Exponential Stability for the Schlögl System by Pyragas Feedback [original]

Vietnam Journal of Mathematics

https://doi.org/10.1007/s10013-020-00382-7

ORIGINAL ARTICLE

Exponential Stability for the Schl¨

ogl System by Pyragas

Feedback

Martin Gugat1·Mariano Mateos2·Fredi Tr¨

oltzsch3

Received: 22 July 2019 / Accepted: 24 October 2019 /

©The Author(s) 2020

Abstract

The Schl¨

ogl system is governed by a nonlinear reaction-diffusion partial differential equa-

tion with a cubic nonlinearity. In this paper, feedback laws of Pyragas-type are presented

that stabilize the system in a periodic state with a given period and given boundary traces.

We consider the system both with boundary feedback laws of Pyragas type and distributed

feedback laws of Pyragas and classical type. Stabilization to periodic orbits is impor-

tant for medical applications that concern Parkinson’s disease. The exponential stability

of the closed loop system with respect to the L2-norm is proved. Numerical examples are

provided.

Keywords Lyapunov function ·Boundary feedback ·Robin feedback ·Parabolic partial

differential equation ·Exponential stability ·Stabilization of periodic orbits ·Periodic

operation ·Stabilization of desired orbits ·Poincar´

e–Friedrichs inequality ·Delay

differential equations

Mathematics Subject Classiﬁcation (2010) 49J20 ·93B52 ·93C20

Dedicated to Volker Mehrmann on the occasion of his 65th birthday.

Martin Gugat

[email protected]

Mariano Mateos

[email protected]

Fredi Tr¨

oltzsch

[email protected]berlin.de

1Department Mathematik, Friedrich-Alexander-Universit¨

at Erlangen-N¨urnberg,

Cauerstr. 11, 91058 Erlangen, Germany

2Departmento de Matematicas E.P.I. Gij´

on Universidad de Oviedo, 33203, Gij´

on, Spain

3Institut f¨ur Mathematik, Technische Universit¨

at Berlin, Str. des 17. Juni 136,

10623 Berlin, Germany

(2020) 48:769–790

Published online: 25 February 2020

M. Gugat et al.

1 Introduction

The Schl¨

ogl system introduced in [28] is a model for chemical reactions for non-equilibrium

phase transitions that describes the concentration of a substance in dimension 1. In neurol-

ogy, the same system is known as Nagumo equation (or Newell–Whitehead–Segel equation,

see [26,29]) and models an active pulse transmission through an axon [8,25]. This sys-

tem is governed by a parabolic partial differential equation with a cubic nonlinearity that

determines three constant equilibrium states u1<u

2<u

3,whereu2is unstable.

The Sch¨

ogl model serves as a simplified model problem for more complicated equations

such as the bidomain system in heart medicine, cf. [19]. Here, the goal of stabilization is to

extinguish undesired spiral waves as fast as possible and hereafter to control the system to

a desired state. However, there are similarities between these models and it is therefore rea-

sonable to consider related questions for the Sch¨

ogl system. A finite-dimensional dynamic

compensator for the Schl¨

ogl model is designed in [3]. Extensive Monte Carlo calculations

of Schl¨

ogl’s model for chemical reactions are presented in [15].

The control functions can act in the domain (distributed control) or on its boundary.

In this paper, the problem of feedback stabilization towards periodic orbits is studied. For

instance, the construction of Lasers with a desired oscillation behavior is an interesting issue

for feedback stabilization.

A possible application of the boundary stabilization of time-periodic orbits in the

Nagumo system is the boundary stabilization of a periodic pulse transmission. Another

important motivation is the development of new treatments for Parkinson’s disease. In [17],

linear combinations of a fixed number of Dirac measures are used in experiments that are

related to the treatment of Parkinson’s disease. There are also interesting applications in

Theoretical Physics such as the control of cluster synchronization in [20] and time-delayed

stabilization of solutions to nonlinear differential equations in [9]. The existence of peri-

odic solutions of nonlinear parabolic equations is studied in [27]. Stabilization towards

desired periodic orbits is also considered in [1] for a pendulum system and in [23,24]

for discrete time chaotic systems. Optimal control problem for systems governed by the

stochastic FitzHugh–Nagumo equation with a Gaussian noise are studied in [2]. Similar

optimal control problems with recovery variable are considered in [9].

In [16] we have studied the boundary stabilization to given desired orbits with linear

Robin-feedback laws. In this new paper, we concentrate on the exponential stabilization

towards a periodic orbit with feedback laws of Pyragas type. In particular, we are inter-

ested in periodic orbits with desired oscillations. We consider both boundary and distributed

feedback control. In our analysis we study the exponential decay of the Lyapunov function

V(t)=1

2L

(u(t, x) −u(t −T,x)

)2dx, (1)

where u(t, x) denotes the system state at time tand position xand Tstands for a desired

period length.

To show that the system is exponentially stable, we verify that Vis a strict Lyapunov

function. The construction of strict Lyapunov functions for semilinear parabolic partial dif-

ferential equations has also been studied in [22]. In [22], it is assumed that the feedback

is space-periodic or the boundary conditions are chosen in such a way that the product of

the state and the normal derivative vanishes at the boundary. This assumption implies that

the boundary terms occurring after partial integration in the time derivative of the Lya-

punov function become nonpositive. For the state feedback laws that are presented in this

paper, this assumption holds. But due to the time delay in the Pyragas feedback term, the

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Exponential Stability for the Schl¨

ogl System by Pyragas Feedback

analysis from [22] is not directly applicable. Therefore a different approach is used in the

analysis: For the boundary control, a Poincar´

e–Friedrichs inequality is used to show that the

Lyapunov function is strict.

