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Journal of Theoretical Biology 241 (2006) 295–306

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Stabilization of inhomogeneous patterns in a diffusion–reaction systemunder structural and parametric uncertainties

Carlos Vilas, Mıriam R. Garcıa, Julio R. Banga, Antonio A. Alonso�

Process Engineering Group, IIM-CSIC, Eduardo Cabello 6, 36208 Vigo, Spain

Received 15 June 2005; received in revised form 18 November 2005; accepted 28 November 2005

Available online 10 January 2006

Abstract

Many phenomena such as neuron firing in the brain, the travelling waves which produce the heartbeat, arrythmia and fibrillation in the

heart, catalytic reactions or cellular organization activities, among others, can be described by a unifying paradigm based on a class of

nonlinear reaction–diffusion mechanisms. The FitzHugh–Nagumo (FHN) model is a simplified version of such class which is known to

capture most of the qualitative dynamic features found in the spatiotemporal signals. In this paper, we take advantage of the dissipative

nature of diffusion–reaction systems and results in finite dimensional nonlinear control theory to develop a class of nonlinear feedback

controllers which is able to ensure stabilization of moving fronts for the FHN system, despite structural or parametric uncertainty. In the

context of heart or neuron activity, this class of control laws is expected to prevent cardiac or neurological disorders connected with

spatiotemporal wave disruptions. In the same way, biochemical or cellular organization related with certain functional aspects of life

could also be influenced or controlled by the same feedback logic. The stability and robustness properties of the controller will be proved

theoretically and illustrated on simulation experiments.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear reaction–diffusion systems; Excitable medium; Robust nonlinear control; FitzHugh–Nagumo; Moving front stabilization; Limit

cycles control; Reduced order models; Cardiac disorders; Neuron transmission

1. Introduction

The spatiotemporal evolution of chemical or electro-chemical signals is at the origin of many biologicalphenomena related with cell growth and distribution aswell as with cell communication (see Murray, 2002b;Stelling et al., 2004; Jin et al., 2005). At the level of tissuesand organs, neurological activity or cardiac cycles areamong the best known examples of relevant biologicalphenomena also induced by spatiotemporal changes in ionconcentrations. In this way, nervous signals are transmittedin the form of periodic flat pulses (fronts) travelling alongneural axons and tissues (Hodgkin and Huxley, 1952).Normal heart activity is also sustained by regular electro-chemical fronts which being produced by the heart’s

e front matter r 2005 Elsevier Ltd. All rights reserved.

i.2005.11.030

ing author. Tel.: +34986 231 930x251;

2762.

ess: antonio@iim.csic.es (A.A. Alonso).

natural peacemaker spread throughout the cardiac tissueand induce contraction (Fenton and Karma, 1998;Witkowski et al., 1998).Many nervous or cardiac disfunctions are originated by

dynamic instabilities that cause front disruption and evenbreakup. Heart arrhythmia (Witkowski et al., 1998) is oneof such disorders produced by a regular front being brokeninto a wandering spiral. In the last instance and after achain breaking process, the spiral waves would transforminto a disorganized set of multiple spirals continuallycreated and destroyed, being this pattern characteristic offibrillation (Keener, 2004). The same class of mechanisms isresponsible of disruptions in nervous signals leading toneurological disorders as observed by Gorelova and Bures(1983) and Dahlem and Muller (2000).In the purpose of devising means for preventing such

disorders it seems reasonable to find a unifying mechanisticapproach able to describe the same class of behaviorobserved in such a varied class of biological systems. As we

ARTICLE IN PRESSC. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306296

will discuss in the next paragraphs, such activity hasreceived considerable attention over the last decades. Inaddition, it is also desirable to investigate ways ofinteracting with the system through external feedbackcontrol systems so to prevent instabilities and therefore topreserve a normal biological activity. The scope of thepresent contribution must be inscribed into this secondaspect.

The underlying mechanisms that govern the spatiotem-poral evolution of this type of signals in many differentbiological contexts are now well understood and des-cribed by a general paradigm based on the interplaybetween diffusion and reaction. The essentials of thisparadigm were first proposed by Turing in 1952 in aseminal article that set up the chemical basis of morpho-genesis (Turing, 1952). Decades later, experimental workin chemical systems such as the Belousov–Zhabotinskiior catalytic driven surface reactions confirmed this hypo-thesis by being able to predict a rich variety of stationaryas well as oscillatory spatial patterns as discussed in, forinstance, Zimmermann et al. (1997), Beaumont et al.(1998), Lebiedz and Brandt-Pollmann (2003) or Lebiedzand Maurer (2004).

