Sampled-data control of 2D Kuramoto-Sivashinsky equation

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAC.2021.3070070, IEEETransactions on Automatic Control

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Sampled-data control of 2D Kuramoto-Sivashinskyequation

Wen Kang Member, IEEE and Emilia Fridman, Fellow, IEEE

Abstract—This paper addresses sampled-data control of 2DKuramoto-Sivashinsky equation over a rectangular domain Ω.We suggest to divide the 2D rectangular Ω into N sub-domains, where sensors provide spatially averaged or pointstate measurements to be transmitted through communicationnetwork to the controller. Note that differently from 2D heatequation, here we manage with sampled-data control underpoint measurements. We design a regionally stabilizing controllerapplied through distributed in space characteristic functions.Sufficient conditions ensuring regional stability of the closed-loop system are established in terms of linear matrix inequalities(LMIs). By solving these LMIs, we find an estimate on the setof initial conditions starting from which the state trajectoriesof the system are exponentially converging to zero. A numericalexample demonstrates the efficiency of the results.

I. INTRODUCTION

In recent decades, Kuramoto-Sivashinsky equation (KSE)has drawn a lot of attention as a nonlinear model of patternformations on unstable flame fronts and thin hydrodynamicfilms (see e.g. [1], [2]). KSE arises in the study of thinliquid films, exhibiting a wide range of dynamics in differ-ent parameter regimes, including unbounded growth and fullspatiotemporal chaos. For 1D KSE, distributed control (seee.g. [3]–[6]) has been considered. Boundary stabilization ofKSE has been studied in [7]–[10].

Sampled-data control of PDEs became recently an activeresearch area (see e.g. [11]–[14]) for practical applicationof finite-dimensional controllers for PDEs, where LMI con-ditions for the exponential/regional stability of the closed-loop systems were derived in the framework of time-delayapproach by employing appropriate Lyapunov functionals.Most of the existing results deal with 1D PDEs system.Sampled-data observers for ND and 2D heat equations withglobally Lipschitz nonlinearities have been suggested in [15]and [16]. However, the above results were confined to diffusionequations. Sampled-data control of various classes of highdimensional PDEs is an interesting and challenging problem.

In our recent paper [14], we have suggested sampled-datacontrol of 1D KSE, where both point and averaged state

W. Kang ([email protected]) is with School of Automation andElectrical Engineering, University of Science and Technology Beijing, P. R.China.

E. Fridman ([email protected]) is with School of Elec-trical Engineering, Tel Aviv University, Israel.

This work was supported by National Natural Science Foundation ofChina (Grant No. 61803026), Israel Science Foundation (grant No. 673/19),Beijing Science Foundation for the Excellent Youth Scholars (Grant No.2018000020124G067), Fundamental Research Funds for the Central Universi-ties (Grant No. FRF-TP-20-039A2Z), Outstanding Chinese and Foreign YouthExchange Program of China Association of Science and Technology.

measurements were studied. In this paper we aim to extendresults of [14] to 2D case. Extension from 1D ( [14], [30])to 2D is far from being straightforward. Thus, in the caseof heat equation, sampled data extension under the pointmeasurements seems to be not possible (see [16]). This is dueto the fact that stability analysis of the closed-loop systemis based on the bound of the L2-norm of the differencebetween the state and its point value. However, accordingto Friedrich’s inequality [16], [17], this bound depends onthe L2-norm of the second-order spatial derivatives of thestate. Differently from the heat equation, stability analysis forKSE in H2 allows to compensate such terms. We establishstability analysis of the closed-loop sampled-data system byconstructing an appropriate Lyapunov-Krasovskii functional.Some preliminary results under averaged measurements werepresented in [29], where the sampled-data case is limited toaveraged measurements and there is no detailed proof of thewell-posedness.

In the present paper, we design a sampled-data controllerfor 2D KSE under averaged/point measurements based onLMIs. In comparison to the existing known results, new specialchallenges of this work are the following:1) The present paper gives the first extension to 2D PDE inthe case of sampled-data point measurements. The results from[16] cannot be extended to the case of point measurements.This is due to the second order spatial derivative in Lemma 3that cannot be compensated in Lyapunov analysis.2) Here we have the nonlinear term “zzx1

” which is locallyLipschitz in D((−A)

12 ) (defined in Section III below). Due

to the nonlinear term, the challenge is to find a bound onthe domain of attraction. This bound is based on the new 2DSobolev inequality (Lemma 4) that we have derived. Lemma 4bounds a function in the C0-norm using L2-norms of its firstand second spatial derivatives. In [15] and [16], the nonlinearterm is subject to sector bound inequality which holds globallyand leads to global results.3) The well-posedness is challenging. We have provided moredetailed proof for this, and shown that A generates an analyticsemigroup even for rectangular domain Ω (non C1 boundary).

The remainder of this work is organized as follows. Usefullemmas are introduced and the problem setting is reportedin Section II. Sections III-IV are devoted to constructionof continuous static output-feedback/sampled-data controllersunder the averaged or point measurements. In Section V, anumerical example is carried out to illustrate the efficiencyof the main results. Finally, some concluding remarks andpossible future research lines are presented in Section VI.

Notation The superscript “T” stands for matrix transposi-

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tion, Rn denotes the n-dimensional Euclidean space with thenorm | · |. Ω ⊂ R2 denotes a computational domain, L2(Ω)denotes the space of measurable squared-integrable functionsover Ω with the corresponding norm ‖z‖2L2(Ω) =

∫Ω|z(x)|2dx.

Let ∂Ω be the boundary of Ω. The Sobolev space Hk(Ω) isdefined as Hk(Ω) = z : Dαz ∈ L2(Ω), ∀ 0 ≤ |α| ≤ k

with norm ‖z‖Hk(Ω) =

( ∑0≤|α|≤k

‖Dαz‖2L2(Ω)

) 12

. The space

Hk0 (Ω) is the closure of C∞c (Ω) in the space Hk(Ω) with the

norm ‖z‖Hk0 (Ω) =

( ∑|α|=k

‖Dαz‖2L2(Ω)

) 12

.

II. PROBLEM FORMULATION AND USEFUL LEMMAS

Denote by Ω the two dimensional (2D) unit square

Ω = [0, 1]× [0, 1] ⊂ R2.

Consider the biharmonic operator:

∆2 =∂4

∂x41

+ 2∂4

∂x21∂x

22

+∂4

∂x42

.

As in [18], we consider the following 2D Kuramoto-Sivashinsky equation (KSE) over Ω under the Dirichlet bound-ary conditions:

zt + zzx1+(1− κ)zx1x1

−κzx2x2+∆2z=

N∑j=1

χj(x)Uj(t),

(x, t) ∈ Ω× [0,∞),

z|∂Ω = 0,∂z

∂n|∂Ω = 0,

z(x, 0) = z0(x),(1)

where x = (x1, x2) ∈ Ω, z ∈ R is the state of KSE,∂z

∂nis

the normal derivative, and Uj(t) ∈ R, j = 1, 2, · · · , N are thecontrol inputs. Here the parameter κ denotes the angle of thesubstrate to the horizontal:for κ > 0 we have overlying film flows, a vertical film flowfor κ = 0, and hanging flows when κ < 0.

Motivated by [6], [12], [13], [15], [19] we suggest to divideΩ into N square sub-domains Ωj covering the whole region∪Nj=1Ωj = Ω (see Fig. 1) with an actuator and a sensor placedin each Ωj . Here

Ωj = x = (x1, x2)T ∈ Ω|xi ∈ [xmini (j), xmax

i (j)], i = 1, 2,j = 1, 2, · · · , N.