The Schl¨

ogl system has the interesting property that it allows traveling wave solutions

(i.e., uniformly translating solutions moving with a constant velocity) which have the shape

of the hyperbolic tangent (see [18]). The traveling wave solutions connect the two stable

constant stationary states u1and u3. The problem to steer associated wave fronts to rest by

distributed optimal control methods with finite time horizon was considered in [5]forthe

Schl¨

ogl model and in [6] for the FitzHugh–Nagumo system, where spiral waves occur. In

the present paper, we propose control laws that stabilize the system exponentially fast to a

periodic orbit.

In this paper, a spatially 1-d system of length Lis studied. In the reaction-diffusion

equation, the diffusion coefficient is normalized to 1. The parameter ρdetermines the size

of the reaction term.

In particular, we are interested in answering the following question: Let a state function

ube given that exhibits a stable oscillatory behavior in a fixed bounded time horizon [0,T].

Will this function approach a periodic and oscillating orbit as t→∞?

This paper has the following structure: In Section 2, a system is studied where the Pyra-

gas terms appear in the boundary conditions. The model is defined and a result about the

well-posedness is given. We give conditions that guarantee that the system converges to a

periodic orbit exponentially fast.

In Section 3, the result about feedback stabilization where the Pyragas terms appear as

distributed controls is presented. The feedback gain can be chosen in such a way that the

system converges exponentially fast to a periodic state with the desired period T. Similar

results are shown for distributed feedback laws of classical type.

For the boundary feedback law, the Pyragas terms acts through the boundary condi-

tions. If the desired T-periodic orbit is determined uniquely by these boundary traces,

under appropriate assumptions the system states converges exponentially fast to the desired

T-periodic orbit.

For the distributed feedback law, the Pyragas term acts directly in the partial differential

equation (pde, for short) whereas in the boundary conditions, desired Neumann bound-

ary traces for the T-periodic orbit are prescribed. Numerical experiments that illustrate the

behavior of the system are presented in Section 5. Section 6contains conclusions.

2 The Schl¨

ogl Model with Pyragas Boundary Feedback Control

2.1 Deﬁnition of the Schl¨

ogl Model

Let real numbers u1≤u2≤u3be given and define the polynomial

R(u) =(u −u1)(u −u2)(u −u3).

Due to its definition, Rhas the property

mR=inf

u∈(−∞,∞)R(u) > −∞,(2)

hence the derivative of Ris bounded from below. The infimum mR≤0 is attained at the

point (u1+u2+u3)/3.

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The system that is considered in this paper is governed by the semilinear parabolic partial

differential equation

ut=uxx −ρR(u) (3)

with a constant ρ>0 complemented by appropriate initial and boundary conditions. In

the reaction diffusion equation (3), the diffusion coefficient is equal to 1 and the constant ρ

determines the size of the reaction term. If ρequals zero, the reaction term vanishes and the

partial differential equation (3) models a pure linear diffusion process.

Let the length L>0, a desired period T>0, and a feedback gain γ∈Rbe given. We

consider two versions of feedback laws.

First, we consider distributed feedback of the form

ut=uxx −ρR(u) +γ(u−udesi), (4)

where udesi is a desired state function.

Second, we define also a boundary feedback law. For the stabilization of (4), for (t, x) ∈

(0,∞)×(0,L), with some real constant C≥1

2L, we consider the Pyragas–Robin boundary

conditions

ux(t, 0)=C[u(t, 0)−u(t −T,0)]+ux(t −T,0), (5)

ux(t, L) =−C[u(t, L) −u(t −T,L)

]+ux(t −T,L).(6)

In order to start the system, for t∈(−T,0)we prescribe some sufficiently regular initial

state. With the feedback laws (5)and(6), if

2ρ|mR|+2γ< 1

(see (10) in Theorem 2) the Lyapunov function V(t)defined in (1) decays exponentially.

2.2 Existence and Uniqueness of the Solutions

In [5], the well-posedness of the system governed by (3) is studied for homogeneous Neu-

mann boundary conditions. It is shown that for initial data in L∞(0,L), the system has a

unique weak solution that is continuous for t>0. If the initial state is continuous, the solu-

tion of the system is continuous for all times. In the associated theorem below, we use the

standard Sobolev space

W(0,T)=L2(0,T,H1(0,L))∩H1(0,T;H1(0,L)

)

and write QT=(0,T)×(0,L).

Let us first consider the following semilinear parabolic problem with inhomogeneous

Robin boundary conditions but without time delay:

ut(t, x) −ρuxx(t, x) +R(u(t, x)) =f(t,x) in QT,

ux(t, 0)−Cu(t, 0)=g1(t) in (0,T),

ux(t, L) +Cu(t, L) =g2(t) in (0,T),

u(0,x) =u0(x) in (0,L).

(7)

Lemma 1 Suppose that it holds K>0and u1≤u2≤u3. Then, for all f∈L2(QT),

u0∈L∞(0,L),gi∈Lp(0,T),i=1,2,p>2, the parabolic initial-boundary value

problem (7)has a unique solution uin

L∞(QT)∩W(0,T)∩C((0,T]×[0,L]).

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Exponential Stability for the Schl¨

ogl System by Pyragas Feedback

If u0∈C[0,1],thenuis also continuous on [0,T]×[0,L].

Proof We apply the standard substitution u(t, x) =eλt v(t,x) and obtain the problem

vt(t, x) −ρvxx(t, x) +λv(t, x) +e−λt R(eλt v(t,x)) =e−λt f(t,x) in QT,

vx(t, 0)−Cv(t,0)=e−λt g1(t) in (0,T),

vx(t, L) +Cv(t,L) =e−λt g2(t) in (0,T),

u(0,x) =u0(x) in (0,L).

If λis taken sufficiently large, then the function d:v→ λv+e−λt R(eλt v) is monotone non-

decreasing. Since Cis non-negative, the same holds for the function b:v→ Cv. Therefore,

the problem fits to a general semilinear parabolic problem with monotone nonlinearities that

is discussed in [32, Theorem 5.5]. This theorem ensures the existence of a unique bounded

solution vin W(0,T)with the claimed regularity properties.