The reaction–diffusion (RD) paradigm is also central inmodelling a number of physiological systems such as thosedescribing neuron communication (Murray, 2002a), cardi-ac rhythms (Fenton and Karma, 1998) or visual perceptionin the retina (Gorelova and Bures, 1983). One such modelis that proposed by Hodgkin and Huxley (1952) to describethe neuron firing in the nerve axon of the giant squid andits subsequent simplified versions, among which the so-called FitzHugh–Nagumo (FHN) model (FitzHugh, 1961;Nagumo et al., 1962) is probably the best known. Slightvariations of this model have been employed to representtravelling waves that induce the heartbeat or the formationof spirals and irregular fronts, responsible of arrhythmia orfibrillation phenomena of the heart.

This paradigm has also been extensively employed tounderstand the routes that drive these systems to instabil-ity. In this way, dynamic analysis of diffusion–reactionsystems and in particular the FHN system, has been thesubject of intensive research, especially in what refers to thedetection of instability conditions and bifurcation analysisleading to moving fronts, spiral waves and patternformation (Zimmermann et al., 1997; Beaumont et al.,1998; Gear et al., 2002; Sweers and Troy, 2003; Bouzat andWio, 2003).

Regarding work oriented to the control and stabilizationof spatiotemporal fronts, a number of simple, thoughefficient, feedback control schemes have been recentlyproposed in the context of biological systems (Pumir andKrinsky, 1999; Peletier et al., 2003; Zykov and Engel, 2004)to ‘‘unpin’’ spiral waves or to command the evolution ofmeandering spiral waves. In Rappel et al. (1999) adistributed feedback controller has been proposed tostabilize the spatiotemporal evolution of electrical wavesin cardiac tissue.

Alternatively, and in the context of chemical systems,Shvartsman and Kevrekidis (1998) and Shvarstman et al.(2000) made use of a particular reduced order representa-tion of the original one dimensional distributed FHNsystem to carry out bifurcation analysis. The reduced orderdynamic model was also employed to design a simplefeedback controller that drove the system to a pre-definedlimit cycle. In Smagina et al. (2002) the authors made useof the Gershgorin theorem to construct a finite dimensionalregulator to stabilize front solutions of RD systems andthey applied it to an one-dimensional version of the FHNsystem.On the other hand, theoretical work in the control of

nonlinear diffusion–reaction systems has been mostlyfocused on the stabilization of given stationary patternsby techniques which made use of the dissipative nature ofthis class of systems and results in nonlinear control theory(Christofides, 2001; Alonso and Ydstie, 2001). Thisapproach was extended by Alonso et al. (2004) to developrobust nonlinear controllers which were able to stabilizearbitrary modes in diffusion–reaction systems.In this paper, we adapt the results developed in the

context of chemical systems, to propose a class of feedbackcontrollers for the FHN system which will preserve thestability of inhomogeneous patterns despite structural andparametric uncertainties. In the context of biologicalphenomena related with cardiac or nervous activity, thesecontrollers should be understood as external electro-chemical devices able to compensate for signal disruption.The required physical devices needed for implementation,such as sensors to reconstruct or identify the field andactuators to interact with the system will depend on eachparticular application. However, the elements neededshould not differ from those employed in other controlschemes previously discussed, for instance, in the context ofcardiac rhythm or neuron activity control. Nevertheless,and in order to keep generality, in this work, emphasis willbe placed not on the electronic construction details of suchcontrollers but on the underlying robust feedback logicthat the device should have in. In other words, emphasis isplaced on the logic which relates the inputs (or measure-ments) containing the dynamic information of the frontwith actions to take so to ensure stabilization despite lackof detailed (parametric and structural) knowledge aboutthe particular diffusion–reaction mechanism.To that purpose, we take advantage of the dissipative

nature of diffusion–reaction systems and make use ofresults on nonlinear control of finite dimensional systems(Khalil, 1996) to construct a class of robust feedbacknonlinear controllers ensuring limit cycle stabilization,which on a spatially distributed domain, translates intofront stabilization.From this point of view, it is our hope that the results we

have in hand, although theoretical in nature, will con-tribute to open ways to design efficient controllers for awide range of biological systems exhibiting front-typebehavior. Ultimately, these controllers could be used in the

ARTICLE IN PRESSC. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306 297

biomedical field to maintain stable signal activity, thuscorrecting or preventing certain cardiac or neural dis-orders.

In addition, it must be pointed out that our results stateconditions to be satisfied by a given robust feedbackmechanism. Therefore, and although not claiming thatintrinsic robust feedback mechanisms should necessarilyshare the same structure as the one proposed here, theresults presented could shed some light and bring clues inidentifying possible intrinsic feedback mechanisms indifferent spatiotemporal biological phenomena with theproperty of preserving robustness.