The measure of their intersections is zero. Let

0 = t0 < t1 < · · · < tk · · · , limk→∞

tk =∞

be sampling time instants. The sampling sub-domains in timeand in space may be bounded,

0 ≤ tk+1 − tk ≤ h,0 < xmax

i (j)− xmini (j) = ∆j ≤ ∆,

i = 1, 2; j = 1, 2, · · · , N,

where h and ∆ are the corresponding upper bounds.

Remark 1. For simplicity, we consider each sub-domain Ωj isa square (i.e. xmax

1 (j)− xmin1 (j)) = xmax

2 (j)− xmin2 (j), j =

Fig. 1. Unit square Ω and sub-domains Ωj

1, 2, · · · , N). Indeed, Ωj can be a rectangular. For the casethat xmax

1 (j)−xmin1 (j)) 6= xmax

2 (j)−xmin2 (j) for some j (see

Fig. 1 of [16]), ∆j can be chosen as follows

∆j = maxxmax1 (j)− xmin

1 (j), xmax2 (j)− xmin

2 (j).

Thus, the results of this work are applicable to the case ofnonsquare sub-domains.

The spatial characteristic functions are taken asχj(x) = 1, x ∈ Ωj ,χj(x) = 0, x /∈ Ωj ,

j = 1, · · · , N. (2)

We assume that sensors provide the following averaged mea-surements

yjk =

∫Ωjz(x, tk)dx

|Ωj |, j = 1, . . . , N ; k = 0, 1, 2 . . . (3)

or point measurements

yjk = z(xj , tk), j = 1, . . . , N ; k = 0, 1, 2 . . . (4)

where xj locates in the center of the square sub-domain Ωj ,and |Ωj | stands for the Lebesgue measure of the domain Ωj .

We aim to design for (1) an exponentially stabilizingsampled-data controller that can be implemented by zero-orderhold devices:

Uj(t) = −µyjk, j = 1, . . . , N ;t ∈ [tk, tk+1), k = 0, 1, 2 . . . ,

(5)

where µ is a positive controller gain and yjk is given by (3)or (4).

We present below some useful lemmas:

Lemma 1. Let Ω = (0, L1) × (0, L2). Assume f : Ω → Rand f ∈ H1(Ω).(i) (Poincare’s inequality) If

∫Ωf(x)dx = 0, then according

to [20]

‖f‖2L2(Ω) ≤L2

1 + L22

π2‖∇f‖2L2(Ω). (6)

(ii) (Wirtinger’s inequality) [15] If f |∂Ω = 0, then thefollowing inequality holds:

‖f‖2L2(Ω) ≤L2

1 + L22

π2‖∇f‖2L2(Ω). (7)




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The following lemma gives a classical Friedrich’s inequality(Theorem 18.1 of [33]) with tight bounds on the coefficients

of terms∥∥∥∥ ∂f∂x1

∥∥∥∥2

L2(Ω)

,∥∥∥∥ ∂f∂x2

∥∥∥∥2

L2(Ω)

and∥∥∥∥ ∂2f

∂x1∂x2

∥∥∥∥2

L2(Ω)

. The

inequality (8) bounds the L2-norm of a function by the recip-rocally convex combination of the L2-norm of its derivatives.

Lemma 2. (see (2) of [16]) Let Ω = (0, l)2, f ∈ H2(Ω) withf(0, 0) = 0. Then the following inequality holds:

‖f‖2L2(Ω) ≤1

α1

(2l

π

)2∥∥∥∥ ∂f∂x1

∥∥∥∥2

L2(Ω)

+1

α2

(2l

π

)2∥∥∥∥ ∂f∂x2

∥∥∥∥2

L2(Ω)

+1

α3

(2l

π

)4 ∥∥∥∥ ∂2f

∂x1∂x2

∥∥∥∥2

L2(Ω)(8)

where α1, α2, α3 are positive constants satisfying

α1 + α2 + α3 = 1.

Lemma 3. [16] Let Ω = (0, l)2, f ∈ H2(Ω) with f(0, 0) = 0,η > 0. Then

η‖f‖2 ≤ β1

(2l

π

)2 ∥∥∥∥ ∂f∂x1

∥∥∥∥2

+ β2

(2l

π

)2 ∥∥∥∥ ∂f∂x2

∥∥∥∥2

+β3

(2l

π

)4 ∥∥∥∥ ∂2f

∂x1∂x2

∥∥∥∥2 (9)

for any β1, β2, β3 satisfying

diagβ1, β2, β3 ≥ η

1 1 11 1 11 1 1

. (10)

The following version of 2D Sobolev inequality will beuseful:

Lemma 4. Let Ω = (0, 1)2 and w = w(x1, x2) ∈ H2(Ω) ∩H1

0 (Ω), where (x1, x2) ∈ Ω. Then

‖w‖2C0(Ω)

≤ 1

2(1 + Γ)

[‖wx1

‖2L2(Ω) + ‖wx2‖2L2(Ω)

]+

1

Γ‖wx1x2

‖2L2(Ω), ∀Γ > 0.(11)

Proof. Due to w ∈ H10 (Ω), application of 1D Sobolev’s

inequality to w in x2 yields

maxx2∈[0,1]

w2(x1, x2) ≤∫ 1

0

w2x2

(x1, ξ2)dξ2. (12)

Further application of Lemma 4.1 of [14] to wx2 in x1 leadsto

maxx1∈[0,1]

w2x2

(x1, ξ2)

≤ (1 + Γ)

∫ 1

0

w2x2

(ξ1, ξ2)dξ1 +1

Γ

∫ 1

0

w2x1x2

(ξ1, ξ2)dξ1,

∀Γ > 0.(13)

Substitution of (13) into (12) yields

‖w‖2C0(Ω)

= max(x1,x2)∈Ω

w2(x1, x2) = maxx1∈[0,1]

[maxx2∈[0,1]

w2(x1, x2)

]≤ maxx1∈[0,1]

[∫ 1

0

w2x2

(x1, ξ2)dξ2

]≤∫ 1

0

maxx1∈[0,1]

w2x2

(x1, ξ2)dξ2

≤ (1 + Γ)

∫ 1

0

∫ 1

0

w2x2

(ξ1, ξ2)dξ1dξ2

+1

Γ

∫ 1

0

∫ 1

0

w2x1x2

(ξ1, ξ2)dξ1dξ2.

(14)Following the same procedure, we can obtain

‖w‖2C0(Ω)

≤ (1 + Γ)

∫ 1

0

∫ 1

0

w2x1

(ξ1, ξ2)dξ1dξ2

+1

Γ

∫ 1

0

∫ 1

0

w2x1x2

(ξ1, ξ2)dξ1dξ2.

(15)

From (14) and (15) it follows that (11) holds.

Remark 2. In Lemma 4, we give a new 2D Sobolev inequalitywith constants depending on a free parameter Γ > 0. Lemma4 is very useful and plays an important role in the stabilityanalysis leading to LMIs (with Γ as a decision parameter)that guarantee stability and give a bound on the domain ofattraction.

