Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 19 (2000), No. 1, 77–93

Exponential Stability of a Nonlinear Distributed Parameter System

J. Tervo and M. Nihtilä

Abstract. A nonlinear parabolic partial differential equation model describing the behaviour of a distributed parameter fixed-bed bioreactor is studied here. Exponential stability around the steady state solution for exponentially decaying deviations in the input and disturbance are proved via abstract formulation of the model as an evolution equation and by utilizing semigroup theory and asymptotic stability of the corresponding evolution operator. Keywords: Infinite-dimensional nonlinear systems, parabolic partial differential equations, ex-

ponential stability, fixed-bed bioreactors

AMS subject classification: Primary 93 C 20, secondary 93 C 80, 35 K 57

1. Introduction

Several different stability concepts have been developed for linear as well for nonlinear partial differential equations and the corresponding evolution operators (see, e.g., Amann [2: p. 68], Lions [14: p. 172], and Curtain and Zwart [5: p. 215]). As compared with the stability of finite-dimensional systems we are facing more complicated situations. Even in the case of linear partial differential equations the location of the poles of the transfer function of the distributed parameter system does not determine directly the stability as is the case of finite-dimensional systems. More refined funtional-analytic and function-space tools are needed. Moreover, the variety of different partial different equation system models is larger depending, e.g. on the location of inputs and outputs, and on the corresponding operator classes defined on abstract function spaces (Banach or Hilbert spaces).

Here, we consider stability properties of solutions of a nonlinear system related to a distributed parameter fixed-bed bioreactor. The system is a infinite-dimensional one. It is governed by partial differential equations of parabolic type, i.e. they are so-called evolution equations (c.f., e.g., Tanabe [21]).

The distributed parameter model of the system has its background in biological water treatment processes [7, 8, 13]. The goal of the process is to remove harmful nitrogen compounds from drinking water or from communal waste water. The process is modelled by two coupled partial differential equations. They describe the growth and substrate (nitrogen) consumption of certain microorganisms. These are immobilised on a fixed bed in the reactor tube. The water to be treated and which includes the substrate flows through the reactor.

J. Tervo: Dept. Appl. Math. Univ., POB 1627, FIN-70211 Kuopio M. Nihtilä: Dept. Food Techn. Univ., POB 27 (Viikki), FIN-00014 University of Helsinki ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

The spatially one-dimensional model of the fixed-bed bioreactor consists of a pair of nonlinear partial differential equations

\begin{matrix}\frac{∂u_{1}}{∂t} & = −k_{d}u_{1} + μ(u_{1}, u_{2})u_{1} \\ \frac{∂u_{2}}{∂t} & = D \frac{∂^{2}u_{2}}{∂x^{2}} − c(t)\frac{∂u_{2}}{∂x} − k_{1}μ(u_{1}, u_{2})u_{1} \end{matrix} \}  (1)

where the spatial variable x belongs to the interval G = (0, 1) ⊂ ℝ and the evolving time t belongs to the interval ℝ_{+} = (0, ∞) (remark that by \overline{ℝ}_{+} the interval [0, ∞) is denoted). The boundary conditions applied here are due to Danckwerts [6] and they are of the form

\begin{matrix}\frac{∂u_{2}}{∂x} (0, t) & = \frac{c(t)}{D}(u_{2}(0, t) − S(t)) \\ \frac{∂u_{2}}{∂x} (1, t) & = 0 \end{matrix} \}  (t ∈ ℝ_{+}).  (2)

In the equations the states u_{1} = u_{1}(x, t) and u_{2} = u_{2}(x, t) are the concentrations of the biomass of the microorganisms and the substrate, respectively. The specific growth rate of the microorganisms (in biomass)

μ(u_{1}, u_{2}) = μ_{m} \frac{u_{2}}{k_{2}u_{1} + u_{2}}

is due to Contois, 1959. This makes system (1) nonlinear. The input flow c is the control variable and the input substrate concentration S is a disturbance variable in the system. They are generally smooth functions of time, i.e. c = c(t) and S = S(t). The output function y (the measured variable) is the substrate concentration at the end of the reactor, that is,

y(t) = u_{2}(1, t).

The initial condition

\begin{matrix}u_{1}(x, 0) & = u_{10}(x) \\ u_{2}(x, 0) & = u_{20}(x) \end{matrix} \}  (3)

is chosen in such a way that u_{10} and u_{20} are the steady state solutions of problem (1)

- (2) before the simulated step changes the input function c(t) and/or the the initial concentration S(t) of the substrate. In the steady state c and S are independent of time and in that case they are denoted by \overline{c} and \overline{S}. Other parameters are positive. We do not list their meanings here but we refer to the contributions [7, 8], where the model and the parameters from the system-theoretic point of view are explained in more detail.

Simulation studies carried out pointed out that the assumption of equally distributed concentrations on the constant cross-sectional area of the reactor tube was adequate (c.f.

[17, 23]), justifying the use of the single space variable, the scaled distance from the input of the reactor tube (the length of which is scaled equal to 1). Boundary conditions, which are different from (2), have been proposed for dispersion models of our type (c.f. [20]), and applied in [13].

In Section 2 problem (1) - (3) is put into the abstract form

\begin{matrix}v̇ + A(t)v & = F (t, v) \\ v(0) & = v_{0} \end{matrix} \}

where A(t) is a linear unbounded operator in appropriate Banach spaces and F (t, v) is a nonlinear function. This enables us to consider the existence, stability and other issues of solutions as an application of the semigroup theory (see, e.g., [1, 2, 5, 9, 10, 18, 21]). The existence of a positive classical global solution u = (u_{1}, u_{2}) of problem (1)

- (3) such that

\begin{matrix}u_{1} & ∈ C^{1}((0, ∞), C(\overline{G})) ∩ C(\overline{ℝ}_{+}, C(\overline{G})) \\ u_{2} & ∈ C^{1}((0, ∞), C(\overline{G})) ∩ C((0, ∞), C^{2}(\overline{G})) ∩ C(\overline{ℝ}_{+}, C(\overline{G})) \end{matrix} \}

was proved in [11] as an application of the semigroup theory. Numerical computations for the original nonlinear system in the state space seem to support analytical stability result obtained here, not only in the spatially 1-dimensional but also in the spatially 3-dimensional case [17, 23].