The work is structured as follows: in Section 2, wepresent a brief description of the FHN model, highlight itsparametric sensitivity through numerical simulations, andstate the control problem we will be dealing with. Next, inSection 3, we demonstrate that a finite (and low)dimensional approximation of FHN system is alwayspossible to be found and show that such representation isin fact a good approximation of the original infinitedimensional system. Finally, we describe the control law inSection 4, demonstrate its stabilizing and robustnessproperties at a theoretical level, and illustrate its perfor-mance on a numerical simulation experiment.

2. Description of the reaction–diffusion system and control

problem statement

In this work we will consider a 2D version of the classicalFHN model (Murray, 2002b). The spatial domain coversthe square surface V ¼ fðx; yÞ=� 1oðx; yÞo1g withboundary B ¼ fðx; yÞ=ðx ¼ �1 8y 2 ½�1; 1�Þ; ðy ¼ �1 8x 2½�1; 1�Þg and the model equations consist of a pair ofcoupled (diffusion–reaction) partial differential equations(PDE) of the form

qv

qt¼ k

q2vqx2þ

q2vqy2

� �þ f ðvÞ � wþ p; f ðvÞ ¼ v� v3, (1)

qw

qt¼ dk

q2w

qx2þ

q2wqy2

� �þ evþ gðwÞ; gðwÞ ¼ �eðg1wþ g0Þ,

(2)

where v and w represent the fast and the slow variable,respectively. In describing cardiac or neural activity, forinstance, the fast variable, also referred to as the activator, isdirectly related to the membrane potential, whereas w, alsoknown as the inhibitor, collects the contributions of ionssuch as sodium or potassium to the membrane current(Murray, 2002b). Parameters k and d are employed torepresent the diffusion coefficients associated to v and w,being d the diffusion ratio between the activator andinhibitor. In addition, e denotes the time-scale ratio atwhich reactions take place and g0 and g1 are given kineticparameters. p in Eq. (1) represents the control variablewhich physically might correspond with a set of spatiallydistributed electrodes supplying predefined electric currents.

In the context of biochemical or cell organization controlproblems, p would represent the effect on the system of aspatially distributed drug delivering mechanism.Finally, system’s description is completed with the

following initial and boundary conditions:

vð0; x; yÞ ¼ v0; wð0;x; yÞ ¼ w0, (3)

dv

dn

����B

¼ 0;dw

dn

����B

¼ 0, (4)

with n being a unitary vector pointing outwards thesurface.As discussed by many authors before (see for instance

Shvartsman and Kevrekidis, 1998 and references therein)the FHN system has a rich variety of dynamic behaviors asits solution is highly dependent on the values of thediffusion and kinetic parameters. In order to illustrate thisfact and to motivate the control problem we are dealingwith in this contribution, system (1)–(4) has been numeri-cally solved by a finite element discretization scheme, withthe following front-type initial conditions:

v0 ¼1 if � 1pxp� 0:9; 8y

0 if � 0:9oxp1; 8y

(and w0 ¼ 0 8 ðx; yÞ,

for two different combinations of system parameters:Case 1: With parameter values: e ¼ 0:017; g0 ¼ �0:03;

g1 ¼ 2; d ¼ 4; k ¼ 0:01, the solution has the form of aplane front which keeps bouncing off some left and rightlimits in the x-axis. We will refer to this solution—illustrated in Fig. 1(a)—as the ‘‘front-type’’ behavior.

Case 2: With parameter values: e ¼ 0:03; g0 ¼ 0;g1 ¼ 2; d ¼ 2:5; k ¼ 10�4, the initial front breaks into alarge number of irregular oscillating fronts as shown in Fig.1(b). Due to the form of these fronts we will refer to thissolution as the ‘‘fingerprint’’ behavior.In the context of heart activity (see for instance the web

page of the center of arrhythmia research (http://arrhyth-mia.hofstra.edu/) for a comparison with experiments), case1 will be associated with normal heart activity while case 2corresponds with the well known ‘‘squirming worms’’behavior characteristic of the fibrillation (Murray, 2002a).Similar patterns have been observed in other biochemicaland biological systems (Gorelova and Bures, 1983; Lebiedzand Brandt-Pollmann, 2003; Jin et al., 2005).Figs. 1(a) and (b) are representative of the two extreme

class of behaviors exhibited by the system. In this context,the control problem we will refer to as the robust front

stabilization can be stated as follows:Design a feedback control scheme that by coupling

measurements of the fields and actuations will be able to

produce and to maintain fronts on a system exhibiting a

fingerprint-type behavior. Furthermore, the feedback control

law must be robust, or in other words ‘‘do the job’’ without

knowledge of the actual values of the parameters or the

structure of the reaction terms.

ARTICLE IN PRESS

Fig. 1. Snapshots corresponding to different solutions in the FHN system. (a) ‘‘Front’’ behavior. (b) ‘‘Fingerprint’’ behavior.

C. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306298

In solving this problem, we will essentially adapt themethodology developed by Christofides (2001) and Alonsoand Ydstie (2001) which takes advantage of the dissipativenature of diffusion–reaction systems, to develop a class ofrobust stabilizing nonlinear controllers based on lowdimensional approximations of the original PDE set. Thisis the purpose of the following sections.

3. Low dimensional approximation of the

FitzHugh–Nagumo system

In this section, we first demonstrate that it is alwayspossible to obtain a finite dimensional model whichcaptures the key dynamic features of the original spatiallydistributed FHN. Secondly, we show through numericalsimulations that for the ‘‘front’’ case this representation is,in fact, low dimensional.

3.1. Finite dimensional approximation via spectral

decomposition

Let us consider the reference ðv�;w�Þ, and define thefields in deviation form as v ¼ v� v� and w ¼ w� w� sothat system (1)–(4) is rewritten as

qv

qt¼ k

q2vqx2þ

q2vqy2

� �þ f ðvÞ � wþ p; f ðvÞ ¼ f ðvÞ � f ðv�Þ,

(5)

qw

qt¼ dk

q2w

qx2þ

q2wqy2

� �þ gðwÞ þ ev; gðwÞ ¼ �eðg1wÞ, (6)

vð0;x; yÞ ¼ v0; wð0;x; yÞ ¼ w0, (7)

dv

dn

����B

¼ 0;dw

dn

����B

¼ 0. (8)

As it was shown in Alonso et al. (2004) from standardresults on Hilbert spaces (Courant and Hilbert, 1989), thefields in (5)–(8) can be described by a convergent infiniteseries of the form

vðx; y; tÞ ¼X1i¼1

cviðtÞfiðx; yÞ; wðx; y; tÞ ¼X1i¼1

cwiðtÞfiðx; yÞ,

(9)

where the spatially dependent functions fiðx; yÞ (eigenfunc-tions) are solutions of the following eigenvalue problem:

Df‘m ¼ �l‘mf‘m, (10)

df‘mdn

����B

¼ 0 (11)

being l‘m the corresponding (positive) eigenvalue asso-ciated to each eigenfunction f‘m and D ¼ ððq2=qx2Þþ

ðq2=qy2ÞÞ. For a rectangular domain, the analytical solutionof (10) and (11) is known and has the form

f‘m ¼

1ffiffiffiffiffiffiffiffiLxLy

p cos ‘pðx�x0Þ

Lx

� �cos

mpðy�y0ÞLy

� �if ‘ ¼ m ¼ 0;ffiffi

2pffiffiffiffiffiffiffiffiLxLy

p cos ‘pðx�x0Þ

Lx

� �cos

mpðy�y0ÞLy

� �if ‘ ¼ 0 or m ¼ 0;

2ffiffiffiffiffiffiffiffiLxLy

p cos ‘pðx�x0Þ

Lx

� �cos

mpðy�y0ÞLy

� �otherwise;

8>>>>><>>>>>:

(12)

l‘m ¼ p2‘2

L2x

þm2

L2y

!; ‘;m ¼ 0; 1; 2; 3; . . . , (13)

where ðLx;LyÞ are the lengths of the rectangle edges andðx0; y0Þ are the coordinates of the left-bottom point of therectangle, which in our case correspond with Lx ¼ Ly ¼ 2and x0 ¼ y0 ¼ �1. One interesting property of the eigen-values is that l‘m !1 as ‘;m!1 (see Eq. (13)). Thisfact allows us to order the eigenvalues (and therefore their

ARTICLE IN PRESSC. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306 299

corresponding eigenfunctions) along the set of naturalnumbersN so that liXlj for i4j. In this way, we define theeigenset ðS;L;NÞ as that with elements being SðDÞ ¼ffigi2N and LðDÞ ¼ fligi2N. In particular, the set ofeigenfunctions (12) form a complete orthonormal basisset on a Hilbert space equipped with the following innerproduct and L2 norm:

hg; hiV ¼

ZV

gTh dV; kgkV ¼ ðhg; giVÞ1=2 (14)

so that

hfi;fjiV ¼1 if i ¼ j;

0 if iaj:

((15)

Before proceeding with the finite dimensional approxima-tion, let us define for convenience the following subsets ofnatural numbersNa andNb, withNa being a finite subsetof arbitrary natural numbers and Nb ¼ NnNa, itscomplement. These subsets will allow us to partition theeigenset ðS;L;NÞ in two disjoint sets ðSa;La;NaÞ andðSb;Lb;NbÞ with Sa ¼ ffigi2Na

, La ¼ fligi2Naand Sb and

Lb their corresponding complements.By defining a particular set Na as that containing the

first n natural numbers, we can split fields (9) as follows:

vðx; y; tÞ ¼Xi2Na

cviðtÞfiðx; yÞ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}va

þXi2Nb

cviðtÞfiðx; yÞ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}vb

, (16)

wðx; y; tÞ ¼Xi2Na

cwiðtÞfiðx; yÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}wa

þXi2Nb

cwiðtÞfiðx; yÞ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}wb

. (17)

Since, in addition, f ðvÞ is Lipschitz, and therefore bounded,the previous partition can be extended to the nonlinearterm, so that

f ¼ f a þ f b ¼Xi2Na

cfifi þXi2Nb

cfifi.