Theorem 1 Let 0<T <T

<∞be given. Assume that udesi belongs to Lr(QT)with

some r>3/2and u0∈C([−T,0]×[0,L])possesses the partial (one-sided) derivatives

(u0)x(t, 0):= (u0)x(t, +0)and (u0)x(t, L) =(u0)x(t, −L) for all t∈[−T,0].Letthe

functions t→ (u0)x(t, 0)and t→ (u0)x(t, L) belong to Lp(−T,0)with some p>2.

Then the parabolic problem with delay T∈(0,T],

ut−uxx +ρR(u) =γ(u−udesi)in QT,

u(t, x) =u0(t, x) in (−T,0)×(0,L),

ux(t, 0)−Cu(t, 0)=ux(t −T,0)−Cu(t −T,0)in (0,T),

ux(t, L) +Cu(t, L) =ux(t −T,L)+Cu(t −T,L) in (0,T)

(8)

has a unique solution usuch that u|[0,T ]∈W(0,T)∩C([0,T]×[0,L]).

Proof We apply the step method that is a classical tool for proving existence and uniqueness

of solutions to delay equations. First, we solve the problem on the time interval [0,T]. Here,

we have u(t −T,x) =u0(t −T,x), hence the problem is one with given Robin boundary

data

g1(t) =(u0)x(t −T,0)−Cu0(t −T,0), g2(t) =(u0)x(t −T,L)+Cu0(t −T,L)

that fits in (7). Thanks to the assumptions imposed on u0, the functions g1and g2belong

to Lp(0,T)with p>2. Therefore and since udesi belongs to Lr(QT), Lemma 1 ensures

existence and uniqueness of uin W(0,T)∩C(QT).

Now we extend the solution uto [T,min{2T,T}]. Without stronger assumptions on the

regularity of g1,g2,andu0, we cannot expect that ux(t −T,0)and ux(t −T,L) exist

as functions on [T,2T]. However, from the boundary conditions we know that ux(t, 0)−

Cu(t, 0)=g1(t) and ux(t, L) +Cu(t, L) =g2(t) exist as real functions in Lp(0,T).

Therefore, on [T,2T]the functions ux(t −T,0)−Cu(t −T,0)=g1(t −T)and ux(t −

T,L)+Cu(t −T,L) =g2(t −T)belong to Lp(T , 2T).

In this way, on [T,min{2T,T}] the boundary conditions of (8) can be re-written as

ux(t, 0)−Cu(t, 0)=g1(t −T),

ux(t, L) +Cu(t, L) =g2(t −T),

T≤t≤min{2T,T}.

Moreover, the new initial function x→ u(T , x) is continuous on [0,L], again thanks

to Lemma 1. Therefore, we can apply Lemma 1 again, now on [T,min{2T,T}], to obtain

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M. Gugat et al.

existence and uniqueness of uuntil min{2T,T}. Repeating this procedure again, we finally

can extend the solution up to T.

Remark 1 As we have pointed out in the preceding proof, we cannot guarantee that ux(t, 0)

and ux(t, L) exist as measurable functions. We resolved this obstacle by using the boundary

conditions

ux(t, 0)−Cu(t, 0)=g1(t −(k −1)T ),

ux(t, L) +Cu(t, L) =g2(t −(k −1)T ),

on [(k −1)T , kT ],k=1,2,.... This trick avoids the explicit use of ux(t −T,0)and

ux(t −T,L) in the right-hand side of the boundary conditions. By results on maximal

parabolic regularity of [21, Theorems 5.1.17 and 5.1.20], we are able to show on [0,T]the

existence of the solution uin C1+α/2,2+α(QT)with some 0 <α<1, provided that u0and

udesi have higher regularity. Then the functions t→ ux(t, 0)and t→ ux(t, L) belong to

C(1+α)/2[0,T]. For the definition of these spaces, we refer to [21]. Analogously, we can

proceed in later time intervals.

2.3 Exponential Stability

In this section, we present our main result about the exponential stability in the L2-sense

of our system with Pyragas boundary control. A boundary feedback law is constructed that

stabilizes the system around a given desired T-periodic state udesi.

An essential tool in the analysis is the 1-d POINCAR´

E-FRIEDRICHS inequality (see also

[32] for the general case) in the following form: let L>0 be given. For all u∈H1(0,L),

the following inequality holds:

L

u2(x)dx ≤Lu(0)2+u(L)2+2L2L

(∂xu(x))2dx.(9)

Now the stabilization result for periodic orbits is given. In what follows, we use the notation

Q2T=(0,2T)×(0,L).

Theorem 2 (Exponential stability) Let a period T>0and a T-periodic state udesi ∈

H2(Q2T)be given. Assume that L>0is sufficiently small in the sense that

L2>2ρ|mR|+2γ. (10)

Let a feedback parameter C≥1

2Land an initial state u0∈C1([−T,0]×[0,L])be

given. Then the solution uof (4)subject to the initial condition

u(t, x) =u0(t, x)

for t∈(−T,0)and x∈(0,L)and to the boundary conditions

ux(t, 0)=C(u(t, 0)−u(t −T,0))+ux(t −T,0),

ux(t, L) =−C(u(t, L) −u(t −T,L)

)+ux(t −T,L)

for t≥0becomes T-periodic exponentially fast in the following sense: With

μ=1

L2−2ρ|mR|−2γ,

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Exponential Stability for the Schl¨

ogl System by Pyragas Feedback

the inequality

L

(u(t, x) −u(t −T,x)

)2dx ≤exp (−μ(t −T)

)L

(u(T , x) −u0(0,x)

)2dx (11)

holds for all t≥T.

There exist a T-periodic function u∗∈L2(Q2T)and a constant P0>0such that for all

t≥0we have u∗(t, ·)∈L2(0,L)and the inequality

L

0u(t, x) −u∗(t, x)2dx ≤P0exp(−μt) (12)

is satisfied for all t≥T.