In Section 3 we show that the steady state solution of problem (1) - (3) is attractive. The proof is based on the fact that the evolution operator U(t, τ) of the operator A(t) can be shown to be asymptotically stable in the relevant spaces considered here. The asymptotic stability of U(t, τ) follows from the results for quasilinear parabolic equations on the interpolation-extrapolation spaces studied in detail in [2]. It is well-known that exponential stability of the solutions both in finite-dimensional and infinite-dimensional cases influences on the input-output stability of the system (c.f. [5]). Consequently, it is a central issue of our system study for control.

We give some preliminary notations applied here:

Let G be an open set in ℝ^{n}. C(\overline{G}) is the space of continuous functions on \overline{G} equipped with the norm ∥w∥_{C(\overline{G})} (= ||w||_{∞}) = sup_{x∈\overline{G}} |w(x)|. Further, L_{p}(G) (1 ≤ p < ∞) is the Lebesgue space of p^{th}-power integrable functions f : G ↦ C and W^{s,p}(G) (s ∈ ℝ, 1 ≤ p < ∞) is the Sobolev-Slobodeckii space (see [2]).

Let Δ be an interval in ℝ. Then C^{l}(Δ, X) is the space of all l times continuously differentiable functions f : Δ ↦ X, when X is a normed space. The space C^{ρ−}(Δ, X) (0 < ρ ≤ 1) is the space of Hölder continuous functions f : Δ ↦ X equipped with the usual norm (see, e.g., [2: p. 40]). In the product space X_{1} × X_{2} of Banach spaces X_{1} and X_{2} we use the norm ∥(w_{1}, w_{2})∥_{X_{1}×X_{2}} = ∥w_{1}∥_{X_{1}} + ∥w_{2}∥_{X_{2}}.

2. Abstract formulation of the partial differential equation system

The original partial differential equation problem (1) - (3) will be converted into the abstract form

\begin{matrix}v̇ + A(t)v & = F (t, v) \\ v(0) & = v_{0} \end{matrix} \}  (4)

where A(t) is a linear unbounded operator and F (t, v) is quadratically bounded by v. This means that the linearized version of the original partial differential equation problem is given by

\begin{matrix}v̇ + A(t)v & = F (t, 0) \\ v(0) & = v_{0} \end{matrix} \}

2.1 Preliminaries. Consider the problem

\begin{matrix}\frac{∂u_{1}}{∂t} & = f(u_{1}, u_{2}) − k_{d}u_{1} \\ \frac{∂u_{2}}{∂t} & = D \frac{∂^{2}u_{2}}{∂x^{2}} − c(t)\frac{∂u_{2}}{∂x} − k_{1}f(u_{1}, u_{2}) \end{matrix} \}  (5)

for (x, t) ∈ G × ℝ_{+},

\begin{matrix}\frac{∂u_{2}}{∂x} (0, t) & = \frac{c(t)}{D}(u_{2}(0, t) − S(t)) \\ \frac{∂u_{2}}{∂x} (1, t) & = 0 \end{matrix} \}  (6)

for t ∈ ℝ_{+}, and

u(x, 0) = u_{0}(x) = (u_{10}(x), u_{20}(x))  (7)

for x ∈ G. Above we denoted

G = (0, 1)  and  f(u_{1}, u_{2}) = μ(u_{1}, u_{2})u_{1} = μ_{m} \frac{u_{1}u_{2}}{k_{2}u_{1} + u_{2}} .

Occasionally we also use the notation

f(u) = f(u_{1}, u_{2}) for u = (u_{1}, u_{2}).

Throughout the paper we assume that c and S are positive C^{1}(\overline{ℝ}_{+})-functions and that ċ and Ṡ are locally Lipschitz continuous. In addition we assume that u_{10} and u_{20} are positive C^{1}(\overline{G})-functions.

Let \overline{u}_{1} and \overline{u}_{2} be solutions of the steady state equations

\begin{matrix}0 & = f(\overline{u}_{1}, \overline{u}_{2}) − k_{d}\overline{u}_{1} \\ 0 & = D \frac{∂^{2}\overline{u}_{2}}{∂x^{2}} − \overline{c}\frac{∂\overline{u}_{2}}{∂x} − k_{1}f(\overline{u}_{1}, \overline{u}_{2}) \end{matrix} \}  (8)

for S(t) = \overline{S} and c(t) = \overline{c}, for x ∈ G, with the two-point boundary conditions

\begin{matrix}\frac{∂\overline{u}_{2}}{∂x} (0) & = \frac{\overline{c}}{D} (\overline{u}_{2}(0) − \overline{S}) \\ \frac{∂\overline{u}_{2}}{∂x} (1) & = 0 \end{matrix} \}  (9)

where \overline{c} = c(0). The solutions \overline{u}_{1} and \overline{u}_{2} can be computed in closed form and they have the expressions

\begin{matrix}\overline{u}_{2}(x) & = S \frac{q cosh(q(1 − x)) + p sinh(q(1 − x))}{q cosh(q) + (a + p) sinh(q)} e^{px} \\ \overline{u}_{1}(x) & = \frac{μ_{m} − k_{d}}{k_{2}k_{d}} \overline{u}_{2}(x) \end{matrix} \}  (10)

where

P = \frac{\overline{c}}{D} , p = \frac{P}{2} , a = \frac{k_{1}(μ_{m} − k_{d})}{k_{2}\overline{c}}, q = \sqrt{\frac{P^{2}}{4} + P a}.

The pair (\overline{u}_{1}, \overline{u}_{2}) is the equilibrium point of the dynamical system (5) - (7) when c ≡ \overline{c} and S ≡ \overline{S}. The constants P, p, a, q are positive for the original relevant parameter values.

Remark 1. The maximum principle for parabolic systems [19] implies the following comparison result for the solutions: Suppose that \overline{c} and \overline{S} are positive constants such that

\begin{matrix}c(t) & ≤ \overline{c} \\ S(t) & ≤ \overline{S} \end{matrix} \} (t ∈ \overline{ℝ}_{+}).

Let \overline{u} be the steady state solution corresponding to constant values \overline{c} and \overline{S} of the control c(t) and disturbance S(t). If u_{0} ≤ \overline{u}, then

u ≤ \overline{u}.  (11)

This result implies especially that the solutions of problem (5) - (7) are bounded when the control c(t) and the disturbance S(t) are bounded. In fact, estimate (11) is natural from the physical viewpoint.

