chemical plants include transport processes that can best be described by partial differential equations (PDEs): see Dochain (1994), Christofides (2001) and Laabissi et al. (2001). When diffusive transport is negligible and convective transport is dominant, processes are commonly described by first-order hyperbolic PDEs (see Example 4.1).
I. Aksikas, J.F. Forbes / Automatica 46 (2010) 1042–1046
Our focus in this section is to apply the results of the previous section to the following semi-linear first-order PDEs system in one spatial dimension:
∂x ∂x = − + Mx(t ) + f (x(t )) ∂t ∂z
(7)
subject to the boundary and initial conditions given by: x(0, t ) = 0,
and
x(z , 0) = x0 (z )
(8)
where x(z , t ) = [x1 (z , t ) · · · xn (z , t )]T ∈ H := L2 (0, l)n denotes the vector of state variables, z ∈ [0, l] and t ∈ [0, ∞) denote position and time, respectively. f = [f1 · · · fn ] is a continuous vector function defined on a closed convex subset F ⊂ H. M is a diagonal matrix with nonpositive entries, i.e. M = diag(−αi ), i = 1, . . . , n, where αi ≥ 0 and x0 ∈ H. Without loss of the generality assume that l = 1. Actually, the equivalent state space description of the model (7)–(8) is given by (3), where A0 is the linear (unbounded) operator defined on its domain D(A0 ) :=
x ∈ H : x is a.c
dx dz
∈ H and x(0) = 0 ,
(9)
Note that if there exists i = 1, . . . , n such that αi = 0, then the constant α = 0. In this case, the inequality l0 < α cannot be satisfied because l0 is positive. The following theorem is stated to avoid the above situation. Theorem 15. Consider a semi-linear system given by (3), (9)–(11) and satisfying condition (H). Assume that f is Lipschitz continuous on F , with a Lipschitz constant l0 such that l0 < e−1 . Then for any x0 ∈ F , lim d(x(t , x0 ), x) = 0,
t →∞
i.e. the equilibrium point x is asymptotically stable on F . We shall prove Theorem 15 by using the following lemma (see Curtain & Zwart, 1995 Theorem 2.1.6 (c) and its proof for more details). Lemma 16. Let S0 (t ) be a C0 -semigroup and ω0 < 0 denotes its growth constant. For all ω ∈ (0, −ω0 ), there exists a constant Mω such that
∀t ≥ 0,
kS0 (t )k ≤ Mω e−ωt ,
where Mω is given by the following expression Mω = eωtω M0,ω
by d
+M ·I dx and the nonlinear operator N0 is defined on F by
(10)
N0 (x) := [f1 (x) · · · fn (x)]T .
(11)
A0 := −
1045
and the constants tω and M0,ω are given, respectively, by
tω := inf τ ≥ 0, kS0 (t )k ≤ e−ωt , ∀t ≥ τ
The following lemma focuses on the properties of the linear operator A0 (see e.g. Aksikas et al., 2007 and Winkin, Dochain, & Ligarius, 2000).
M0,ω = sup{kS0 (t )k, t ∈ [0, tω ]}. Proof of Theorem 15. Observe that S0 (t ) satisfies the following
kS0 (t )k =
1, 0,
0≤t <1 t ≥ 1.
Lemma 12. Let us consider the linear operator A0 given by (9)–(10). The following properties hold: (i) A0 is m-dissipative and generates an exponentially stable C0 semigroup S (t ). Moreover, for all t ≥ 0,
Then, (a) the growth constant is equal −∞, (b) the constant tω defined in Lemma 16 is equal 1 and (c) the constant M0,ω is equal 1. Therefore, the condition ω > Ml0 in Theorem 10 can be replaced by the weaker condition
kS (t )k ≤ e−αt ,
l0 < sup ωMω−1 = sup ωe−ω = e−1 .