Proof By Theorem 1 the state uis continuous. Hence, Vis well-defined by (1). The

definition of Vimplies that V(t)≥0. For t≥T, we can write (4) in the form

ut(·−T,·)=uxx(·−T,·)−ρR(u(·−T,·)) +γu(·−T,·)−udesi.

Hence, the pde (4) implies that for all t≥Twe have

ut(t, x) −uxt(t −T,x)−uxx(t, x) +uxx(t −T,x)

=−ρ(R(u(t, x)) −R(u(t −T,x)

)+γ (u(t, x) −u(t −T,x)) (13)

in the sense of the solution of (4) that is given in Theorem 1. Notice that the term with

udesi cancels out by subtracting the pdes for u(t, x) and u(t −T,x). We multiply (13)by

[u(t, x) −u(t −T,x)]and integrate to obtain for t0>T

t0

TL

[u(t, x) −u(t −T,x)][ut(t, x) −ut(t −T,x)]−[uxx(t, x) −uxx(t −T,x)]dxdt

=t0

TL

−ρ[u(t, x) −u(t −T,x)]R(u(t,x)) −R(u(t −T,x))

+γ[u(t, x) −u(t −T,x)]2dxdt.

For initial data u0in H2((−T,0)×(0,L)), integration by parts yields

L

2[u(t, x) −u(t −T,x)]2

t=Tdx +t0

TL

[ux(t, x) −ux(t −T,x)]2dxdt

=t0

[u(t, x) −u(t −T,x)][ux(t, x) −ux(t −T,x)]|L

x=0dt

+t0

TL

−ρ[u(t, x) −u(t −T,x)]R(u(t, x)) −R(u(t −T,x))

+γ[u(t, x) −u(t −T,x)]2dxdt.

Due to (2), for all v1,v2∈(−∞,∞)it holds

(v2−v1)(R(v2)−R(v1)) ≥(v2−v1)2mR. (14)

The definition of Vand the boundary conditions imply

V(t

0)−V(T)+t0

TL

[ux(t, x) −ux(t −T,x)]2dxdt

≤−t0

C[u(t, L) −u(t −T,L)]2+C[u(t, 0)−u(t −T,0)]2dt

+t0

TL

(−ρmR+γ)[u(t, x) −u(t −T,x)]2dxdt.

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M. Gugat et al.

Thanks to C≥1

2L, due to the Poincar´

e–Friedrichs inequality (9) this implies

V(t

0)−V(T)≤t0

TL

0−1

2L2−ρmR+γ[u(t, x) −u(t −T,x)]2dxdt.

Now (10) implies V(t

0)≤V(T).Infact,wehave0≤V(t

0)≤V(T)−t0

TμV (τ)dτ.Now

Gronwall’s lemma implies the inequality

V(t

0)≤V(T)exp (−μ(t0−T)

Since H2((−T,0)×(0,L))is dense in L∞((−T,0)×(0,L)), the same estimate remains

true (by continuous extension) for any initial state u0in L∞(0,L). Thus Vis a strict Lya-

punov function and the assertion (11) follows. (Similar classical Lyapunov analysis can be

found for example in [10]or[11]).

Inequality (11) implies that for all t∈(0,T), the sequence (u(t +jT,·))∞

j=1is a Cauchy

sequence in L2(0,L). This can be seen as follows. Define

C0=L

0u(T , x) −u0(x)2dx.

Then for all j∈{1,2,3,...}we have

L

0u(t +jT,x)−u(t +(j −1)T , x)2dx ≤C0exp −μ

2(t +(j −1)T ).

We use the notation L

0(f (x))2dx =fL2(0,L).

For all k∈{0,1,2,...}the triangle inequality implies

u(t +(j +k)T, ·)−u(t +(j −1)T , ·)L2(0,L)

≤



l=0

u(t +(l +j)T,·)−u(t +(l +j−1)T , ·)L2(0,L)

≤C0



l=0

exp −μ

2(t +(l +j−1)T )

≤C0

exp −μ

2(t +(j −1)T )

1−exp −μ

2T.

Hence, for all t∈(0,T), the sequence (u(t +jT,·))∞

j=1converges in L2(0,L).Thereexist

limit functions u∗(t, ·)∈L2(0,L)such that, for all t∈(0,T),

lim

j→∞ L

(u(t +jT,x)−u∗(t, x))2dx =0.

Then (11) implies that u∗(t, ·)generates a T-periodic orbit. More precisely, we have

u∗(t, ·)−u(t +(j −1)T , ·)L2(0,L) ≤C0

exp −μ

2(t +(j −1)T )

1−exp −μ

2T

which implies (12).

Remark 2 Note that γ<0 can always be chosen such that (10) holds.

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Exponential Stability for the Schl¨

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Example 1 Let u1=−1, u2=0, u3=π.ThenwehaveR(u) =(u2+u)(u −π) and

mR=−

π2+π+1

3>−5.

Let L=1andC=1

2; the constants ρand γwill be specified below. Consider the desired

state udesi(t, x) =(sin(t) +sin(2t))3sin(πx) which is T-periodic with T=2π.The

corresponding Robin–Pyragas-feedback is

ux(t, 0)=ux(t −T,0)+1

2[u(t, 0)−u(t −T,0)],

ux(t, L) =ux(t −T,L)−1

2[u(t, L) −u(t −T,L)

– With the choice ρ=1

10 and γ=0, (10) holds. Theorem 2 is applicable and implies

the convergence to a T-periodic state.

– Consider now ρ=1. Then for γ≥0, the inequality (10) is not satisfied. Hence in this

case, Theorem 2 is not applicable.

–Forρ=1andγ<−9

2, the inequality (10) is satisfied and Theorem 2 is applicable.