2.2 Linearization. The nonlinear problem (5) - (7) is linearized around the steady state (10). Consequently, a careful analysis for the nonlinear term f has to be carried out. Formally f can be linearized by using the decomposition

f(\overline{u} + w) − f(\overline{u}) = \frac{∂f}{∂u_{1}} (\overline{u})w_{1} + \frac{∂f}{∂u_{2}} (\overline{u})w_{2} + g(w)

where the residual g(w) must have certain appropriate properties. From (8) we get

μ(\overline{u}_{1}, \overline{u}_{2}) = μ_{m} \frac{\overline{u}_{2}}{k_{2}\overline{u}_{1} + \overline{u}_{2}} = k_{d}  (12)

which also implies the last equation in (10), i.e. \overline{u}_{1} = \overline{u}_{2} \frac{μ_{m}−k_{d}}{k_{2}k_{d}}. Applying (10) and (12) we find by a routine computation that

\begin{aligned}\frac{∂f}{∂u_{1}} (\overline{u}) & = μ(\overline{u}_{1}, \overline{u}_{2}) − μ_{m}k_{2} \frac{\overline{u}_{1}\overline{u}_{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} \\ & = k_{d} − μ_{m}k_{2} \frac{\overline{u}_{2}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} \frac{μ_{m} − k_{d}}{k_{2}k_{d}} \\ & = \frac{k_{d}^{2}}{μ_{m}} \\ & =: a_{1}. \end{aligned}  (13)

It can be seen also that

μ_{m} \frac{\overline{u}_{2}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} ≡ a_{1}.  (14)

Similar computations show that

\frac{∂f}{∂u_{2}} (\overline{u}) = μ_{m}k_{2} \frac{\overline{u}_{1}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} ≡ \frac{(μ_{m} − k_{d})^{2}}{μ_{m}k_{2}} =: a_{2}.  (15)

Based on these calculations the following lemma, which guarantees a successful linearization, is proved.

Lemma 1. There exist constants C > 0 and δ > 0 such that

f(\overline{u} + w) − f(\overline{u}) = a_{1}w_{1} + a_{2}w_{2} + g(w)  (16)

where a_{1} and a_{2} are given above and where g(w) satisfies

∥g(w)∥_{C(\overline{G})} ≤ C∥w∥_{C(\overline{G})×C(\overline{G})}^{2}

for all w ∈ C(\overline{G}) × C(\overline{G}) such that ∥w∥_{C(\overline{G})×C(\overline{G})} ≤ δ.

The norm abbreviations ∥ · ∥_{1} = ∥ · ∥_{C(\overline{G})} and ∥ · ∥_{2} = ∥ · ∥_{C(\overline{G})×C(\overline{G})} are used in the proof, which is given in Parts A and B.

Proof of Lemma 1. Part A: Denote u = \overline{u} + w. Then by using (13) - (15) we have

\begin{matrix}∥f(\overline{u} + w) − f(\overline{u}) − (a_{1}w_{1} + a_{2}w_{2})∥_{1} \\ = μ_{m} ∥\frac{u_{1}u_{2}}{k_{2}u_{1} + u_{2}} − \frac{\overline{u}_{1}\overline{u}_{2}}{k_{2}\overline{u}_{1} + \overline{u}_{2}} − (\frac{\overline{u}_{2}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} w_{1} + k_{2} \frac{\overline{u}_{1}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} w_{2})∥_{1} \\ = μ_{m} ∥k_{2} \frac{u_{1}\overline{u}_{1}}{(k_{2}u_{1} + u_{2})(k_{2}\overline{u}_{1} + \overline{u}_{2})}w_{2} + \frac{\overline{u}_{2}u_{2}}{(k_{2}u_{1} + u_{2})(k_{2}\overline{u}_{1} + \overline{u}_{2})}w_{1} \\ − (\frac{\overline{u}_{2}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} w_{1} + k_{2} \frac{\overline{u}_{1}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} w_{2})∥_{1} \\ ≤ μ_{m} \{∥\frac{\overline{u}_{2}u_{2}}{(k_{2}u_{1} + u_{2})(k_{2}\overline{u}_{1} + \overline{u}_{2})} − \frac{\overline{u}_{2}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} ∥_{1} \\ + k_{2} ∥\frac{\overline{u}_{1}u_{1}}{(k_{2}u_{1} + u_{2})(k_{2}\overline{u}_{1} + \overline{u}_{2})} − \frac{\overline{u}_{1}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} ∥_{1}\} (∥w_{1}∥_{1} + ∥w_{2}∥_{1}). \end{matrix} (17)

Part B: Furthermore, we have

\begin{matrix}∥\frac{\overline{u}_{1}u_{1}}{(k_{2}u_{1} + u_{2})(k_{2}\overline{u}_{1} + \overline{u}_{2})} − \frac{\overline{u}_{1}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} ∥ & 1 \\ = ∥\frac{1}{(k_{2}u_{1} + u_{2})(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} (\overline{u}_{1}\overline{u}_{2}w_{1} + \overline{u}_{1}^{2}w_{2})∥_{1} \\ ≤ \{∥\frac{1}{k_{2}\overline{u}_{1} + \overline{u}_{2}} ∥_{1} ∥\frac{\overline{u}_{1}\overline{u}_{2}}{(k_{2}u_{1} + u_{2})(k_{2}\overline{u}_{1} + \overline{u}_{2})}∥_{1} \\ + ∥\frac{1}{k_{2}u_{1} + u_{2}} ∥_{1} ∥\frac{\overline{u}_{1}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} ∥_{1}\} (∥w_{1}∥_{1} + ∥w_{2}∥_{1}). \end{matrix}  (18)

Suppose that

∥w_{1}∥_{1} + ∥w_{2}∥_{1} < \frac{1}{2} inf_{x∈G} min\{\overline{u}_{1}(x), \overline{u}_{2}(x)\} =: δ .  (19)

Then we find that

u_{j}(x) = u_{j}(x) − \overline{u}_{j}(x) + \overline{u}_{j}(x)

= \overline{u}_{j}(x) − w_{j}(x)