with α = min αi . 1≤i≤n
(ii) There exists a positive constant λ such that the operator (I − λA0 )−1 is compact. In what follow, we need to assume the following condition: (H) For all x0 ∈ D(A0 ) ∩ F , system (7)–(8) has at least one solution denoted by x(t , x0 ). Now we are in a position to state the following corollaries about the asymptotic behavior of the system (3), (9)–(11). Two scenarios are presented. First we are interested in a dissipative nonlinearity (Corollary 13) and second we assume that the nonlinearity is not necessarily dissipative, but only Lipschitz continuous (Corollary 14). Corollary 13. Consider a semi-linear system given by (3), (9)–(11) and satisfying condition (H). Assume that f is Lipschitz continuous dissipative on F . Then for any x0 ∈ F , lim d(x(t , x0 ), ω(x0 )) = 0.
t →∞
If in addition f is strictly dissipative, then the equilibrium point x is asymptotically stable on F . Corollary 14. Consider a semi-linear system given by (3), (9)–(11) and satisfying condition (H). Assume that f is Lipschitz continuous on F , with a Lipschitz constant l0 such that l0 ≤ α . Then for any x0 ∈ F , lim d(x(t , x0 ), ω(x0 )) = 0.
t →∞
If in addition l0 < α , then the equilibrium point x is asymptotically stable on F .
ω>0
ω>0
Remark 17. The exponential stability of the linearized model of (7)–(8) is proved in Aksikas, Fuxman, Forbes, and Winkin (2009, Theorem 2) under some regularity conditions while our analysis imposes some dissipativity conditions in order to show the asymptotic stability of the nonlinear system. However, the linearization-based technique can be used only if the initial profile is close enough to the equilibrium profile, our approach guarantees the stability for any initial state on the domain of the system generator. Example 4.1. Plug flow reactor Let us consider a chemical plug flow reactor with the following two endothermic reactions: A −→ B and A −→ D. If the kinetics of the first reaction is first order with respect to the reactant concentration CA (mol/l) and has an Arrheniustype dependence on temperature T (K ), and if the kinetics of the second reaction is first-order kinetics with respect to the reactant concentration, the dynamics of the process are described by the following energy and mass balance PDEs:
α ∂T ∂T = −v + k1 1HCA exp − − β(T − Te ) ∂τ ∂z T α ∂ CA ∂ CA = −v − k1 CA exp − − k2 CA ∂τ ∂z T with the boundary conditions given for all τ ≥ 0, by T (0, τ ) = Tin and CA (0, τ ) = CA,in .
(12) (13)
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I. Aksikas, J.F. Forbes / Automatica 46 (2010) 1042–1046
In the equations above, 1H is the heat of reaction and Te represents the equilibrium profile of jacket temperature. Let us consider the following new variables x1 =
T − Tin
and x2 =
Tin
CA − CA,in CA,in
and consider the new time variable t = τ v . Then the plug flow reactor PDE model can be written under the form (7) such that the matrix M is defined by
1 −β M = v 0
0 −k2
and the function f is defined on its domain F = (x1 , x2 ) ∈ L2 (0, 1)2 : x1 ≥ −1 and − 1 ≤ x2 ≤ 0
by
−λ a(1 + x2 ) exp + β xe 1 1+ x1 f (x1 , x2 ) := −λ v −k (1 + x ) exp 1 2 1 + x1 where a = k1 1H
CA,in Tin
,
λ=
α
and xe =
Tin
Te − Tin Tin
.
The Lipschitz constant of f can be computed by using similar calculations to Aksikas et al. (2007), and it is given by
l0 = max 1, −
1HCA,in
k1
1+
v
Tin
4
λe2
.