With these preliminaries, we are now in the position ofconstructing a finite dimensional approximation of thedistributed FHN system by projecting (5) and (6) over theset S of eigenfunctions. To that purpose, we first note thatfrom (10) and (15), it follows for v, and similarly for w, that

hFb; viV ¼ Fb;X1i¼1

cvifi

* +V

¼ cvb,

hFb;DviV ¼ Fb;X1i¼1

cviDfi

* +V

¼ Fb;X1i¼1

�licvifi

* +V

¼ �Ubcvb,

where cvb is a vector containing the modes of the subsetðSb;Lb;NbÞ, Fb and Ub are matrices containing theeigenfunctions of Sb and eigenvalues of Lb, respectively.

Projecting Eqs. (5) and (6), with p ¼ 0, over the set Sb thenleads to

dcvb

dt¼ �kUbcvb � cwb þ hFb; f biV, (18)

dcwb

dt¼ �kdUbcwb þ ecvb � eg1cwb. (19)

In order to show that it is always possible to obtain a finitedimensional model which captures the key features of theoriginal spatially distributed FHN, we will combine theresults which just have been discussed, with Lyapunovarguments. In this way, we will show that for a given subsetNa with a sufficiently large number n of elements, themodes associated to the subset ðSb;Lb;NbÞ are exponen-tially stable, or in other words, that they will convergeexponentially fast to zero.Let us define a quadratic function Bb ¼

12ðecTvbcvb þ

cTwbcwbÞ and compute its time derivative along trajectories(18) and (19) so that

dBb

dt¼ ecTvb

dcvb

dtþ cTwb

dcwb

dt¼ �k ecTvbUbcvb

�þdcTwbUbcwb þ

eg1k

cTwbcwb

�þ ehvb; f biV. ð20Þ

As mentioned before, the nonlinear function is Lipschitz sothere exists a positive parameter m so that hvb; f biVpmhvb; vbiV ¼ mcTvbcvb. Thus, by choosing l‘ ¼ minl Lb, weobtain

dBb

dtp� kl‘ e 1�

mkl‘

� �cTvbcvb þ dþ

eg1kl‘

� �cTwbcwb

� . (21)

Since, li !1 as i!1, there exists a large enough n

which makes m=kl‘o1. For this value we can always finda positive parameter r, so that rp inf ½1

2ð1� ðm=kl‘ÞÞ;

12ðdþ ðeg1=kl‘ÞÞ�. Therefore, inequality (21) can be rewrit-ten as

dBb

dtp� kl‘rBb (22)

and the result follows since by Gronwall–Bellman Lemma(Khalil, 1996) we have that BbðtÞpBbð0Þ expð�kl‘rtÞ andthus cvb ! 0 at a exponential rate. Furthermore, the largerthe value of n, the larger the value of l‘, this implying fasterconvergence rates for the corresponding modes which inturns implies shorter relaxation times.This property, which is characteristic of dissipative

systems, makes it always possible to find a large enoughnumber n of elements in ðSa;La;NaÞ so the contribution ofthe modes associated to ðSb;Lb;NbÞ to the solution isnegligible in the large (slow) time-scale. Consequently, thefield can be approximated by truncated a series of the form

vðx; y; tÞ ffi va ¼Xi2Na

cviðtÞfiðx; yÞ;

wðx; y; tÞ ffi wa ¼Xi2Na

cwiðtÞfiðx; yÞ.

ARTICLE IN PRESSC. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306300

3.2. Reduced order model of the FHN system

The reduced order representation to be employed incontrol design is obtained by projecting Eqs. (5) and (6) onthe finite set of eigenfunctions Sa ¼ ffigi2Na

so to obtainthe following ODE system:

_cva

_cwa

" #¼�kUa �I

eI �kdUa

!cva

cwa

" #þ

Fa

Ga

" #þ

pa

0

� ,

(23)

where Ua is a diagonal matrix containing the eigenvalues ofLa, I is the identity matrix, whereas Fa, Ga and pa

correspond with the projection of the nonlinear andcontrol terms, respectively,

Fa ¼ hFa; f iV; Ga ¼ hFa; giV; pa ¼ hFa; piV

with Fa ¼ ½fNað1Þ;fNað2Þ

; . . . ;fNaðnÞ�. The initial conditions

for Eq. (23) now become

c0va ¼ hFa; v0iV; c0wa ¼ hFa;w0iV.