Remark 3 The feedback strategies of this section can generate timely constant periodic

functions, because the standard solutions of the Schl¨

ogl model without time delay are con-

stant. In Sections 3and 4, we investigate the stabilization of the Schl¨

ogl model with different

stabilization strategies, that are able to generate non-constant periodic limit functions of a

desired period.

2.4 Another Application: A Parabolic Model for Gas Pipeline Flow

In this section we consider another application of the presented boundary feedback law,

namely the flow of ideal gas through a horizontal pipe of length L>0. Let numbers α>0,

q0>0 be given. Let Pdenote the pressure and q0+qthe flow rate, where we assume

that |q|/q0is sufficiently small. The number c>0 denotes the sound speed in the gas. The

following system of partial differential equations can be used as a model for the flow for

x∈[0,L]and t>0:

Pt+c2qx=0,P

x=−αq2

0+2q0q

(see [4,12,31]). From the second equation we obtain qx=− 1

2αq0(1

2P2)xx.Inorderto

obtain a single partial differential equation we insert this in the first equation. This yields

Pt=− c2

2αq01

2P2xx

Let p0(x) > 0 denote a function such that 1

2p2

0x=−αq2

0. We introduce a new variable p

by P=p0+p. If we neglect the lower order terms with px,p2

xwe obtain pt=− c2

2αq0(p0+

p)pxx. By neglecting the quadratic term this yields the linear model

pt=− c2

2αq0

p0pxx (15)

which has the same form as the Schl¨

ogl model with zero reaction term, that is ρ=0. So we

can apply the methods for boundary stabilization from Section 2. This is of interest for the

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M. Gugat et al.

operation of gas pipelines since often, the customer demand has a periodic structure (see

[14]). If the variations of the demand curves are sufficiently small, that is if the gas flow

remains close to the stationary state (q0,p

0(x)), they can be modeled with (15).

3 The Schl¨

ogl Model for Distributed Pyragas Control

3.1 Deﬁnition of the Model

Again we introduce a given desired T-periodic measurable state udesi ∈L∞(Q2T),

udesi =udesi(t, x)

that has well-defined T-periodic Neumann traces udesi

x(·,0)∈Lp(0,T) and

udesi

x(·,L)∈Lp(0,T), p > 2. We extend udesi to a T-periodic function on [0,∞)×[0,L].

Moreover, we assume that an initial state u0∈C([−T,0],L

∞(0,L))is given. Let a real

number

κ>0

be given. For (t, x) ∈[0,∞)×(0,L), our system state is defined as the solution of the

initial-boundary value problem

(S) :

ut(t, x) =uxx(t, x) −uxx(t −T,x)

−ρ[R(u(t, x)) −R(u(t −T,x))

]+ut(t −T,x)−κ[u(t, x)

−u(t −T,x)],

ux(t, 0)=udesi

x(t, 0)for t>0,

ux(t, L) =udesi

x(t, L) for t>0,

u(t, x) =u0(t, x) for (t, x) ∈(−T,0)×(0,L).

While the above form of the equations nicely shows the action of Pyragas feedback stabi-

lization, the following re-written partial differential equation fits better to the discussion of

its analysis:

ut−uxx +ρR(u)+κu =ut(·−T,·)−uxx(·−T,·)+ρR(u(·−T,·))+κu(·−T,·). (16)

3.2 Existence and Uniqueness of the Solutions

Theorem 3 Suppose that ρ≥0,u1≤u2≤u3, and u0∈L∞((−T,0)×(0,L))are given,

such that the (classical) partial derivatives (u0)t,(u0)xx exist a.e. in (−T,0)×(0,L)and

(u0)t−(u0)xx +ρR(u0)+κu0∈L2((−T,0)×(0,L)).

Suppose further that udesi obeys the properties assumed above. Then for all T≥Tthe

parabolic initial-boundary value problem with time delay

ut−uxx +ρR(u) +κu =ut(·−T,·)−uxx(·−T,·)

+ρR(u(·−T,·)) +κu(·−T,·)in QT,

ux(t, 0)=udesi

x(t, 0)in (0,T),

ux(t, L) =udesi

x(t, L) in (0,T),

u(t, x) =u0(t, x) in [−T,0]×(0,L),

has a unique solution usuch that

u|[0,T ]∈L∞(QT)∩W(0,T)∩C((0,T]×[0,L]).

If u0belongs to C([−T,0]×[0,L]),thenuis also continuous on [0,T]×[0,L].

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Exponential Stability for the Schl¨

ogl System by Pyragas Feedback

Proof Again, we apply the step method. In the interval [0,T], the right-hand side of the

parabolic equation is

(u0)t(t −T,x)−(u0)xx(t −T,x)+ρR(u0(t −T,x))+κu0(t −T,x) =: F(t,x).

Thanks to our assumption, Fbelongs to L2(QT). Now Theorem 5.5 of [32] can be applied

that yields existence and uniqueness of a solution u∈W(0,T) ∩C((0,T]×[0,L])∩

L∞(QT)of

ut−uxx +ρR(u) +κu =Fin QT,

ux(t, 0)=udesi

x(t, 0)in (0,T),

ux(t, L) =udesi

x(t, L) in (0,T),

u(0,x) =u0(0,x) in (0,L).

Next, we extend the solution to [T,min{2T,T}]. Here, the new system reads

ut−uxx +ρR(u) +κu =F(·−T,·)in (T , min{2T,T})×(0,L),

ux(t, 0)=udesi

x(t, 0)in (T , min{2T,T}),

ux(t, L) =udesi

x(t, L) in (T , min{2T,T}),

u(T , x) =u(T −0,x) in (0,L).

Clearly, F(·−T,·)belongs to L2((T , min{2T,T})×(0,L)). Again we obtain existence,

uniqueness, and regularity of uon [T,min{2T,T}] as above. If T>2T, then we repeat

the same procedure with 2Tas new initial time. After finitely many steps, we arrive at the

final time T.