≥ inf_{x∈G} \overline{u}_{j}(x) − ∥w_{j}∥_{1}

≥ \frac{1}{2} inf_{x∈G} \overline{u}_{j}(x)

≥ \frac{1}{2} inf_{x∈G} min_{j=1,2}\{\overline{u}_{j}(x)\}

=: c_{1}

for all x ∈ G. Hence when (19) holds, we see that

k_{2}u_{1}(x) + u_{2}(x) ≥ (k_{2} + 1)c_{1} (x ∈ G)

which implies that

∥\frac{1}{k_{2}u_{1} + u_{2}} ∥_{1} ≤ \frac{1}{(k_{2} + 1)c_{1}} =: C_{1}.  (20)

Combining (18) and (20) we find that for ∥w∥_{2} = ∥w_{1}∥_{1} + ∥w_{2}∥_{1} < δ

\begin{matrix}∥\frac{\overline{u}_{1}u_{1}}{(k_{2}u_{1} + u_{2})(k_{2}\overline{u}_{1} + \overline{u}_{2})} − \frac{\overline{u}_{1}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} ∥ & 1 \\ ≤ \{C_{1} ∥\frac{1}{k_{2}\overline{u}_{1} + \overline{u}_{2}} ∥_{1} ∥\frac{\overline{u}_{1}\overline{u}_{2}}{k_{2}\overline{u}_{1} + \overline{u}_{2}} ∥_{1} + C_{1} ∥\frac{\overline{u}_{1}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} ∥_{1}\} \\ × (∥w_{1}∥_{1} + ∥w_{2}∥_{1}) \\ =: C_{2}∥w∥_{2} \end{matrix}  (21)

where C_{2} does not depend on w. Similarly we find that

‖\frac{\overline{u}_{2}u_{2}}{(k_{2}u_{1} + u_{2})(k_{2}\overline{u}_{1} + \overline{u}_{2})} − \frac{\overline{u}_{2}^{2}}{(k_{2}\overline{u}_{1} + \overline{u}_{2})^{2}} ‖_{1} ≤ C_{3}∥w∥_{2}  (22)

for ∥w∥_{2} < δ. Hence we find from (17) that

f(\overline{u} + w) − f(\overline{u}) − (a_{1}w_{1} + a_{2}w_{2}) = g(w)

where ∥g(w)∥_{1} ≤ C_{4}∥w∥_{2}^{2} for ∥w∥_{2} < δ which completes the proof

Remark 2. Consequently, the above considerations show that the mapping f : C(\overline{G}) × C(\overline{G}) ↦ C(\overline{G}) is differentiable in the neighbourhood of \overline{u}.

Denote U = (U_{1}, U_{2}) = u − \overline{u}. Substracting equations (5) - (6) and (8) - (9) side by side we get

\begin{matrix}\frac{∂U_{1}}{∂t} & = f(U + \overline{u}) − f(\overline{u}) − k_{d}U_{1} \\ \frac{∂U_{2}}{∂t} & = D \frac{∂^{2}U_{2}}{∂x^{2}} − c(t)\frac{∂(U_{2} + \overline{u}_{2})}{∂x} + \overline{c}\frac{∂\overline{u}_{2}}{∂x} − k_{1}f(\overline{u} + U) + k_{1}f(\overline{u}) \end{matrix} \}  (23)

for (x, t) ∈ G × ℝ_{+},

\begin{matrix}\frac{∂U_{2}}{∂x} (0, t) & = \frac{c(t)}{D}[(U_{2} + \overline{u}_{2})(0, t) − S(t)] − \frac{\overline{c}}{D} [\overline{u}_{2}(0) − \overline{S})] \\ \frac{∂U_{2}}{∂x} (1, t) & = 0 \end{matrix} \}  (24)

for t ∈ ℝ_{+}, and

\begin{matrix}U_{1}(x, 0) & = u_{10}(x) − \overline{u}_{1}(x) \\ U_{2}(x, 0) & = u_{20}(x) − \overline{u}_{2}(x) \end{matrix} \}  (25)

for x ∈ G. Due to Lemma 1 problem (23) - (25) can be put into the form

\begin{matrix}\frac{∂U_{1}}{∂t} & = (a_{1} − k_{d})U_{1} + a_{2}U_{2} + g(U) \\ \frac{∂U_{2}}{∂t} & = D \frac{∂^{2}U_{2}}{∂x^{2}} − c(t)\frac{∂U_{2}}{∂x} − [c(t) − \overline{c}]\frac{∂\overline{u}_{2}}{∂x} − k_{1}a_{1}U_{1} − k_{1}a_{2}U_{2} − k_{1}g(U) \end{matrix} \} (26)

for (x, t) ∈ G × ℝ_{+},

\begin{matrix}\frac{∂U_{2}}{∂x} (0, t) & = \frac{c(t)}{D}U_{2}(0, t) + \frac{c(t) − \overline{c}}{D}\overline{u}_{2}(0) − \frac{c(t)}{D}S(t) + \frac{\overline{c}}{D} \overline{S} \\ \frac{∂U_{2}}{∂x} (1, t) & = 0 \end{matrix} \}  (27)

for t ∈ ℝ_{+},

\begin{matrix}U_{1}(x, 0) & = u_{10}(x) − \overline{u}_{1}(x) \\ U_{2}(x, 0) & = u_{20}(x) − \overline{u}_{2}(x). \end{matrix} \}  (28)