Lemma 5. (Halanay’s Inequality [24]) Let V : [−h,∞) →[0,∞) be an absolutely continuous function. If there exist 0 <δ1 < 2δ such that for all t ≥ 0 the following inequality holds

V (t) + 2δV (t)− δ1 sup−h≤θ≤0

V (t+ θ) ≤ 0, (16)

then we have

V (t) ≤ e−2σt sup−h≤θ≤0

V (θ), t ≥ 0, (17)

where σ is a unique solution of

σ = δ − δ12e2σh. (18)

III. GLOBAL STABILIZATION: CONTINUOUS STATICOUTPUT-FEEDBACK

In this section, we will establish the well-posedness andstability analysis for the system (1) under the continuous-timeaveraged measurements

yj(t) =

∫Ωjz(x, t)dx

|Ωj |, j = 1, . . . , N (19)

or point measurements

yj(t) = z(xj , t), j = 1, . . . , N (20)

via a controller

Uj(t) = −µyj(t), j = 1, . . . , N. (21)

The closed-loop system can be represented in the followingform:

zt + zzx1 + (1− κ)zx1x1−κzx2x2 + ∆2z

= −µN∑j=1

χj(x)[z − fj ], (x, t) ∈ Ω× [0,∞),

z|∂Ω = 0,∂z

∂n|∂Ω = 0,

z(x, 0) = z0(x),

(22)




4

where for (19)

fj(x, t) = z(x, t)−

∫Ωjz(ζ, t)dζ

|Ωj |, (23)

for (20)fj(x, t) = z(x, t)− z(xj , t). (24)

Now we establish the well-posedness of the system (22)subject to (23) or (24). Define the spatial differential operatorA : D(A) ⊂ L2(Ω)→ L2(Ω) as follows:

Af = −∆2f, ∀f ∈ D(A),

D(A) = f ∈ H4(Ω) : f |∂Ω = 0,∂f

∂n|∂Ω = 0.

(25)

Note that A∗ = A and Re〈Af, f〉 ≤ 0, ∀f ∈ D(A).Thus, the operator A is self-adjoint and dissipative. Moreover,the inverse A−1 is bounded, and hence 0 ∈ ρ(A). By theLumer-Phillips theorem [28], A generates a C0-semigroup.Since the resolvent of A is compact on L2(Ω), the spectrumof A consists of isolated eigenvalues only, and a sequenceof corresponding eigenfunctions of A forms an orthonormalbasis of L2(Ω). Let λn be the eigenvalues of A and letφn be the corresponding eigenfunctions, i.e. Aφn = λnφn.Since A is negative, all the eigenvalues are located on thenegative real axis, i.e. λn < 0. For any x0 ∈ L2(Ω), it

can be presented in the following form: x0 =∞∑n=1

anφn with∞∑n=1|an|2 = ‖x0‖2L2(Ω). Therefore, for any τ ∈ R,

‖(iτI −A)−1x0‖2L2(Ω) =∞∑n=1

|an|2

|iτ − λn|2=∞∑n=1

|an|2

τ2 + λ2n

≤∞∑n=1

|an|2

τ2=

1

τ2‖x0‖2L2(Ω),

which implies

‖(iτI −A)−1‖ ≤ 1

|τ |, ∀τ ∈ R.

Hence,lim|τ |→∞

‖τ(iτI −A)−1‖ <∞.

Then from Theorem 1.3.3 of [36], it follows that A generatesan analytic semigroup. Since −A is positive, (−A)

12 is also

positive and

D((−A)12 ) = f ∈ H2(Ω) : f |∂Ω = 0,

∂f

∂n|∂Ω = 0.

The norm of D((−A)12 ) is given by

‖f‖2D((−A)

12 )

=

∫Ω

|∆f |2dx.

Throughout the paper, we assume that z0 ∈ D((−A)12 ). We

can rewrite the system (22) subject to (23) or (24) as theevolution equation:

d

dtz(·, t) = Az(·, t) + F (z(·, t)),

z(·, 0) = z0(·)(26)

subject to

F (z(·, t)) = −z(x, t)zx1(x, t)− (1− κ)zx1x1

(x, t)

+κzx2x2(x, t)− µ

N∑j=1

χj(x)[z(x, t)− fj(x, t)].

Note that the nonlinear term F is locally Lipschitz continuous,that is, there exists a positive constant l(C) such that thefollowing inequality holds:

‖F (z1)− F (z2)‖L2(Ω) ≤ l(C)‖z1 − z2‖D((−A)

12 )

for any z1, z2 ∈ D((−A)12 ) with ‖z1‖

D((−A)12 )≤ C,

‖z2‖D((−A)

12 )≤ C. Here we prove the nonlinear term F is

locally Lipschitz continuous for the case of point measure-ments. From the expression of F (z(·, t)), using Minkowskii’sinequality we have

‖F (z1)− F (z2)‖L2(Ω) ≤ ‖z1z1x1− z2z2x1

‖L2(Ω)

+|1− κ| · ‖z1x1x1− z2x1x1

‖L2(Ω)

+|κ| · ‖z1x2x2− z2x2x2

‖L2(Ω) + µ‖z1 − z2‖L2(Ω)

+µN∑j=1

‖∫ xxjz1ξ(ξ, t)− z2ξ(ξ, t)dξ‖L2(Ωj).

(∗)Note that D((−A)

12 ) = f ∈ H2(Ω) : f |∂Ω = 0,

∂f

∂n|∂Ω =

0. Since z1, z2 ∈ D((−A)12 ) with ‖z1‖

D((−A)12 )≤ C,

‖z2‖D((−A)

12 )≤ C, we obtain that z1 − z2 ∈ D((−A)

12 ),

and there exist some positive constants M1 and M2 such that

‖z1‖L2(Ω) ≤M1

[‖z1x1‖L2(Ω) + ‖z1x2‖L2(Ω)

],

‖z1x1‖L2(Ω) + ‖z1x2

‖L2(Ω) ≤M2‖z1‖D((−A)

12 ),

|z2‖L2(Ω) ≤M1

[‖z2x1‖L2(Ω) + ‖z2x2‖L2(Ω)

],

‖z2x1‖L2(Ω) + ‖z2x2‖L2(Ω) ≤M2‖z2‖D((−A)

12 ),

‖z1−z2‖L2(Ω)≤M1

[‖z1x1

−z2x1‖L2(Ω)+‖z1x2

−z2x2‖L2(Ω)

],

‖z1x1−z2x1‖L2(Ω)+‖z1x2−z2x2‖L2(Ω)≤M2‖z1−z2‖D((−A)

12 ).

Hence, the following holds:

‖z1z1x1 − z2z2x1‖L2(Ω)

= ‖z1z1x1 − z2z1x1 + z2z1x1 − z2z2x1‖L2(Ω)

≤ ‖z1 − z2‖L2(Ω)‖z1x1‖L2(Ω)+‖z2‖L2(Ω)‖z1x1

− z2x1‖L2(Ω)

≤M1M2‖z1 − z2‖D((−A)

12M2‖z1‖

D((−A)12

+M1M2‖z2‖D((−A)

12M2‖z1 − z2‖

D((−A)12

≤ 2M1M22C‖z1 − z2‖

D((−A)12,

|1− κ|·‖z1x1x1− z2x1x1

‖L2(Ω)+|κ|·‖z1x2x2− z2x2x2

‖L2(Ω)

≤ max |1− κ|, |κ| ‖z1 − z2‖D((−A)

12,

µ‖z1 − z2‖L2(Ω) ≤ µM1M2‖z1 − z2‖D((−A)

12.

Moreover, Lemma 3 implies that there exist some positiveconstants M3, M4 and M5 such that

µN∑j=1

‖∫ xxjz1ξ(ξ, t)− z2ξ(ξ, t)dξ‖L2(Ωj)

≤ µ[M3‖z1x1

− z2x1‖L2(Ω) +M4‖z1x2

− z2x2‖L2(Ω)

+M5‖z1x1x2− z2x1x2

‖L2(Ω)

]≤ µM3M2‖z1 − z2‖

D((−A)12

+ µM4M2‖z1 − z2‖D((−A)

12

+µM5‖z1 − z2‖D((−A)

12.




5

Substitution of the above inequalities into the right-hand sideof (*) yields

‖F (z1)− F (z2)‖L2(Ω) ≤ l(C)‖z1 − z2‖D(−A)

12,

where

l(C) = 2M1M22C + max |1− κ|, |κ|

+µ(M1M2 +M3M2 +M4M2 +M5).