3.3 Exponential Decay with Distributed Pyragas Control

Now the stabilization result for periodic orbits is given.

Theorem 4 (Exponential stability) Let T>0and a T-periodic state udesi ∈H2(Q2T)be

given. Let the assumptions of Theorem 3 hold. Define

μ=2κ−2ρ|mR|.

Assume that μ>0. The function

V(t)=1

2L

0u(t, x) −u(t −T,x)

2dx

is a strict Lyapunov function for the system (S) in the sense that for t≥Tit satisfies the

inequality

V(t)≤exp(−μ(t −T ))V (T ). (17)

Moreover, there exists a T-periodic function u∗∈L2(Q2T)and a constant P0>0such

that for all t≥0we have u∗(t, ·)∈L2(0,L)and the inequality

L

0u(t, x) −u∗(t, x)2dx ≤P0exp(−μt)

is satisfied for all t≥T.

Proof By Theorem 3 the state uis continuous. Hence, Vis well-defined by (1).

The definition of Vimplies that V(t) ≥0. The pde (16) implies that for all

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t≥Tin the sense of the solution of the system (S) presented in Theorem 3 we

have

ut(t, x) −ut(t −T,x)−[uxx(t, x) −uxx(t −T,x)

]

=−ρ[R(u(t, x)) −R(u(t −T,x))

]−κ[u(t, x) −u(t −T,x)]. (18)

We multiply (18)by[u(t, x) −u(t −T,x)]and integrate to obtain for t0>T

t0

TL

[u(t, x) −u(t −T,x)][ut(t, x) −ut(t −T,x)]−[uxx(t, x) −uxx(t −T,x)]dxdt

=t0

TL

−ρ[u(t, x) −u(t −T,x)]R(u(t,x)) −R(u(t −T,x))

−κ[u(t, x) −u(t −T,x)]2dxdt.

For initial data u0in H2((−T,0)×(0,L)), integration by parts yields

L

2[u(t, x) −u(t −T,x)]2

t=Tdx

=t0

TL

−[ux(t, x) −ux(t −T,x)]2dxdt

+t0

[u(t, x) −u(t −T,x)][ux(t, x) −ux(t −T,x)]

x=0dt

+t0

TL

−ρ[u(t, x) −u(t −T,x)]R(u(t,x))−R(u(t −T,x))

−κ[u(t, x) −u(t −T,x)]2dxdt.

Due to (2), for all v1,v2∈(−∞,∞)it holds (14). With the definition of Vand since due

to the periodic boundary conditions the terms at x=0andx=Lcancel we obtain

V(t

0)−V(T) ≤t0

TL

−[ux(t, x) −ux(t −T,x)]2dxdt

+t0

TL

(−ρmR−κ)[u(t, x) −u(t −T,x)]2dxdt.

Thus we have

V(t

0)≤V(T)+t0

TL

−μ1

2[u(t, x) −u(t −T,x)]2dxdt.

Since μ>0 this implies V(t

0)≤V(T). In fact, we have 0 ≤V(t

0)≤V(T) −

t0

TμV (τ)dτ. Now, Gronwall’s lemma implies the inequality

V(t

0)≤V(T)exp (−μ(t0−T)

With a density argument, this implies (17). The last part of the assertion follows as in the

proof of Theorem 2.

Example 2 Let T=2πand L=1. Assume that

udesi

x(t, 0)=udesi

x(t, L) =0.

For (t, x) ∈(−T,0)×(0,L),let

u0(t, x) =sin(t).

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Exponential Stability for the Schl¨

ogl System by Pyragas Feedback

Then system (S) is

u(t, x) =sin(t) for (t, x) ∈(−2π,0)×(0,L),

ux(t, 0)=0fort>0,

ux(t, L) =0fort>0,

ut=uxx −uxx(·−T,·)−ρ[R(u) −R(u(·−T,·))]

+ut(·−T,·)−κ[u−u(·−T,·)]for t>0.

(19)

The solution of the pde in (19) is in fact independent of x,sofort>0 the pde reduces to

the ordinary differential equation

ut=ut(·−2π,·)−ρ[R(u) −R(u(·−2π, ·))]−κ[u−u(·−2π, ·)].

Since u(t, x) =sin(t) is 2π-periodic, it satisfies the above ordinary differential equation.

Thus it is the 2π-periodic solution of (19).

4 Stabilization to a Desired State

In order to complete the picture, we also consider non-Pyragas distributed feedback. The

analysis is completely analogous to Section 3. Let a desired state udesi ∈C2([0,∞)×

[0,L])be given. Let initial data u0in C2([0,L])be given. Consider the system

(V) :

ut−uxx +ρR(u) +κu =udesi

t−udesi

xx +ρR(udesi)+κudesi for (t, x) ∈(0,∞)×(0,L),

ux(t, 0)=udesi

x(t, 0)for t>0,

ux(t, L) =udesi

x(t, L) for t>0,

u(0,x) =u0(x) for x∈(0,L).

For the semilinear system (V), existence results from [16] apply.

Theorem 5 (Exponential stability) Define

μ=2κ−2ρ|mR|.

Assume that μ>0. The function

V(t)=1

2L

0u(t, x) −udesi(t, x)2

is a strict Lyapunov function for the system (V) in the sense that for t≥0we have

V(t)≤exp(−μt)V (0). (20)

Proof The pde in (V) implies that we have

ut−udesi

t−uxx −udesi

xx =−ρR(u) −R(udesi)−κ[u−udesi]. (21)

We multiply (21)by[u(t, x) −udesi(t, x)]and integrate to obtain for t0>0

t0

0L

[u(t, x) −udesi(t, x)][ut(t, x) −udesi

t(t, x)]−[uxx(t, x) −udesi

xx (t, x)]dxdt

=t0

0L

−ρ[u(t, x) −udesi(t, x)]R(u(t,x)) −R(udesi(t, x))−κ[u(t, x) −udesi(t, x)]2dxdt.