By substituting the transformed variable v = (v_{1}, v_{2}) = (κU_{1}, U_{2} + s(t)) with κ = (k_{1} \frac{a_{1}}{a_{2}} )^{\frac{1}{2}} into system (26) - (28) and by making the definition

s(t) = \frac{D}{c(t)} [\frac{c(t) − \overline{c}}{D}\overline{u}_{2}(0) − \frac{c(t)}{D}S(t) + \frac{\overline{c}}{D} \overline{S}]  (29)

we obtain the system

\begin{matrix}\frac{∂v_{1}}{∂t} & = (a_{1} − k_{d})v_{1} + κa_{2}v_{2} − κa_{2}s(t) + κg(\frac{1}{κ}v_{1}, v_{2} − s(t)) \\ \frac{∂v_{2}}{∂t} & = D \frac{∂^{2}v_{2}}{∂x^{2}} − c(t)\frac{∂v_{2}}{∂x} − (c(t) − \overline{c})\frac{∂\overline{u}_{2}}{∂x} − \frac{k_{1}a_{1}}{κ} v_{1} \\ & − k_{1}a_{2}v_{2} + k_{1}a_{2}s(t) − k_{1}g(\frac{1}{κ}v_{1}, v_{2} − s(t)) + ṡ(t) \end{matrix} \}  (30)

for (x, t) ∈ G × ℝ_{+},

\begin{matrix}\frac{∂v_{2}}{∂x} (0, t) & = \frac{c(t)}{D}v(0) \\ \frac{∂v_{2}}{∂x} (1, t) & = 0 \end{matrix} \}  (31)

for t ∈ ℝ_{+}, and

\begin{matrix}v_{1}(x, 0) & = κ(u_{10}(x) − \overline{u}_{1}(x)) \\ v_{2}(x, 0) & = u_{20}(x) − \overline{u}_{2}(x) + s(0). \end{matrix} \}  (32)

Define a linear operator

A(t) : C(\overline{G}) × L_{p}(G) ↦ C(\overline{G}) × L_{p}(G)

by

\begin{aligned}D(A(t)) & = C(\overline{G}) × \{w_{2} ∈ W^{2,p}(G) |(\frac{∂w_{2}}{∂x} − \frac{c(t)}{D}w_{2}) (0) = 0, \frac{∂w_{2}}{∂x} (1) = 0\} \\ A(t)w & = − ((a_{1} − k_{d})w_{1} + κa_{2}w_{2}, D \frac{∂^{2}w_{2}}{∂x^{2}} − c(t)\frac{∂w_{2}}{∂x} − \frac{k_{1}a_{1}}{κ} w_{1} − k_{1}a_{2}w_{2}) \end{aligned}

for w ∈ D(A(t)). Since W^{2,p}(G) ⊂ C^{1}(\overline{G}) (because n = 1) for p > 1 the terms w_{2}(0), w_{2}(1), and \frac{∂w_{2}}{∂x} (0) are well-defined. Furthermore, define

\begin{aligned}F (t, w) & = (−κa_{2}s(t) + κg(\frac{1}{κ}w_{1}, w_{2} − s(t)) , \\ & − [c(t) − \overline{c}]\frac{∂\overline{u}_{2}}{∂x} + k_{1}a_{2}s(t) − k_{1}g(\frac{1}{κ}w_{1}, w_{2} − s(t)) + ṡ(t)) . \end{aligned}

Then problem (28) - (32) gets the abstract form

\begin{matrix}\frac{dv}{dt} + A(t)v & = F (t, v) \\ v(0) & = v_{0} \end{matrix} \}  (33)

where

v_{0}(x) = (κ[u_{10}(x) − \overline{u}_{1}(x)], u_{20}(x) − \overline{u}_{2}(x) + s(0)).

3. Stability of solutions

3.1 The evolution operator of A(t). We use the notations and theory of [2]. Let for s ∈ [−1, 1] and p ∈ (1, ∞) be

W_{B}^{s,p}(G) = \{ \begin{matrix}W^{s,p}(G) & if 0 ≤ s ≤ 1 \\ (W^{−s,p′}(G))′ & if −1 ≤ s ≤ 0. \end{matrix}

Define time-dependent Sobolev spaces W_{B(t)}^{2,p}(G) by

W_{B(t)}^{2,p}(G) = \{w ∈ W^{2,p}(G) | D \frac{∂w}{∂x} (0) − c(t)w(0) = 0 and \frac{∂w}{∂x} (1) = 0\} .

In addition we define Banach spaces

E(t) = C(\overline{G}) × W_{B(t)}^{2,p}(G)  and  E_{α+ \frac{1}{2}} = C(\overline{G}) × W_{B}^{2α,p}(G)

where −\frac{1}{2} ≤ α ≤ \frac{1}{2}. The operator A(t) can be interpreted as a bounded operator E_{1} ↦ E_{0} (that is, A(t) ∈ L(E_{1}, E_{0})) by

A(t) = ( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22}(t) \end{matrix} )

where for w = (w_{1}, w_{2}) ∈ E_{1}

A_{11}w_{1} = (k_{d} − a_{1})w_{1},  A_{12}w_{2} = −κa_{2}w_{2},  A_{21}w_{1} = \frac{k_{1}a_{1}}{κ} w_{1}

and where

A_{22}(t) : W^{1,p}(G) ↦ W_{B}^{−1,p}(G) = (W^{1,p′}(G))′

is the operator corresponding to the antilinear form a_{22}(t) : W^{1,p}(G) × W^{1,p′}(G) ↦ C such that

a_{22}(t)(w_{2}, v_{2}) = ∫_{0}^{1} (D \frac{∂w_{2}}{∂x} \overline{\frac{∂v_{2}}{∂x}} + k_{1}a_{2}w_{2}\overline{v}_{2} + c(t)\frac{∂w_{2}}{∂x} \overline{v}_{2})dx + c(t)w_{2}(0)\overline{v}_{2}(0),

that is,

⟨A_{22}(t)w_{2}, v_{2}⟩ := ∫_{0}^{1} (−D \frac{∂^{2}w_{2}}{∂x^{2}} + c(t)\frac{∂w_{2}}{∂x} + k_{1}a_{2}w_{2}) \overline{v}_{2}dx = a_{22}(t)(w_{2}, v_{2})

(which follows by partial integration).

For the definition of the (parabolic) evolution operator

U(t, τ) : C(\overline{G}) × L_{p}(G) ↦ C(\overline{G}) × L_{p}(G)

for A(t) we refer to [2: pp. 45 - 47]. The solution v of problem (33) satisfies the (Volterra) integral equation

v(t) = U(t, 0)v_{0} + ∫_{0}^{t} U(t, τ)F (τ, v(τ)) dτ.  (34)

For this operator the following estimates are proved.