From Theorem 6.3.1 of [28], it follows that the system (22)subject to (23) or (24) has a unique local classical solution z ∈C([0, T ), L2(Ω)) ∩C1((0, T ), L2(Ω)) for any initial functionz0 ∈ D((−A)

12 ).

Remark 3. The above mentioned well-posedness result isdependent on the initial condition z0 ∈ D((−A)

12 ). If the

initial function z0 ∈ L2(Ω), the solution of the system (22)subject to (23) or (24) may become a mild solution or weaksolution.

A. Distributed controller under averaged measurements

Proposition 1. Consider the closed-loop system (22) subjectto (23). Given positive scalars ∆, if there exist δ > 0, µ > 0and λi ≥ 0 (i = 1, 2) such that the following LMI holds:

Υ ,

−2µ+ 2δ − λ1π2

2Υ12 µ

∗ −2 0∗ ∗ −λ2

≤ 0

(27)where

Υ12 = −λ1

2− λ2

∆2

π2− (1− κ),

then the closed-loop system is globally exponentially stable inthe L2-sense:∫

Ω

z2(x, t)dx ≤ e−2δt

∫Ω

z2(x, 0)dx, ∀t ≥ 0. (28)

Furthermore, if the strict LMI (27) is feasible for δ = 0, thenthe closed-loop system is exponentially stable with a smallenough decay rate.

Proof. The proof is divided into three parts.Step 1: We have shown that there exists a local clasicalsolution to (22) subject to (23), where T = T (z0). ByTheorem 6.23.5 of [27], we obtain that the solution exists forany T > 0 if this solution admits a priori estimate. In Step3, it will be shown that the feasibility of LMI (27) guaranteesthat the solution of (22) subject to (23) admits a priori bound,which can further guarantee the existence of the solution forall t ≥ 0.

Step 2: Assume formally that there exists a classical solutionof (22) subject to (23) for all t ≥ 0. We consider the followingLyapunov-Krasovskii functional:

V (t) = ‖z(·, t)‖2L2(Ω). (29)

Since z|∂Ω = 0 and∂z

∂n|∂Ω = 0, integration by parts leads to∫Ω

z2zx1dx = 0, (30)

∫Ωz2x1x2

dx =∫

Ωzx1x1zx2x2dx. (31)

Furthermore, from (31) it follows that

−2∫

Ω[z2x1x1

+ z2x2x2

+ 2z2x1x2

]dx= −2

∫Ω

[zx1x1+ zx2x2

]2dx = −2∫

Ω|∆z|2dx. (32)

Differentiating (29) along (22), integrating by part and using(30), (32) we obtain

V (t) + 2δV (t) = 2∫

Ωzztdx+ 2δ

∫Ωz2dx

= 2(1− κ)∫

Ωz2x1dx− 2κ

∫Ωz2x2dx− 2

∫Ω|∆z|2dx

−(2µ− 2δ)∫

Ωz2dx+ 2µ

N∑j=1

∫Ωjzfjdx.

(33)

From Lemma 1, the Wirtinger’s inequality yields

λ1

[∫Ω∇zT∇zdx− π2

2

∫Ωz2dx

]≥ 0, (34)

where λ1 ≥ 0.For the case of averaged measurements, fj is given by (23).Since

∫Ωjfj(x, t)dx = 0, from Lemma 1, the Poincare

inequality leads to∫Ωjf2j dx ≤ 2∆2

π2

∫Ωj∇zT∇zdx.

Hence,

λ2

N∑j=1

[2∆2

π2

∫Ωj∇zT∇zdx−

∫Ωjf2j dx

]≥ 0, (35)

where λ2 ≥ 0.Integration by parts yields

−∫

Ω∇zT∇zdx =

∫Ωz∆zdx. (36)

Applying S-procedure [31], we add to V (t)+2δV (t) the left-hand side of (34), (35) and use (36). Then it follows that

V (t) + 2δV (t)

≤ V (t) + 2δV (t) + λ1


2

∫Ωz2dx

]+λ2

[2∆2

π2

∫Ω∇zT∇zdx−

N∑j=1

∫Ωjf2j dx

]≤ 2(1− κ)

∫Ωz2x1dx− 2κ

∫Ωz2x2dx− 2

∫Ω|∆z|2dx

−(2µ− 2δ + λ1π2

2 )∫

Ωz2dx+ 2µ

N∑j=1

∫Ωjzfjdx

−(λ1 + λ22∆2

π2 )∫

Ωz∆zdx− λ2

N∑j=1

∫Ωjf2j dx.

Note that

2(1− κ)∫

Ωz2x1dx− 2κ

∫Ωz2x2dx

= 2(1− κ)∫

Ω∇zT∇zdx− 2

∫Ωz2x2dx

≤ 2(1− κ)∫

Ω∇zT∇zdx.

(37)

Then substitution of (36) into (37) yields

2(1− κ)∫

Ωz2x1dx− 2κ

∫Ωz2x2dx

≤ −2(1− κ)∫

Ωz∆zdx.

(38)

Hence,

V (t) + 2δV (t) ≤N∑j=1

∫Ωj

[z ∆z fj

]Υ

z∆zfj

dx ≤ 0




6

if Υ ≤ 0 holds. Therefore,

V (t) ≤ e−2δtV (0), ∀t ≥ 0.

Note that the feasibility of the strict LMI (27) with δ = 0implies its feasibility with a small enough δ0 > 0. Therefore,if the strict LMI (27) holds for δ = 0, then the closed-loopsystem is exponentially stable with a small decay rate δ0 > 0.

Step 3: The feasibility of LMI (27) yields that the solutionof (22) subject to (23) admits a priori estimate V (t) ≤e−2δtV (0). By Theorem 6.23.5 of [27], continuation of thissolution under a priori bound to entire interval [0,∞).

B. Distributed controller under point measurementsProposition 2. Consider the closed-loop system (22) subjectto (24). Given positive scalars ∆, if there exist δ > 0, µ > 0,η > 0, λ1 ≥ 0, λ2 ∈ R and βi > 0 (i = 1, 2, 3) such that(10) is satisfied and the following LMIs hold:

2(1− κ) + β1

(∆

π

)2

+ λ1 − λ2 ≤ 0, (39)

−2κ+ β2

(∆

π

)2

+ λ1 − λ2 ≤ 0, (40)

Λ=

−2µ+ 2δ − λ1

π2

2−λ2

2−λ2

2µ

∗ −2 −2 +β3

2

(∆

π

)4

0

∗ ∗ −2 0∗ ∗ ∗ −η

≤0,

(41)the closed-loop system is globally exponentially stable satis-fying (28). Furthermore, if the strict LMI (41) is feasible forδ = 0, then the closed-loop system is exponentially stable witha small enough decay rate.

Proof. Step 1: We have shown that there exists a local clasicalsolution to (22) subject to (24), where T = T (z0). ByTheorem 6.23.5 of [27], we obtain that the solution existsfor any T > 0 if this solution admits a priori estimate. InStep 3, it will be shown that the feasibility of LMIs (39)-(41)guarantees that the solution of (22) subject to (24) admits apriori bound, which can further guarantee the existence of thesolution for all t ≥ 0.

Step 2: Assume formally that there exists a classical solutionof (22) subject to (24) for all t ≥ 0. Consider V given by (29).Differentiating V along (22) and integrating by parts, we have(33).For the case of point measurements, fj is given by (24). FromLemma 3, we have

η‖fj‖2L2(Ωj) ≤ β1

(∆π

)2

‖zx1‖2L2(Ωj)+ β2

(∆π

)2

‖zx2‖2L2(Ωj)

+β3

(∆π

)4

‖zx1x2‖2L2(Ωj)

for any scalars β1, β2, β3 such that (10) holds.Hence,

N∑j=1

[β1

(∆π

)2

‖zx1‖2L2(Ωj)+ β2

(∆π

)2

‖zx2‖2L2(Ωj)

+β3

(∆π

)4

‖zx1x2‖2L2(Ωj) − η‖fj‖

2L2(Ωj)

]≥ 0.