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M. Gugat et al.

Integration by parts yields

L

2[u(t, x) −udesi(t, x)]2|t0

t=0dx

=t0

0L

−[ux(t, x) −udesi

x(t, x)]2dxdt +t0

[u(t, x) −udesi(t, x)][ux(t, x) −udesi

x(t, x)]|L

x=0dt

+t0

0L

−ρ[u(t, x) −udesi(t, x)]R(u(t,x)) −R(udesi(t, x))−κ[u(t, x) −udesi(t, x)]2dxdt.

Due to (2), for all v1,v2∈(−∞,∞)it holds (14). With the definition of Vand since due

to the boundary conditions the terms at x=0andx=Lcancel we obtain

V(t

0)−V(0)≤t0

0L

−[ux(t, x) −udesi

x(t, x)]2dxdt

+t0

0L

(−ρmR−κ)[u(t, x) −udesi(t, x)]2dxdt.

Thus we have

V(t

0)≤V(0)+t0

0L

−μ1

2[u(t, x) −udesi(t, x)]2dxdt.

Since μ>0 this implies V(0)≤V(T).Infact,wehave0≤V(t

0)≤V(0)−t0

0μV (τ)dτ.

Now, Gronwall’s lemma implies the inequality V(t

0)≤V(0)exp(−μt0). Thus we have

shown (20).

5 Numerical Experiments

5.1 Numerical Results for Distributed Pyragas Feedback

For simplicity we concentrate here on ordinary differential equations. They are equivalent

to the Schl¨

ogl model, where spatially constant initial data functions u0(x, t) ≡u0(t) are

given. Then the solution uof the Schl¨

ogl model satisfies uxx =0 and it is spatially constant

for all times. Moreover, it obeys the homogeneous Neumann boundary conditions.

The feedback strategies of the previous sections partially generated periodic functions

that are in fact constant. This is not a surprise, because certain standard solutions of

the Schl¨

ogl model without time delay are constant. In the next following examples, we

investigate the stabilization of delay equations that exhibit periodic solutions. By different

stabilization strategies, we are able to generate as time limit functions of a desired period.

However, in contrast to the preceding part, we were not yet able to prove this periodic limit

behavior. Therefore, this subsection has an experimental character.

We define

ρ=1,u

1=0,u

2=0.25,u

3=1, R(u) =(u −u1)(u −u2)(u −u3).

For delay s=1.240683838477202 and weight ω=−1.766552137106608, let v:

[−s,∞)→Rbe the solution of the nonlinear delay equation

v(t) =−ρR(v(t)) +ωv(t −s) for t∈[0,∞),

v(t) =1ift≤0. (22)

These special numbers sand ωwere determined such that the solution vof the nonlinear

delay ode (22) in the interval [0,160]minimizes the L2(0,160)-distance to the solution of

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Exponential Stability for the Schl¨

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-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

0.8

1.2

[156,160]

[956,960]

Fig. 1 The solution vof the delay ode (22) does not seem to be 4-periodic

the linear delay ode

U(t) =−

2U(t −1)for t∈[0,∞),

U(t) =1ift≤0;

we refer to the computational examples in [7].

The function Uis known to be periodic with period T=4, cf. [13]. At first glance,

the function vseems to have this property, too. However, if we solve equation (22)inthe

interval [0,960]and then compare the solution in [156,160]with the solution in [956,960],

we can see that the obtained vis not periodic of period 4; see Fig. 1. Most likely, some

numerical instabilities are the reason for this behavior. On the other hand, the function v

might not be periodic.

Therefore, to obtain a periodic solution, we try to stabilize the solution of (22)bysome

feedback.

Strategy 1. We want to stabilize (22) using a Pyragas feedback stabilization. To obtain the

target period T=4, we solve the equation

u(t) =−ρR(u(t)) +ωu(t −s) +κ(u(t −4)−u(t)) for t∈[160,960],

u(t) =v(t) if 156 ≤t≤160 (23)

with some positive κ. For a sufficiently large weight κ,κ=100 in our computations,

we obtain a very fine numerical adjustment, cf. Fig. 2.Indeed,theL2(−4,0)difference

between u(t +960)and v(t +160)is 2 ×10−2.SincevL2(156,160)=1.4927, we report

only on absolute errors. In all the figures, the graph of the error is magnified 40 times with

respect to the scale of the function.

Strategy 2. We try to stabilize the solution vof (22) by adding the difference between the

periodic target function

udesi =v(mod(t −160,4)+160 −4)

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M. Gugat et al.

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

-1

-0.5

0.5

-0.02

-0.01

0.01

0.02

0.03

error

Original in [156,160]

Stabilized in [956,960]

error

Fig. 2 The solution uof the delay ode (23) with Pyragas feedback seems to be 4-periodic

and the solution uof

u(t) =−ρR(u(t)) +ωu(t −s) +κ(udesi(t) −u(t)) for t∈[160,960],

u(t) =v(t) if 156 ≤t≤160. (24)

For κ=100, the L2(−4,0)-norm of the difference between u(t +960)and v(t +160)is

9×10−3;seeFig.3.

Stabilization in the presence of small perturbations of parameters and data. The solu-

tion of (22) is very sensitive w.r.t. small perturbations of s,ωand the initial history v(t) for

t≤0. For instance, we consider the round off of sand ωwith two digits to the right of the

decimal point,

ˆs=1.24,ˆω=−1.77.

Let ˆvbe the solution of the perturbed problem

ˆv(t) =−ρR(ˆv(t)) +ˆωˆv(t −ˆs) for t∈[0,∞),

ˆv(t) =1ift≤0. (25)

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

-1

-0.5

0.5

-0.02

-0.01

0.01

0.02

0.03

error

Original in [156,160]

Stabilized in [956,960]

error

Fig. 3 The solution uof the delay ode (24) with forcing term seems to be 4-periodic

784

Exponential Stability for the Schl¨

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-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

-1

-0.5

0.5

Original in [156,160]

Perturbed in [156,160]

Fig. 4 The solution ˆvof the perturbed delay equation (25) is quite different from the original solution vof (22)

We obtain a solution that considerably differs from the original one in the last interval

[156,160]. We have an error of v−ˆvL2(156,160)=9×10−1;seeFig.4.