Stability of a Distributed System 87

Theorem 1. There exists a constant δ > 0 such that when [c]_{C^{ρ−}(\overline{ℝ}_{+})} < δ, where 0 < ρ ≤ 1, then the operator A(t) has an evolution operator U(t, τ) : C(\overline{G}) × L_{p}(G) ↦ C(\overline{G}) × L_{p}(G) and there exist α ∈ (\frac{1}{2}, 1] and β > 0 such that for w ∈ C(\overline{G}) × W^{α,p}(G) and for 0 < τ < t

∥U(t, τ)w∥_{3} ≤ Me^{−β(t−τ)}∥w∥_{3}  (35)
∥U(t, τ)w∥_{3} ≤ M(t − τ)^{−(α− \frac{1}{2} )}e^{−β(t−τ)}∥w∥_{4}  (36)

where we use the abbreviations ∥ · ∥_{3} = ∥ · ∥_{C(\overline{G})×W^{α,p}(G)} and ∥ · ∥_{4} = ∥ · ∥_{C(\overline{G})×L_{p}(G)}. For the definition of the quantity [·]_{C^{ρ−}(\overline{ℝ}_{+})} see [2: p. 40]. Proof of Theorem 1. We shall only outline the proof and omit many details. The

proof is given in Parts A, B and C.

Part A: Let A_{0}(t) = A(t)_{|E(t)}. It is well-known that the operator A_{0}(t) (t ≥ 0) generates an analytic semigroup on E_{\frac{1}{2}} with domain E(t) (see, e.g., [1, 9, 10, 18, 21]) and then with the notations of [2] A_{0}(t) ∈ H(E(t), E_{\frac{1}{2}} ).

Part B: Using the relation κ = (k_{1} \frac{a_{1}}{a_{2}} )^{\frac{1}{2}} and the fact that, for w ∈ L_{2}(G) × W_{B(t)}^{2,2}(G),

D \frac{∂w_{2}}{∂x} (0)\overline{w}_{2}(0) = −c(t)w_{2}(0)\overline{w}_{2}(0) and \frac{∂w_{2}}{∂x} (1) = 0

we find that, for all w ∈ L_{2}(G) × W_{B(t)}^{2,2}(G),

\begin{matrix}ℜ⟨A(t)w, w⟩_{L_{2}(G)^{2}} \\ = ℜ [(k_{d} − a_{1}) ∫_{0}^{1} w_{1}\overline{w}_{1}dx + k_{1}a_{2} ∫_{0}^{1} w_{2}\overline{w}_{2}dx \\ − κa_{2} ∫_{0}^{1} w_{2}\overline{w}_{1}dx + \frac{k_{1}a_{1}}{κ} ∫_{0}^{1} w_{1}\overline{w}_{2}dx \\ + c(t)w_{2}(0)\overline{w}_{2}(0) + D ∫_{0}^{1} \frac{∂w_{2}}{∂x} \overline{\frac{∂w_{2}}{∂x}} dx + c(t) ∫_{0}^{1} \frac{∂w_{2}}{∂x} \overline{w}_{2}dx] \\ = (k_{d} − a_{1}) ∫_{0}^{1} |w_{1}|^{2}dx + k_{1}a_{2} ∫_{0}^{1} |w_{2}|^{2}dx \\ − κa_{2}ℜ ∫_{0}^{1} w_{2}\overline{w}_{1}dx + \frac{k_{1}a_{1}}{κ} ℜ ∫_{0}^{1} w_{1}\overline{w}_{2}dx \\ + c(t)|w_{2}(0)|^{2} + D ∫_{0}^{1} \frac{∂w_{2}}{∂x} \overline{\frac{∂w_{2}}{∂x}} dx + c(t)|_{0}^{1} \frac{1}{2}|w_{2}|^{2} \\ = (k_{d} − a_{1}) ∫_{0}^{1} |w_{1}|^{2}dx + k_{1}a_{2} ∫_{0}^{1} |w_{2}|^{2}dx \\ + D ∫_{0}^{1} |\frac{∂w_{2}}{∂x} |^{2} dx + \frac{c(t)}{2} [|w_{2}(1)|^{2} + |w_{2}(0)|^{2}] . \end{matrix}  (37)

Equation (37) immediately gives that for w ∈ L_{2}(G) × W_{B(t)}^{2,2}(G) we have the estimate

ℜ⟨A(t)w, w⟩_{L_{2}(G))^{2}} ≥ min{k_{d} − a_{1}, k_{1}a_{2}}∥w∥_{L_{2}(G)×W^{1,2}(G)}^{2}.  (38)

Part C: Equation (38) and Part A imply that for p = 2

σ(A_{0}(t)) ⊂ {z ∈ C| ℜz ≥ γ}  (t ≥ 0)  (39)

where γ = min{k_{d} − a_{1}, k_{1}a_{2}}. Here γ is positive since

k_{d} − a_{1} = \frac{k_{d}(μ_{m} − k_{d})}{μ_{m}}  and  k_{1}a_{2} = k_{1} \frac{(μ_{m} − k_{d})^{2}}{μ_{m}k_{2}}

are positive for relevant parameter values. Since G is bounded we know (from elliptic theory) that inclusion (39) is valid for any p > 1 .

The continuity of c, inclusion (39) and well-known estimates for elliptic operators (cf. [1: p. 42]) imply that for any σ > −γ and ω > 0

σI + A(·) ∈ C(\overline{ℝ}_{+}, H(E_{1}, E_{0}, κ, ω))

where κ > 0. The space C(\overline{ℝ}_{+}, H(E_{1}, E_{0}, κ, ω)) is defined as in [2: p. 11]. Furthermore, we find that (for 0 < ρ ≤ 1)

[A(·)]_{C^{ρ−}(\overline{ℝ}_{+},L(E_{1},E_{0}))} ≤ [c]_{C^{ρ−}(\overline{ℝ}_{+})}.

Hence when [c]_{C^{ρ−}(\overline{ℝ}_{+})} is small enough we find that [2: Assumption (II. 5.0.1)] holds. Choosing β = β_{−} = \frac{1}{2}, [2: Lemma II.5.1.3] implies the assertion

Remark 3. Based on the assumption ∥ċ∥_{C(\overline{ℝ}_{+})} < δ it can be concluded that [c]_{C^{ρ−}(\overline{ℝ}_{+})} < δ when ρ = 1.