The degree of accuracy of the reduced approximation isillustrated in Fig. 2 for an increasing number of dimen-sions. As it can be seen in Fig. 2(a), projection over theeight most significant modes is enough to reproduce thequalitative aspects of the front as compared with the finiteelement solution reproduced in Fig. 1(a). Increasing thedimension of the ODE set improves the approximation asdepicted in Figs. 2(b) and 2(c). In fact, projection over themost dominant 20 eigenfunctions essentially reproduces the‘‘real front’’. This model, as compared with the finiteelement discretization alternative implies a dimensionreduction of two orders of magnitude.

Finally, it must be noted that the front developedcorresponds with a limit cycle-type evolution whentrajectories are represented in either the v and w L2 norms(Fig. 3(a)) or in terms of the eigenmodes (Fig. 3(b)). Thisfact suggests the statement of the robust front stabilizationproblem, as that of limit cycle stabilization. Such problemwill be discussed next.

4. Front stabilization in the presence of uncertainties

In designing the control law, we make use of results innonlinear control of finite dimensional systems (see forinstance Khalil, 1996) and extend them to infinite dimen-sional systems by employing the techniques in Alonso et al.(2004). First, a reference is defined which corresponds withthe front-type, or equivalently, the limit cycle behavior.The system is then expressed in error form with respect tosuch a reference. This error PDE equation, which containsuncertain terms associated with the diffusivity and thereaction kinetics, is combined with a suitable Lyapunovfunction to construct a nonlinear control law which ensuresboundedness of the error despite uncertainty, i.e. provingrobustness.

A schematic representation of the dynamic behavior ofthe system under such a controller is depicted in Fig. 4.Given a certain state in deviation form, the control lawdrives the error exponentially fast to a region arbitrarilyclosed to the set-point or reference. Once the state enters inthis region it will remain there in the future; that is, theresponse becomes ultimately bounded. It is worth mention-ing that the size of the region can be made arbitrarily smallby appropriate tuning, although at the expenses of largercontrol efforts.

4.1. The control law

As we did in Section 3, let us partition the eigensetassociated to our system into a finite set ðSa;La;NaÞ,whereNa contains as elements the indexes of the modes wewant to drive into the desired limit cycle (front-typebehavior), and define its complement as ðSb;Lb;NbÞ. Inaddition, let us separate the fields and control, respectively,as v ¼ va þ vb, w ¼ wa þ wb, and p ¼ pa þ pb.The controller now is designed so that the modes in Na

converge to those describing the limit cycle, which isexpressed as

dc�va

dt¼ �k�Uac�va þ hFa; f

�aiV � c�wa þ hFa; p

�aiV, (24)

while the rest, which are collected in Nb, are forced torelax exponentially fast to reach the following reference:

c�vb ¼ 0; c�wb ¼ 0. (25)

Projecting Eq. (5) over the sets ðSa;La;NaÞ andðSb;Lb;NbÞ we obtain

dcva

dt¼ �kUacva þ hFa; f aiV � cwa þ hFa; paiV, (26)

dcvb

dt¼ �kUbcvb þ hFb; f biV � cwb þ hFb; pbiV (27)

subtracting Eqs. (24) and (25) from Eqs. (26) and (27),respectively, we arrive at an expression for mode evolutionin deviation form

dcva

dt¼ �kUacva þ wUac�va þ hFa; f aiV � cwa þ hFa; paiV,

(28)

dcvb

dt¼ �kUbcvb þ hFb; f biV � cwb þ hFb; pbiV, (29)

where w ¼ k� � k.As we will see in this section, the control objectives are

attained by a control law of the form

pb ¼wb � ovb � Z vb

kvbkVif ZkvbkVXE;

wb � ovb � Z2 vb

E if ZkvbkVoE;

8<: (30)

pa ¼wa � b�va � Z� va

kvakVif Z�kvakVXy;

wa � b�va � ðZ�Þ2 va

y if Z�kvakVoy;

8<: (31)

ARTICLE IN PRESS

Fig. 2. Snapshots obtained by using the reduced order model with (a) 8 eigenfunctions, (b) 15 eigenfunctions and (c) 20 eigenfunctions.

-0.4 -0.2 0 0.2 0.4 0.6

1.4

1.45

1.5

1.55

1.6

1.65

cv1

c v2

1.72 1.725 1.73 1.735 1.74 1.7450.2

0.21

0.22

0.23

0.24

0.25

0.26

|| v|| v

||w||

v

(a) (b)

Fig. 3. Limit cycle behavior for the FHN system. (a) On L2 norm of fields v and w. (b) On the modes corresponding to the two smallest eigenvalues of v.

C. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306 301

ARTICLE IN PRESS

Time

Open loop

Exponentially stable

Ultimately bounded

Stat

e

g(θ)

System under control

Setpoint

Fig. 4. A schematic representation of the system dynamic evolution under control law (30) and (31).

Table 1

Functions and parameters employed in the control law

E y w lq o

0.01 0.01 9:9� 10�3 315.17 0.01

b Z Z� b�

maxVð1þ 3jvajjv�ajÞ kvkV 0:01wlqkv

�akV bþ 0:01b

C. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306302

where Z, b� and Z� are given functions bounding theuncertain terms (see Table 1 for details). o, E and y aretuning parameters which control the convergence rate andthe size of the region where the fields will remain ultimatelybounded. The fields va, wa, vb and wb are computed frommeasurements of the fields vðt;x; yÞ and wðt; x; yÞ as follows:

va ¼Xi2Na

ficvi; wa ¼Xi2Na

ficwi; vb ¼ v� va,

wb ¼ w� wa.

In order to show that the control law (30) and (31) will infact enforce limit cycle stabilization, we make use ofLyapunov arguments. Define a quadratic function Bb ¼12ðcTvbcvbÞ and compute its time derivative along (29) so that

dBb

dt¼ �kcTvbUbcvb þ hvb; f biV � cTvbcwb þ hvb; pbiV. (32)

Defining l1b ¼ minðLbÞ, using the Schwartz inequality(Courant and Hilbert, 1989) and substituting the expres-sion given in Eq. (30), we have the following two cases:

(1)

For ZkvbkVXs, in Eq. (32) becomes

dBb

dtp� kl1b2Bb þ kvbkVkf bkV � o2Bb � ZkvbkV.

Since we can choose Z to be greater than kf bkV wehave

dBb

dtp� 2ðkl1b þ oÞBb

which by using the Gronwall–Bellman Lemma (Khalil,1996) implies that Bb (and thus cvb) exponentially tendto zero as t!1.

(2)

For ZkvbkVoE, Eq. (32) becomes

dBb

dtp� 2ðkl1b þ oÞBb þ ZkvbkV � Z2

kvbk2V

Efunction cb ¼ ZkvbkV � Z2ðkvbk

2V=EÞ has a maximum

value of cb ¼ E=4 so that

dBb

dtp� 2ðkl1b þ oÞBb þ

E4) lim

t!1BbðtÞ ¼

E8ðkl1b þ oÞ

which implies that Bb is ultimately bounded.

The same line of arguments can be employed in astraightforward manner to show that (31) stabilizes modescva. After some manipulations in Eq. (28) we obtain (see theAppendix for details)

dBa

dtp� kl1a2Ba þ Z�kvakV þ bkvak

2V � cTvacwa þ hva; paiV,

where Z� ¼ wlqkv�akV with lq ¼ maxðLaÞ and b4maxV

ð1þ 3jvajjv�ajÞ. Substituting Eq. (31) in the above inequality,

we have again two cases

(1)

For Z�kvakVXy:

dBa

dtp� kl1a2Ba þ ðb� b�Þkvak

2V

since we can choose b� to be greater than b weconclude, as before, that Ba (and then cva) exponen-tially tend to zero.

(2)

For Z�kvakVoy

dBa

dtp� kl1a2Ba þ ðb� b�Þkvak

2V þ Z�kvakV

� ðZ�Þ2kvak

2V

y

ARTICLE IN PRESS

Fig. 5. v-field evolution under the control law (30) and (31). (a) Before entering the control, (b) and (c) in the transition period, (d) when the field is

ultimately bounded.

C. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306 303

function ca ¼ Z�kvakV � ðZ�Þ2ðkvak

2V=yÞ have a max-

imum value of ca ¼ y=4 so

dBa

dtp� kl1a2Ba þ

y4) lim

t!1BaðtÞ ¼

y8kl1a

which implies that Ba (and then cva) are ultimatelybounded.

4.2. Numerical simulation experiment

The effect of the control law defined by Eqs. (30) and(31) on a FHN system exhibiting a fingerprint-typebehavior (see case 2 in Section 2) is presented in Fig. 5.Under this controller, the system, which initially evolves asin the ‘‘fingerprint’’ case (Fig. 5(a)), is forced to follow areference of front type which is reached (Fig. 5(d)) after ashort transition period (Figs. 5(b) and (c)). The slight

differences between the reference (Fig. 2(c)) and thecontrolled field (Fig. 5(d)) are caused by the presence ofuncertainty. Such divergences could be arbitrarily reducedby setting up stricter limits on the y and � parameters, asshown previously, although at the expenses of an increas-ing control effort.The parameters and functions that we have used in these

controls are summarized in Table 1. Fig. 6 illustrates theefficiency of the control law on a mode representation. Forclarity reasons, only a few (the most significant) modes wererepresented, four associated to the set ðSa;La;NaÞ (contin-uous lines) and one associated to the set ðSb;Lb;NbÞ (dashedline). Figs. 6(a) and (b) represent the open loop evolution ofmodes for the ‘‘fingerprint’’ and the ‘‘front’’ (reference) cases,respectively. The effect of the control law on the selectedmodes can be seen in Fig. 6(c). Once the controller is switchedon at 40 time units, there is a transition period after which