Let us test the behavior of both stabilization strategies in this case.

Strategy 1. We try to stabilize (25) using a Pyragas feedback stabilization. For the target

period T=4, we solve the equation

u(t) =−ρR(u(t)) +ˆωu(t −ˆs) +κ(u(t −4)−u(t)) for t∈[160,960],

u(t) =1.01v(t) if 156 ≤t≤160 (26)

for some positive κ. Notice that we have also perturbed the initial history. For κ=100, the

L2(−4,0)-norm of the difference between u(t +960)and v(t +160)is 1.5 ×10−2.Thisis:

the stabilized function not only appears to be periodic, but also is very close to the solution

of the original unperturbed problem; cf. Fig. 5.

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

-1

-0.5

0.5

-0.02

-0.01

0.01

0.02

0.03

error

Original in [156,160]

Stabilized in [956,960]

error

Fig. 5 The solution uof the perturbed delay ode (26) with Pyragas feedback seems to be 4-periodic and is

quite similar to the original solution vof (22)

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M. Gugat et al.

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

-1

-0.5

0.5

-0.02

-0.01

0.01

0.02

0.03

error

Original in [156,160]

Stabilized in [956,960]

error

Fig. 6 The solution of the perturbed delay ode with forced feedback also seems to be 4-periodic and is quite

similar to the original solution vof (22)

Strategy 2. Using the target udesi(t) =v(mod(t −160,4)+160 −4)instead of u(t −4)

yields also a successful stabilization. The error in this case is again 9 ×10−3;cf.Fig.6.

Comment about perturbation. One could think that the periodic behavior is imposed only

or mainly by the Pyragas feedback term because it has this big weight κandweimposea

delay equal to the searched period. Nevertheless, if the ode we are trying to stabilize does

not have a solution that is periodic (or is at least close to be periodic), then the Pyragas

feedback term will not help to obtain such a periodic solution.

Our first motivation for this example was to mimic the solution of the linear delay equa-

tion (23), that has delay ˆs=1 and weight ˆω=−π/2, with a the solution of the nonlinear

ode (22). By an optimization method, we found the “strange” pair s=1.240683838477202

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

-1

-0.5

0.5

Original in [156,160]

Not stabilized in [956,960]

Fig. 7 The solution uof the delay ode (27) cannot be steered to match the solution vof the original delay

ode (22) using a Pyragas feedback

786

Exponential Stability for the Schl¨

ogl System by Pyragas Feedback

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

-1

-0.5

0.5

-0.02

-0.01

0.01

0.02

0.03

error

Original in [156,160]

error

Fig. 8 The solution uof an ode without delay, (28), is forced to match a periodic pattern by Strategy 2

and ω=−1.766552137106608, cf. [7]. If we follow Strategy 1 and try to stabilize the

perturbed equation inserting a Pyragas feedback term, i.e., solving the following equation,

u(t) =−ρR(u(t)) −π

2u(t −1)+κ(u(t −4)−u(t)) for t∈[160,960],

u(t) =v(t) if 156 ≤t≤160,(27)

we are not successful; see Fig. 7.

However, Strategy 2 leads to a function that is very close to the solution vof (22). In

Fig. 8we show, how the solution of

u(t) =−ρR(u(t)) +κ(udesi(t) −u(t)) for t∈[160,960],

u(t) =v(t) if 156 ≤t≤160 (28)

corresponding to ω=0 (no delay) is lead to a periodic pattern by feedback Strategy 2. The

error in this case is 2 ×10−2.

3012 45678910

-2

Fig. 9 The function u(t) −sin(t) for κ=7

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M. Gugat et al.

0 5 10 15 20 25 30

Fig. 10 The function u(t) −sin(t) for κ=3

5.2 Numerical Results for Distributed (non-Pyragas) Feedback

As in Example 1, we select the numbers u1=−1, u2=0, u3=π.Thenwehave

R(u) =(u2+u)(u−π)and also mRis as in Example 1. Let ρ=1andudesi(t, x) =sin(t).

For constant initial data u0, the solutions of system (V) do not depend on xand are the

solutions of the Cauchy problem with the initial condition u(0)=u0and the ordinary

differential equation

ut+(u2+u)(u −π)+κu =cos(t) +(sin2(t) +sin(t))(sin(t) −π)+κsin(t).

Let κ=7. Since κ>ρ|mR|, Theorem 4 implies that for any constant initial state the

function t→ (u(t) −sin(t))2decays exponentially. This is illustrated by Fig. 9. The initial

value for t=0isu0=10.

Numerical simulations indicate that for κ=3, the function (u(t) −sin(t))2does not

converge to zero for t→∞. This is illustrated by Fig. 10. Note however that the function

u(t) −sin(t) appears to be periodic.

6 Conclusion

In this paper, feedback laws with Pyragas terms have been discussed that stabilize the

Schl¨

ogl system globally to a T-periodic state for a given period T>0 under appropriate

assumptions.

Both the case of Pyragas boundary feedback control on both ends of the interval and

of distributed Pyragas feedback control were considered. A strict Lyapunov function was

constructed to show the exponential stability of the resulting closed-loop system in the L2-

sense. We have presented inequalities that guarantee the exponential stability of the systems

and can easily be verified.

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Exponential Stability for the Schl¨

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Acknowledgements Open Access funding provided by Projekt DEAL. This work was supported by DFG

in the framework of the Collaborative Research Center Transregio 154, project C03. The second author was

partially supported by Spanish Ministerio de Econom´

ıa y Competitividad under research project MTM2017-

83185-P.

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