The evolution operator of A(t) − β_{1}I is

U(t, τ) = e^{tβ_{1}I} ∘ U(t, τ) ∘ e^{−τβ_{1}I}

for 0 < β_{1} < β. This evolution operator satisfies for w ∈ C(\overline{G}) × W^{α,p}(G) and for 0 < τ < t the estimates

∥U(t, τ)w∥_{3} ≤ Me^{−(β−β_{1})(t−τ)}∥w∥_{3}  (40)
∥U(t, τ)w∥_{3} ≤ M(t − τ)^{−(α− \frac{1}{2} )}e^{−(β−β_{1})(t−τ)}∥w∥_{4}  (41)

where the norms are given in Theorem 1.

3.2. Local asymptotic stability. Here we prove a result about the long-time behaviour for the solutions of problem (5) - (7). The basic idea of the proof is standard (cf., e.g., [3]). However, the verification contains some special estimates which must be computed. In the proof we establish the required estimates. The proof is given in Parts A, B, and C.

Theorem 2. Let max\{\frac{1}{p}, \frac{1}{2}\} < α ≤ 1. Then there exists a constant δ > 0 such that if

∥e^{β_{1}(·)}(c − \overline{c})∥_{∞} < δ, ∥e^{β_{1}(·)}ċ∥_{∞} < δ  (42)
∥e^{β_{1}(·)}(S − \overline{S})∥_{∞} < δ, ∥e^{β_{1}(·)}Ṡ∥_{∞} < δ  (43)

and if

∥u_{0} − \overline{u}∥_{C(\overline{G})×W^{α,p}(G)} < δ,  (44)

then

lim_{t→∞} ∥u(·, t) − \overline{u}∥_{C(\overline{G})×W^{α,p}(G)} = 0.  (45)

Proof. Part A: Denote V = e^{β_{1}t}v. Then system (33) becomes

\begin{matrix}\frac{dV}{dt} + [A(t) − β_{1}I]V & = e^{β_{1}t}F (t, e^{−β_{1}t}V) \\ V (0) & = e^{β_{1}0}v_{0} =: V_{0}. \end{matrix} \}  (46)

Consider the nonlinearity \overline{F} (t, V) = e^{β_{1}t}F (t, e^{−β_{1}t}V). We find that

\begin{aligned}\overline{F} (t, V) & = e^{β_{1}t} (−κa_{2}s(t) + κg, −[c(t) − \overline{c}]\frac{∂\overline{u}_{2}}{∂x} + k_{1}a_{2}s(t) − k_{1}g + ṡ(t)) \\ g & = g (\frac{1}{κ}e^{−β_{1}t}V_{1}, e^{−β_{1}t}V_{2} − s(t)) . \end{aligned}

Since G = (0, 1) ⊂ ℝ, the Sobolev inequality implies that

∥u∥_{C(\overline{G})} ≤ C_{1}∥u∥_{W^{α,p}(G)} (u ∈ W^{α,p}(G))

when \frac{1}{p} < α. Hence due to Lemma 1 we find that there exists constants 0 < δ′ < 1 and C_{2} > 0 such that for ∥V ∥_{3} + ∥s∥_{∞} < δ′ (c.f. Theorem 1 for the norms)

∥\overline{F} (t, V)∥_{4} ≤ C_{2}\{∥V ∥_{3}^{2} + |e^{β_{1}t}s(t)| + |e^{β_{1}t}[c(t) − \overline{c}]| + |e^{β_{1}t}ṡ(t)| \}.  (47)

Part B: We find, by adding and substracting a term in (29), that

s(t) = \frac{D}{c(t)} \{\frac{c(t) − \overline{c}}{D}\overline{u}_{2}(0) − \frac{c(t)}{D}[S(t) − \overline{S}] − \frac{\overline{S}}{D} [c(t) − \overline{c}]\} .  (48)

By differentiating this equality we have

\begin{aligned}ṡ(t) & = −\frac{Dċ(t)}{c(t)^{2}} \{\frac{c(t) − \overline{c}}{D}\overline{u}_{2}(0) − \frac{c(t)}{D}[S(t) − \overline{S}] − \frac{\overline{S}}{D} [c(t) − \overline{c}]\} \\ & + \frac{D}{c(t)} \{\frac{ċ(t)}{D}\overline{u}_{2}(0) − \frac{ċ(t)}{D}S(t) − \frac{c(t)}{D}Ṡ(t) − \frac{\overline{S}}{D} ċ(t)\} . \end{aligned}  (49)

The assumptions imply that |c|, |ċ| and |S|, |Ṡ| are bounded from above and that c is bounded from below by a positive constant. Hence there exist constants C_{3} > 0 and C_{4} > 0 such that

|s(t)| ≤ C_{3} \{|c(t) − \overline{c}| + |S(t) − \overline{S}|\}  (50)
|ṡ(t)| ≤ C_{4} \{|c(t) − \overline{c}| + |S(t) − \overline{S}| + |ċ(t)| + |Ṡ(t)|\} .  (51)

In addition, we find that ∥V_{0}∥_{3} < δ′ if

∥u_{0} − \overline{u}∥_{3} < \frac{δ′}{2 max\{κ, 1\}}  and  |s(0)| < \frac{δ′}{2} .  (52)

Part C: Suppose that ∥V_{0}∥_{3} + ∥s∥_{∞} < δ′. Then there exists t′ > 0 such that ∥V (t)∥_{3} + ∥s∥_{∞} < δ′ for t ∈ (0, t′). Due to (47) and (50) - (52) we get for t ∈ (0, t′)

\begin{aligned}∥V (t)∥_{3} & = ∥U(t, 0)V_{0} + ∫_{0}^{t} U(t, τ)\overline{F} (τ, V (τ)) dτ∥_{3} \\ & ≤ ∥U(t, 0)V_{0}∥_{3} + ∫_{0}^{t} ∥U(t, τ)\overline{F} (τ, V (τ))∥_{3}dτ \\ & ≤ Me^{−(β−β_{1})t}∥V_{0}∥_{3} + ∫_{0}^{t} M(t − τ)^{−(α− \frac{1}{2} )}e^{−(β−β_{1})(t−τ)}∥\overline{F} ∥_{4}dτ \\ & ≤ Me^{−(β−β_{1})t}∥V_{0}∥_{3} + ∫_{0}^{t} M(t − τ)^{−(α− \frac{1}{2} )}e^{−(β−β_{1})(t−τ)} \\ & × C_{2}\{∥V (τ)∥_{3}^{2} + |e^{β_{1}τ} s(τ)| + |e^{β_{1}τ} [c(τ) − \overline{c}]| + |e^{β_{1}τ} ṡ(τ)| \}dτ \\ & ≤ Me^{−(β−β_{1})t}∥V_{0}∥_{3} + MC_{2}\{ sup_{t∈[0,t′)} ∥V (τ)∥_{3}^{2} + ∥e^{β_{1}(·)}s∥_{∞} \\ & + ∥e^{β_{1}(·)}(c − \overline{c})∥_{∞} + ∥e^{β(·)}ṡ∥_{∞}\} ∫_{0}^{∞} τ^{−(α− \frac{1}{2} )}e^{−(β−β_{1})τ} dτ \end{aligned}  (53)

where the abbreviation ∥\overline{F} ∥_{4} = ∥\overline{F} (τ, V (τ))∥_{4} was used. From (50) - (53) we are able to conclude that there exists δ > 0 such that if