(42)

From (36), for any λ2 ∈ R we have

λ2

[∫Ω

∇zT∇zdx+

∫Ω

z∆zdx

]= 0. (43)

Similarly, we add to V (t) + 2δV (t) the left-hand side of (34)and (42). Then by taking into account (33), we obtain

V (t) + 2δV (t)

≤ V (t) + 2δV (t) + λ1


2

∫Ωz2dx

]+λ2

[−∫

Ω∇zT∇zdx−

∫Ωz∆zdx

]+

[β1

(∆π

)2

‖zx1‖2L2(Ωj) + β2

(∆π

)2

‖zx2‖2L2(Ωj)

+β3

(∆π

)4


2L2(Ωj)

]≤[2(1− κ) + β1

(∆π

)2

+ λ1 − λ2

] ∫Ωz2x1dx

+

[−2κ+ β2

(∆π

)2

+ λ1 − λ2

] ∫Ωz2x2dx

+β3

(∆π

)4 ∫Ωz2x1x2

dx− 2∫

Ω|∆z|2dx

−(2µ− 2δ + λ1π2

2 )∫

Ωz2dx+ 2µ

N∑j=1

∫Ωjzfjdx

−λ2

∫Ωz∆zdx− η

N∑j=1

∫Ωjf2j dx.

By using (31) and (32), the following inequality holds for allt ≥ 0

V (t) + 2δV (t) ≤N∑j=1

∫Ωj

ψT (x, t)Λψ(x, t)dx (44)

whereψ(x, t) = colz, zx1x1

, zx2x2, fj.

Therefore, the LMIs (39)-(41) yield (28).Step 3: The feasibility of LMIs (39)-(41) yields that the

solution of (22) subject to (24) admits a priori estimate V (t) ≤e−2δtV (0). By Theorem 6.23.5 of [27], continuation of thissolution under a priori bound to entire interval [0,∞).

IV. SAMPLED-DATA REGIONAL STABILIZATION

A. Sampled-data control under averaged measurements

For j = 1, · · · , N ; k = 0, 1, · · · we consider the quantities

fj(x, t) = z(x, t)−

∫Ωjz(ζ, t)dζ

|Ωj |, (45)

gj(t) =1

t− tk

∫Ωj

∫ ttkzs(ζ, s)dsdζ

|Ωj |. (46)

Then the controller (5) subject to (3) leads to the closed-loopsystem

zt + zzx1 + (1− κ)zx1x1−κzx2x2 + ∆2z

= −µN∑j=1

χj(x)[z − fj − (t− tk)gj ],

(x, t) ∈ Ω× [tk, tk+1),

z|∂Ω = 0,∂z

∂n|∂Ω = 0.

(47)




7

Now we use the step method (see e.g. [23], [25]) toestablish the proof of the well-posedness for system (47). Fort ∈ [t0, t1], we consider the following equation:

zt + zzx1+ (1− κ)zx1x1

−κzx2x2+ ∆2z

= −µN∑j=1

χj(x)

∫Ωjz0(x)dx

|Ωj |,

z|∂Ω = 0,∂z

∂n|∂Ω = 0.

(48)

Then system (48) can be represented as an evolution equation(26) subject to

F (z(·, t)) = −z(x, t)zx1(x, t)− (1− κ)zx1x1(x, t)+ κzx2x2(x, t)

−µN∑j=1

χj(x)

∫Ωjz0(x)dx

|Ωj |.

Note that the nonlinearity F (z(·, t)) is locally Lipschitzcontinuous. From Theorem 3.3.3 of [26], it follows thatthere exists a unique local strong solution z(·, t) ∈C([0, T ];D((−A)

12 )) ∩ C1((0, T ];D(A)) of (48) initialized

with z0 ∈ D((−A)12 ) on some interval [0, T ] ⊂ [0, t1], where

T = T (z0) > 0. By Theorem 6.23.5 of [27], we obtainthat if this solution admits a priori estimate, then the solutionexists on the entire [0, t1]. The priori estimate on the solutionsstarting from the domain of attraction will be guaranteed bythe stability conditions that we will provide (see Theorem 1).Then we apply the same line of reasoning step-by-step to thetime segments [t1, t2], [t2, t3], · · · . Following this procedure,we find that the strong solution exists for all t ≥ 0.

In order to derive the stability conditions for (47) we employthe following Lyapunov-Krasovskii functional

V1(t) = p1‖z‖2L2(Ω) + p2‖∆z‖2L2(Ω)

+r(tk+1 − t)∫

Ω

∫ ttke2δ(s−t)z2

s(x, s)dsdx,

t ∈ [tk, tk+1), p1 > 0, p2 > 0, r > 0.

(49)

Remark 4. Note that without delay/sampling behavior, theenergy norm is usually used. In the present work, due tothe sampling terms, we need to use Lyapunov-Krasovskiifunctionals (see e.g. [13], [35]). Therefore, additionally tothe energy norm p1‖z‖2L2(Ω) + p2‖∆z‖2L2(Ω), we employ theterm r(tk+1 − t)

∫Ω


s(x, s)dsdx to deal with thesampling.

For convenience we define

V = D((−A)12 )

with the norm

‖z‖2V = p1‖z‖2L2(Ω) + p2‖∆z‖2L2(Ω).

Here p1 and p2 are positive constants that are related tothe Lyapunov-Krasovskii functional (49). By using Lyapunov-Krasovskii functional (49), in Theorem 1 we provide LMIconditions for regional exponential stability of (47) and for abound on the domain of attraction.

Theorem 1. Consider the closed-loop system (47). Givenpositive scalars C, h, µ, ∆ and δ, let there exist scalars r > 0,

Γ > 0, p1 > 0, p2 > 0, λi ≥ 0 (i = 1, 2) and λ3 ∈ R satisfythe linear matrix inequalities:

Ξi|z=C < 0, Ξi|z=−C < 0, i = 1, 2 (50) p2 − (1 + Γ)1

π2

√1

2∗ Γ

> 0, (51)

where

Ξ1 =

−rhz0

Φ1 −(1− κ)rh

κrh−rh−µrhµrh

∗ −rh

, (52)

Ξ2 =

−rhz0

Φ2 −(1− κ)rh

κrh−rh−µrhµrh

µrh2

∗ −rh

, (53)

Φ1 = φij is a symmetric matrix composed from

φ11 = 2p1(1− κ) + λ1 +2∆2

π2λ2 − λ3, φ15 = −p2z,

φ22 = −2p1κ+ λ1 +2∆2

π2λ2 − λ3,

φ33 = −2p1 + 2δp2, φ34 = 2δp2,

φ35 = −p2(1− κ), φ36 = −λ3

2,

φ44 = −2p1 + 2δp2, φ45 = p2κ, φ46 = −λ3

2,

φ55 = −2p2, φ56 = −p2µ, φ57 = p2µ,

φ66 = −2p1µ+ 2δp1 −π2

2λ1, φ67 = p1µ,

φ77 = −λ2,

Φ2 =

[Φ1 ∗

0 0 0 0 p2µh p1µh 0 −rhe−2δh

]. (54)

Then for any initial state z0 ∈ V satisfying ‖z0‖2V < C2, aunique solution of (47) exists and satisfies

p1‖z‖2L2(Ω) + p2‖∆z‖2L2(Ω)

≤ e−2δt[p1‖z0‖2L2(Ω) + p2‖∆z0‖2L2(Ω)], , t ≥ 0.