ARTICLE IN PRESS

0 50 100 150 200 -1

-0.5

0

0.5

1

1.5

2

Time

c v c v

c v

(a)

0 50 100 150 200 -1

-0.5

0

0.5

1

1.5

2

Time(b)

0 50 100 150 200 -1

-0.5

0

0.5

1

1.5

2

Time(c)

0 50 100 150 200

Time(d)

0.5

1

1.5

2

Con

trol

nor

m

p1

p2

p

Fig. 6. Evolution of some modes of the FHN system. (a) Open loop behavior. (b) Oscillatory reference trajectory. (c) Behavior of the system under

control. The continuous lines represent modes belonging to the subset ðSa;La;NaÞ and dashed lines modes belonging to ðSb;Lb;NbÞ. (d) Control effort.

C. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306304

modes associated to ðSa;La;NaÞ converge to the selected(oscillating) reference while the modes associated toðSb;Lb;NbÞ vanish. Finally, the resulting control effort forp, pa and pb in the L2 norm is presented in Fig. 6(d).

5. Conclusions

In this work, a robust nonlinear feedback controller ableto stabilize inhomogeneous patterns (limit cycle) wasdesigned for a class of reaction–diffusion systems which,based on the FHN model, exhibits the characteristic ofspatiotemporal patterns observed in many biologicalsystems at the level of cell organization and tissues (e.g.heart or neural activity). The proposed control law ensuresstabilization even in the presence of structural or para-metric uncertainties. The underlying control logic was builtbased on classical results in nonlinear robust control of

finite dimensional systems and adapted to infinite dimen-sional systems by exploiting the dissipation properties ofreaction–diffusion systems. These properties allowed us toapproximate the limit cycle (reference trajectory) by a finitenumber of representative modes. The control law wasdesigned so to drive the corresponding modes of the systeminto the desired limit cycle while stabilizing those relatedwith abnormal dynamics. The stability of the system undercontrol was proved first theoretically and then throughnumerical simulations.

Acknowledgements

The authors acknowledge financial support receivedfrom the Spanish Government (MCyT Projects PPQ2001-3643 & DPI2004-07444-C04-03) and Xunta de Galicia(PGIDIT02-PXIC40209PN).

ARTICLE IN PRESSC. Vilas et al. / Journal of Theoretical Biology 241 (2006) 295–306 305

Appendix

In this section we show how to obtain bounds on thesecond and third terms in Eq. (28), required to design thestabilizing control law.

First, let us consider the quadratic function Ba ¼12ðcTvacvaÞ and compute its time derivative along Eq. (28) sothat

dBa

dt¼ � kcTvaUacva þ wcTvaUac�va þ hva; f aiV

� cTvacwa þ hva; paiV. ð33Þ

The second RHS term of Eq. (33) is bounded as follows:

wcTvaUac�va ¼ wXi2Na

cvilic�vipjwjlq

Xi2Na

jc�vicvij,

where lq ¼ maxðLaÞ. Using the Holder inequality

jwjlq

Xi2Na

jc�vicvijpjwjlqkc�vak2kcvak2

¼ jwjlq

Xi2Na

jc�vij2

!1=2 Xi2Na

jcvij2

!1=2

¼ jwjlqkv�akVkvakV,

choosing Z� ¼ jwjlqkv�akV, we have

wcTvaUac�vapZ�kvakV. (34)

On the other hand and in order to find a bound in the thirdterm of the RHS of Eq. (33):

va ¼ va þ v�a ) v3a ¼ v3a þ 3ðv�aÞ2va þ 3ðv�aÞv

2a þ ðv

�aÞ

3.

Inserting this expression in the nonlinear term f a ¼ ðva �

v3aÞ � ðv�a � ðv

�aÞ

3Þ and after some algebraic manipulations,

we obtain

vTa f apvTa vað1þ 3jvajjv�ajÞ,

so by selecting b4maxVð1þ 3jvajjv�ajÞ we have that

hva; f aiVpbkvak2V. (35)

Finally, introducing inequalities (34) and (35) in Eq. (33)and choosing l1a ¼ minðLaÞ we get the following bound onthe time derivative of Ba:

dBa

dtp� kl1a2Ba þ Z�kvakV þ bkvak

2V � cTvacwa þ hva; paiV.

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