∥e^{β_{1}(·)}(c − \overline{c})∥_{∞} < δ, ∥e^{β_{1}(·)}ċ∥_{∞} < δ
∥e^{β_{1}(·)}(S − \overline{S})∥_{∞} < δ, ∥e^{β_{1}(·)}Ṡ∥_{∞} < δ

and if

∥u_{0} − \overline{u}∥_{3} < δ,

then

∥V (t)∥_{3} ≤ δ_{1} (t ∈ [0, t_{1})) ⟹ ∥V (t)∥_{3} ≤ δ_{0} < δ_{1} (t ∈ [0, t_{1})).

This implies that V is bounded (∥V (t)∥_{3} ≤ δ_{1}) on \overline{ℝ}_{+}, that is, v satisfies ∥v(t)∥_{3} ≤ C_{5}e^{−β_{1}t} for all t ∈ \overline{ℝ}. Hence, due to the assumption, we have

∥u(t) − \overline{u}∥_{3} = ∥U(t)∥_{3} = ∥(\frac{1}{κ}v_{1}, v_{2} − s(t))∥_{3} ≤ C_{6} \{∥v(t)∥_{3} + |s(t)|\} ≤ C_{7}e^{−β_{1}t}.

This completes the proof

4. A remark on the output tracking problem

One of the specific questions in the controller design related to problem (5) - (7) is the solvability of the following output tracking problem:

For the given reference function y^{∗} ∈ C(\overline{ℝ}_{+}) with 0 < y^{∗}(t) < S(t) for all t ∈ \overline{ℝ}_{+} find the control c = c(u, y^{∗}) (sufficiently smooth) such that the output

y(t) := u_{2}(1, t) = y^{∗}(t) (t ∈ \overline{ℝ}_{+})

at least approximately and/or asymptotically. Also, the stability (in appropriate spaces) of the associated closed loop system is significant in practical situations. For example, changes in S cause disturbances in the state and output of the system.

In mathematical setting the producing of the prescribed reference output y^{∗}(t) means the non-homogeneous boundary condition

u_{2}(1, t) = y^{∗}(t).  (54)

From

\begin{matrix}\frac{∂u_{1}}{∂t} & = −k_{d}u_{1} + f(u_{1}, u_{2}) \\ u_{1}(x, 0) & = \overline{u}_{1}(x) \end{matrix} \}

we can at least in theory solve the unknown u_{1}, say u_{1} = θ(u_{2}, t). From (6) we obtain

c(t) = \frac{D \frac{∂u_{2}}{∂x} (0, t)}{u_{2}(0, t) − s(t)}.

Hence we find that u_{2} satisfies the problem

\frac{∂u_{2}}{∂t} = D \frac{∂^{2}u_{2}}{∂x^{2}} − \frac{D \frac{∂u_{2}}{∂x} (0, t)}{u_{2}(0, t) − s(t)} \frac{∂u_{2}}{∂x} − k_{1}f(θ(u_{2}, t), u_{2})  (55)
u_{2}(1, t) = y^{∗}(t), \frac{∂u_{2}}{∂x} (1, t) = 0  (56)
u_{2}(x, 0) = \overline{u}_{2}(x).  (57)

Denoting V_{1} = u_{2} and V_{2} = \frac{∂u_{2}}{∂x} problem (55) - (57) gets the form

\begin{aligned}\frac{∂V_{1}}{∂x} & = V_{2} \\ \frac{∂V_{2}}{∂x} & = \frac{1}{D} \frac{∂V_{1}}{∂t} + \frac{V_{2}(0, t)}{V_{1}(0, t) − s(t)}V_{2} + \frac{k_{1}}{D} f(θ(V_{1}, t), V_{1}) \end{aligned}  (58)

V_{1}(x, 0) = \overline{u}_{2}(x)  (59)
V_{1}(1, t) = y^{∗}(t), V_{2}(1, t) = 0.  (60)

Furthermore, we denote W_{1} = V_{1}−\overline{u}_{2}(x) and W_{2} = V_{2}. Then we have for W = (W_{1}, W_{2})

\begin{matrix}\frac{∂W}{∂x} + AW & = F (t, W) \\ W (1) & = (y^{∗}(t) − \overline{u}_{2}(1), 0) \end{matrix} \}  (61)

where A is an operator L_{p}(ℝ_{+})^{2} ↦ L_{p}(ℝ_{+})^{2} such that

D(A) = L_{p}(ℝ_{+}) × W_{0}^{p,1}(ℝ_{+}), AW = − (W_{2}, \frac{1}{D} \frac{∂W_{1}}{∂t} )

and

F (t, W) = (−\frac{∂\overline{u}_{2}}{∂x} , \frac{W_{2}(0)}{W_{1}(0) + \overline{u}_{2}(0) − s(t)}W_{2} + \frac{k_{1}}{D} f(θ(W_{1} + \overline{u}_{2}, t), W_{1} + \overline{u}_{2})) .

In this approach the existence and stability properties of system (61) would be interesting. The corresponding control law is given by

C = \frac{W_{2}(0, t)}{W_{1}(0, t) + \overline{u}_{2}(0) − s(t)}.

The solvability of the output tracking problem given above remains open.

Acknowledgement. Part of this work was supported by the Academy of Finland, Research Council for Natural Sciences and Engineering (Project no: 37462, 1997) which is greatly acknowledged.

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Received 27.02.1999