Furthermore, if the strict LMI (50) is feasible for δ = 0, thenthe closed-loop system is exponentially stable with a smallenough decay rate.

Proof. The proof is divided into three parts.Step 1: We have shown that there exists a unique local strongsolution to (47) on some interval [0, T ] ⊂ [0, t1]. By Theorem6.23.5 of [27], we obtain that the solution exists on the entire




8

interval [0, t1] if this solution admits a priori estimate. In Step3, it will be shown that the feasibility of LMIs (50), (51)guarantees that the solution of (47) admits a priori bound,which can further guarantee the existence of the solution forall t ≥ 0.

Step 2: Assume formally that there exists a solution of (47)for all t ≥ 0. Differentiating V1 along (47), we have

V1(t) + 2δV1(t) = 2p1

∫Ωzztdx+ 2p2

∫Ω

∆z∆ztdx

−r∫

Ω


s(x, s)dsdx

+r(tk+1 − t)∫

Ωz2t (x, t)dx

+2δp1

∫Ωz2dx+ 2δp2

∫Ω|∆z|2dx.

(55)Substitution of zt from (47) leads to

r(tk+1 − t)∫

Ωz2t (x, t)dx

= r(tk+1− t)N∑j=1

∫Ωj

[−zzx1 − (1− κ)zx1x1 + κzx2x2

−∆2z − µz + µfj + µ(t− tk)gj ]2dx

≤ rhN∑j=1

∫Ωj

[−zzx1−(1− κ)zx1x1

+ κzx2x2−∆2z

−µz + µfj + µ(t− tk)gj ]2dx.

(56)Integration by parts yields

2p1

∫Ωzztdx

= 2p1(1− κ)∫

Ωz2x1dx− 2p1κ

∫Ωz2x2dx− 2p1

∫Ω|∆z|2dx

−2p1µ∫

Ωz2dx+ 2p1µ

N∑j=1

∫Ωjz[fj + (t− tk)gj ]dx,

(57)and

2p2

∫Ω

∆z∆ztdx = 2p2

∫Ω

∆2z · ztdx= 2p2

∫Ω

∆2z[−zzx1− (1− κ)zx1x1

+ κzx2x2−∆2z]dx

−2p2µ∫

Ω∆2z[z − fj − (t− tk)gj ]dx.

(58)The Jensen inequality leads to

−r∫

Ω


s(x, s)dsdx

≤ −re−2δh∫

Ω1

t−tk

[∫ ttkzs(x, s)ds

]2dx

≤ −r(t− tk)e−2δhN∑j=1

∫Ωjg2jdx.

(59)

From (36), we obtain

λ3

[−∫

Ω∇zT∇zdx−

∫Ωz∆zdx

]= 0, (60)

where λ3 ∈ R.Set

η1 = colzx1, zx2

, zx1x1, zx2x2

,∆2z, z, fj,η2 = colη1, gj,β ,

[−z 0 −(1− κ) κ −1 −µ µ µ(t− tk)

].

Then

[−zzx1−(1− κ)zx1x1

+ κzx2x2−∆2z − µz + µfj

+µ(t− tk)gj ]2 = ηT2 β

Tβη2.(61)

From (56) and (61) we have

r(tk+1 − t)∫

Ωz2t (x, t)dx ≤ rh

N∑j=1

∫Ωj

ηT2 βTβη2dx. (62)

Applying S-procedure, we add to V1(t)+2δV1(t) the left-handside of (34), (35), (60). Then,

V1(t) + 2δV1(t)

≤ V1(t) + 2δV1(t) + λ1


2

∫Ωz2dx

]+λ2

[2∆2

π2

∫Ω∇zT∇zdx−

N∑j=1

∫Ωjf2j dx

]+λ3

[−∫

Ω∇zT∇zdx−

∫Ωz∆zdx

]≤

N∑j=1

∫Ωj

h− t+ tkh

ηT1 Φ1η1 +t− tkh

ηT2 Φ2η2dx

+rhN∑j=1

∫Ωj

ηT2 βTβη2dx− (4p1 − 4δp2)

∫Ω

z2x1x2

dx.

(63)Note that LMIs (50) imply that φ33 < 0, i.e. p1 > δp2. As in[22], first we assume that

‖z(·, t)‖C0(Ω) < C, ∀t ≥ 0. (64)

Note thath− t+ tk

h+t− tkh

= 1 and t− tk ≤ h. Under theassumption (64), applying Schur complement to (62), from(61)-(63) we obtain

V1(t) + 2δV1(t)

≤N∑j=1

∫Ωj

h− t+ tkh

[ηT1 1

]Ξ1

[η1

1

]dx

+N∑j=1

∫Ωj

t− tkh

[ ηT2 1 ]Ξ2

[η2

1

]dx

≤ 0

(65)

if Ξ1 < 0, Ξ2 < 0 for all z ∈ (−C,C).Matrices Ξ1 and Ξ2 given by (52), (53) are affine in z. Hence,Ξ1 < 0 and Ξ2 < 0 for all z ∈ (−C,C) if these inequalitieshold in the vertices z = ±C hold, i.e. if LMIs (50) are feasible.

We prove next that (64) holds. Lemma 4 and Schur com-plement theorem lead to

‖z‖2C0(Ω)

≤ 1

2(1 + Γ)

[‖zx1‖2L2(Ω)+‖zx2

‖2L2(Ω)

]+

1

Γ‖zx1x2

‖2L2(Ω)

≤[(1 + Γ)

1

π2+

1

2Γ

]‖∆z‖2L2(Ω) ≤ V1(t).

(66)The last inequality in (66) follows from (51), and for thesecond inequality in (66) we use the Wirtinger’s inequality,(31) and (32). Therefore, it is sufficient to show that

V1(t) < C2, ∀t ≥ 0. (67)

Indeed, for t = 0, the inequality (67) holds. Let (67) befalse for some t1. Then V1(t1) ≥ C2 > V1(0). Since V1 iscontinuous in time, there must exist t∗ ∈ (0, t1] such that

V1(t) < C2 ∀t ∈ [0, t∗) and V1(t∗) = C2. (68)

The first relation of (68), together with the feasibility of (50),guarantees that V1(t) + 2δV1(t) ≤ 0 on [0, t∗). Therefore,V1(t) ≤ V1(0) < C2. This contradicts the second relation of(68). Thus, (67) and consequently, (65) is true, which impliesprovided that ‖z0‖V < C.Note that the feasibility of LMI (50) with δ = 0 implies itsfeasibility with a small enough δ0 > 0. Therefore, if LMI (50)




9

holds for δ = 0, then the closed-loop system is exponentiallystable with a small decay rate.

Step 3: The feasibility of LMIs (50), (51) yields that thesolution of (47) admits a priori estimate V1(t) ≤ e−2δtV1(0).By Theorem 6.23.5 of [27], this solution (under a priori bound)can be continued to entire interval [0,∞).

B. Sampled-data control under point measurements

Under the controller (5) subject to (4), the closed-loopsystem becomes

zt + zzx1+ (1− κ)zx1x1

−κzx2x2+ ∆2z

= −µN∑j=1

χj(x)z(xj , tk), (x, t) ∈ Ω× [tk, tk+1),

z|∂Ω = 0,∂z

∂n|∂Ω = 0.