References Aksikas, I., Fuxman, A., Forbes, J. F., & Winkin, J. J. (2009). Lq-control for a class of hyperbolic pde systems: application to a fixed-bed reactor. Automatica, 45, 1542–1548. Aksikas, I., Winkin, J. J., & Dochain, D. (2007). Asymptotic stability of infinitedimensional semi-linear systems: application to a nonisothermal reactor. Systems & Control Letters, 56, 122–132. Belleni-Morante, A., & Mc Bride, A. (1998). Applied nonlinear semigroups: an introduction. Willey. Brezis, H. (1973). Opéateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. In Mathematics studies. North-Holland. Christofides, P. D. (2001). Nonlinear and robust control of partial differential equation systems: methods and application to transport-reaction processes. Birkhauser. Crandall, M. G., & Pazy, A. (1969). Semi-groups of nonlinear contractions and dissipative sets. Journal of Functional Analysis, 3, 376–418. Curtain, R. F., & Zwart, H. J. (1995). An introduction to infinite-dimensional linear systems theory. New York: Springer Verlag. Dafermos, C. M., & Slemrod, M. (1973). Asymptotic behavior of nonlinear contraction semigroups. Journal of Functional Analysis, 13, 97–106. Dochain, D. (1994). Contribution to the analysis and control of distributed parameter systems with application to (bio)chemical processes and robotics. Thèse d’Agrégation de l’Enseignement Supérieur, Université Catholique de Louvain, Belgium. Laabissi, M., Achhab, M. E., Winkin, J., & Dochain, D. (2001). Trajectory analysis of nonisothermal tubular reactor nonlinear models. Systems & Control Letters, 42, 169–184. Lakshmikantham, V., & Leela, S. (1981). Nonlinear differential equations in abstract spaces. Oxford: Pergamon. Luo, Z., Guo, B., & Morgül, O. (1999). Stability and stabilization of infinite dimensional systems with applications. London: Springer Verlag. Martin, R. H. (1976). Nonlinear operators and differential equations in Banach spaces. New York: John Wiley & Sons. Miyadera, I. (1992). Nonlinear semigroups. American Mathematical Society. Pazy, A. (1983). Semigroups of linear operators and application to partial differential equations. In Applied mathematical sciences: Vol. 44. New York: Springer Verlag. Winkin, J., Dochain, D., & Ligarius, P. (2000). Dynamical analysis of distributed parameter tubular reactors. Automatica, 36, 349–361.
The stability condition is
l0 = max 1, −
1HCA,in Tin
k1 1 +
4
λe2
< min(β, k2 ).
Remark 18. Note that if the jacket temperature is used as a state feedback, then the function f will change and then the Lipschitz condition must be checked for the new function. 5. Conclusion In this paper, we have studied the asymptotic stability property of a class of dissipative semi-linear infinite-dimensional systems with nonlinearity not defined everywhere. Different scenarios have been considered. First the stability property was studied for semi-linear systems with dissipative nonlinearity. Secondly, Lipschitz nonlinearity was considered and the asymptotic stability was proved by adding an exponential stability condition on the underlying linear semigroup. Finally, the previous results were applied to a class of transport–reaction systems that can be described by first-order hyperbolic PDEs.
Ilyasse Aksikas was born on September 17, 1977, in Azemmour (El Jadida), Morocco. He received a Bachelor degree from the University of Chouaib Doukkali (Morocco) in 2000. He joined the Université catholique de Louvain (Belgium), where he obtained a Master degree (DEA) and a doctorate degree in Applied Mathematics in 2002 and 2005, respectively. He is currently a Research Associate at the Department of Chemical and Materials Engineering, University of Alberta, Canada. His main research interest is in the area of system and control theory, especially infinite-dimensional (distributed parameter) system theory, linear-quadratic optimal control, spectral factorization techniques and dynamical analysis and control of tubular reactors models. J. Fraser Forbes is a Professor and Chair of the Department of Chemical and Materials Engineering at the University of Alberta, Canada. He has over 25 years of experience in control/systems engineering as a researcher, educator and practitioner, and has worked in the steel, food products, forest products, and petrochemical industries. His current research interests include the use of the control of fixedbed reactor systems and optimization techniques for the design and synthesis of industrial automation systems.