(69)

Theorem 2. Consider the closed-loop system (69). Givenpositive scalars C, h, µ, ∆ and δ1 < 2δ, let there exist scalarsr > 0, Γ > 0, p1 > 0, p2 > 0, η > 0, λi ≥ 0 (i = 1, 2),λ3 ∈ R and βi > 0 (i = 1, 2, 3) such that (10) is satisfied andthe following LMIs hold:

−2δ1p2 + β3

(∆

π

)4

≤ 0, (70)

Θ =

−δ1p2 −β1

2

(∆

π

)2

−β2

2

(∆

π

)2

∗ −δ1p2 0∗ ∗ −δ1p2

≤ 0, (71)

p2 − (1 + Γ)1

π2

√1

2∗ Γ

> 0, (72)

Λi|z=C < 0, Λi|z=−C < 0, i = 1, 2 (73)

where

Λ1 =

[Θ0 Θ1

∗ −rh

],

Λ2 =

[Θ2 Θ3

∗ −rh

],

Θ0 = θij is a symmetric matrix composed from

θ11 = 2p1(1− κ) + λ1 − λ2, θ22 = −2p1κ+ λ1 − λ2,

θ33 = −2p1 + 2δp2, θ35 = −p2(1− κ), θ36 = −λ2

2,

θ44 = −2p1 + 2δp2, θ45 = p2κ, θ46 = −λ2

2,

θ55 = −2p2, θ56 = −p2µ, θ57 = p2µ,

θ66 = −2p1µ+ 2δp1 −π2

2λ1, θ67 = p1µ,

θ77 = −η,

Θ1 = [ −rhz 0 −(1− κ)rh κrh −rh −µrh µrh ]T

(74)

Θ2 =

[Θ0 ∗

0 0 0 0 p2µh p1µh 0 −rhe−2δh

], (75)

Θ3 = [ Θ1 µrh2 ]T (76)

Then for any initial state z0 ∈ V satisfying ‖z0‖2V < C2, aunique solution of (69) exists and satisfies

p1‖z‖2L2(Ω) + p2‖∆z‖2L2(Ω)

≤ e−2σt[p1‖z0‖2L2(Ω) + p2‖∆z0‖2L2(Ω)], t ≥ 0,

where σ is a unique positive solution of (18).

Proof. See Appendix.

V. NUMERICAL EXAMPLE

Consider the system (1) under the sampled-data control law(5) with the averaged measurements (3). Here we choose µ =0.95. By verifying LMI conditions of Theorem 1 with δ = 0.1,κ = −0.5, ∆ = 1/4, C = 2, we find that the closed-loopsystem (47) preserves the exponential stability within a givendomain of initial conditions ‖z0‖2V < 4 for tk+1 − tk ≤ h ≤0.39. Note that ∆ = 1/4 corresponds to N = 16 squaresubdomains with the sides length 1/4. The feasible solutionsof LMIs with h = 0.35 are given as follows: p1 = 80.6354,p2 = 5.145.

We compute the solution of the closed-loop system (47)numerically via finite element method. Let ξ = 0.1 and M =1/ξ = 10. Define (xi1, x

j2) = (iξ, jξ), i, j = 0, 1, 2, . . . ,M .

We divide Ω on M2 = 100 squares Rij defined by

Rij , (x1, x2) ∈ Ω|xi1 ≤ x1 ≤ xi+11 , xj2 ≤ x2 ≤ xj+1

2 .

On the node (xi1, xj2), four finite element basis functions are

selected as

N i,j1 (x) =

(

1− x1 − xi1ξ

)(x2 − xj2

ξ

), x ∈ Rij ,

0, otherwise

N i,j2 (x) =

(

1− x1 − xi1ξ

)(1− x2 − xj2

ξ

), x ∈ Rij ,

0, otherwise

N i,j3 (x) =

(x1 − xi1

ξ

)(1− x2 − xj2

ξ

), x ∈ Rij ,

0, otherwise

N i,j4 (x) =

(x1 − xi1

ξ

)(x2 − xj2

ξ

), x ∈ Rij ,

0, otherwise

We consider the Galerkin approximation solution of theclosed-loop system in finite dimensional space generated bythese basis functions, which takes the form

zM (x, t) =4∑k=1

M∑i,j=1

mi,jk (t)N i,j

k (x),

where mi,jk (t) are determined by standard finite element

Galerkin method to satisfy some ODEs. Fig. 2 shows snap-shots of the state z(x, t) = z(x1, x2, t) at different times forthe closed-loop system (47) with tk+1 − tk = 0.35, N = 16and initial condition z(x1, x2, 0) = 0.236sin(πx1)sin(πx2),(x1, x2)∈ Ω = (0, 1)2. It is seen that the closed-loop systemis stable. Fig. 3 demonstrates the time evolution of V1(t) viathe finite difference method, where the steps of space and timeare taken as 1/4 and 0.00025, respectively.




10

By verifying the LMI conditions of Theorem 1, we obtainthe maximum value h = 0.39 that preserves the exponentialstability. By simulation of the solution to the closed-loopsystem starting from the same initial condition, we find thatstability is preserved for essentially larger values of h tillapproximately h = 2.45.

0

2

1

4

6

1

z(x

1,x

2,t

)

8

x2

0.5

x1

10

0.5

0 0

0

2

1

4

6

1

z(x

1,x

2,t

)

8

x2

0.5

x1

10

0.5

0 0

0

2

1

4

6

1

z(x

1,x

2,t

)

8

x2

0.5

x1

10

0.5

0 0

Fig. 2. Snapshots of state z(x1, x2, t) at different time t ∈ 0, 1.4, 14

Fig. 3. Lyapunov function V1(t)

For the sampled-data controller (5) under the point mea-surement (4), by choosing µ = 0.95, and using Yalmipwe verify LMI conditions of Theorem 2 with δ = 0.2,δ1 = 0.15, κ = −0.5, ∆ = 1/4, C = 2. Then we findthat the resulting closed-loop system is exponentially stablefor tk+1 − tk ≤ h ≤ 0.37 for any initial values satisfying‖z0‖L2(0,1) < 1.

Since point measurements use less information on the state,the point measurements allow smaller sampling intervals thanaveraged measurements. Simulations of the solutions to theclosed-loop system under the point measurements confirm thetheoretical results.

VI. CONCLUSION

The present paper discusses sampled-data control of 2DKSE under the spatially distributed averaged or point mea-

surements. Sufficient LMI conditions have been investigatedsuch that the regional stability of the closed-loop system isguaranteed.

Our results are applicable to sampled-data controller designof high dimensional distributed parameter systems. Our nextstep places its main focus on H∞ filtering problem of highdimensional coupled ODE-PDE/PDE-PDE system.

APPENDIX

Proof of Theorem 2

Step 1: We have shown that there exists a unique local strongsolution to (69) on some interval [0, T ] ⊂ [0, t1]. By Theorem6.23.5 of [27], we obtain that the solution exists on the entireinterval [0, t1] if this solution admits a priori estimate. In Step3, it will be shown that the feasibility of LMIs (50), (51)guarantees that the solution of (69) admits a priori bound,which can further guarantees the existence of the solution forall t ≥ 0.

Step 2: Assume formally that there exists a strong solutionof (69) starting from ‖z0‖V < C for all t ≥ 0. DifferentiatingV1 along (69), we obtain the inequality (55). Denote

fj(x, t) = z(x, t)− z(xj , t) =

∫ x

xj

zξ(ξ, t)dξ, (A.77)

From Lemma 3, we have

η‖fj‖2L2(Ωj) ≤β1

(∆π

)2

‖zx1‖2L2(Ωj)+ β2

(∆π

)2

‖zx2‖2L2(Ωj)

+β3

(∆π

)4

‖zx1x2‖2L2(Ωj)

(A.78)for any scalars β1, β2, β3 such that (10) holds.Hence,

N∑j=1

[β1

(∆π

)2

‖zx1‖2L2(Ωj) + β2

(∆π

)2

‖zx2‖2L2(Ωj)

+β3

(∆π

)4


2L2(Ωj)

]≥ 0.

(A.79)Denote

ρ(x, t) ,1

t− tk

∫ t

tk

zs(x, s)ds.

Then, we have

z(x, t) = z(x, tk) + (t− tk)ρ(x, t).

By using Jensen’s inequality, we have

−r∫

Ω


s(x, s)dsdx

≤ −re−2δh(t− tk)∫

Ωρ2(x, t)dx.

(A.80)

Integration by parts leads to

2p1

∫Ωzztdx

= 2p1(1− κ)∫

Ωz2x1dx− 2p1κ

∫Ωz2x2dx− 2p1

∫Ω|∆z|2dx

+2p1µN∑j=1

∫Ωjz [(t− tk)ρ(x, t) + fj(x, tk)] dx

−2p1µ∫

Ωz2dx,

(A.81)




11

2p2

∫Ω

∆z∆ztdx = 2p2

∫Ω

∆2z · ztdx= 2p2

∫Ω

∆2z[−zzx1− (1− κ)zx1x1


−2p2µN∑j=1

∫Ωj

∆2z[z(x, t)− (t− tk)ρ(x, t)− fj(x, tk)]dx.

(A.82)Then from (A.80)-(A.82), adding (A.79) into V1 + 2δV1 weobtain

V1(t) + 2δV1(t)− δ1 supθ∈[−h,0]

V1(t+ θ)

≤ V1(t) + 2δV1(t)− δ1V1(tk)

+λ1


2

∫Ωz2dx

]+λ2

[−∫

Ω∇zT∇zdx−

∫Ωz∆zdx

]+

[β1

(∆π

)2 ∫Ωz2x1

(x, tk)dx+ β2

(∆π

)2 ∫Ωz2x2

(x, tk)dx

+β3

(∆π

)4 ∫Ωz2x1x2

(x, tk)dx− ηN∑j=1

∫Ωjf2j (x, tk)dx

]≤ 2p1(1− κ)

∫Ωz2x1dx− 2p1κ

∫Ωz2x2dx− 2p1

∫Ω|∆z|2dx

+2p1µN∑j=1

∫Ωjz [(t− tk)ρ+ fj(x, tk)] dx− 2p1µ

∫Ωz2dx

+2p2

∫Ω

∆2z[−zzx1− (1− κ)zx1x1


−2p2µ∫

Ω∆2z[z − (t− tk)ρ− fj(x, tk)dξ]dx

−re2δh(t− tk)∫

Ωρ2(x, t)dx+ r(tk+1 − t)

∫Ωz2t (x, t)dx

+2δp1

∫Ωz2dx+ 2δp2

∫Ω|∆z|2dx− δ1p1

∫Ωz2(x, tk)dx

−δ1p2

∫Ω|∆z(x, tk)|2dx+ (λ1 − λ2)

∫Ω

[z2x1

+ z2x2

]dx

−λ1π2

2

∫Ωz2dx− λ2

∫Ωz∆zdx+ β1

(∆π

)2 ∫Ωz2x1

(x, tk)dx

+β2

(∆π

)2 ∫Ωz2x2

(x, tk)dx+ β3

(∆π

)4 ∫Ωz2x1x2

(x, tk)dx

−ηN∑j=1

∫Ωjf2j (x, tk)dx.

(A.83)Using (32), we have

−2p1

∫Ω|∆z|2dx

= −2p1

∫Ω

[z2x1x1

(x, t) + z2x2x2

(x, t) + 2z2x1x2

(x, t)]dx,(A.84)

−δ1p2

∫Ω|∆z(x, tk)|2dx

= −δ1p2

∫Ω

[z2x1x1

(x, tk) + z2x2x2

(x, tk) + 2z2x1x2

(x, tk)]dx.(A.85)

Set

η0 = zx1(x, t), zx2

(x, t), zx1x1(x, t), zx2x2

(x, t),∆2z(x, t),z(x, t), fj(x, tk),

η1 = zx1(x, t), zx2(x, t), zx1x1(x, t), zx2x2(x, t),∆2z(x, t),z(x, t), fj(x, tk), ρ(x, t),

η = z(x, tk), zx1x1(x, tk), zx2x2

(x, tk)

Substituting (A.84), (A.85) into (A.83), we obtain

V1(t) + 2δV1(t)− δ1 supθ∈[−h,0]

V1(t+ θ)

≤N∑j=1

∫Ωj

h− t+ tkh

ηT0 Θ0η0 +t− tkh

ηT1 Θ1η1dx

+∫

ΩηT Θηdx+ r(tk+1 − t)

∫Ωz2t (x, t)dx

−(4p1 − 4δp2)∫

Ωz2x1x2

(x, t)dx

−[2δ1p2 − β3( ∆π )4]

∫Ωz2x1x2

(x, tk)dx.(A.86)

Substitution of zt from (69) yields

r(tk+1 − t)∫

Ωz2t (x, t)dx

≤ rhN∑j=1

∫Ωj

[−zzx1 −(1− κ)zx1x1 + κzx2x2 −∆2z

−µz + µ(t− tk)ρ+ µfj(x, tk)]2dx.(A.87)

Set ψ = colzx1(x, t), zx1x1

(x, t), zx2x2(x, t),∆2z(x, t),

z(x, t), fj(x, tk), ρ(x, t) and

β ,[−z −(1− κ) κ −1 −µ µ µ(t− tk)

].

Then[−zzx1

−(1− κ)zx1x1+ κzx2x2

−∆2z − µz + µfj(x, tk)+µ(t− tk)ρ]2 = ψTβTβψ.

(A.88)Application of Schur complement theorem to (A.88), togetherwith (A.86) and (A.87), implies

V1(t) + 2δV1(t)− δ1 supθ∈[−h,0]

V1(t+ θ) ≤ 0

if (70)-(72) are satisfied, and Λ1 < 0, Λ2 < 0 hold for allz ∈ (−C,C). Similar to Theorem 1, LMI (73) imply Λ1 < 0,Λ2 < 0 for all z ∈ (−C,C). Thus the result is established viaHalanay’s inequality.

Step 3: The feasibility of LMIs (70)-(73) yields thatthe solution of (69) admits a priori estimate V1(t) ≤e−2σt sup

−h≤θ≤0V1(θ). By Theorem 6.23.5 of [27], continuation

of this solution under a priori bound to entire interval [0,∞).

Acknowledgment

The authors would like to thank Prof. Bao-Zhu Guo for usefuldiscussions related to the proof of well-posedness result.

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Wen Kang received her B.S. degree from WuhanUniversity in 2009, and the Ph.D. degree fromAcademy of Mathematics and Systems Science,Chinese Academy of Sciences, Beijing, China, in2014, both in Mathematics. From 2014 to 2017 shewas a Lecturer at Harbin Institute of Technology.From November 2015 to November 2017 she was aPostdoctoral Researcher at the School of ElectricalEngineering, Tel Aviv University, Israel. Currentlyshe is an Associate Professor at School of Au-tomation and Electrical Engineering, University of

Science and Technology Beijing.She serves as Associate Editor in IMA Journal of Mathematical Control

and Information. Her research interests include distributed parameter systems,time-delay systems, event-triggered control.

Emilia Fridman received the M.Sc.degree fromKuibyshev State University, USSR, in 1981 and thePh.D. degree from Voronezh State University, USSR,in1986, all in Mathematics. From 1986 to 1992 shewas an Assistant and Associate Professor in theDepartment of Mathematics at Kuibyshev Instituteof Railway Engineers, USSR. Since 1993 she hasbeen at Tel Aviv University, where she is currentlyProfessor of Electrical Engineering-Systems. Since2018, she has been the incumbent for Chana andHeinrich Manderman Chair on System Control at

Tel Aviv University. She has held visiting positions at the WeierstrassInstitute for Applied Analysis and Stochastics in Berlin (Germany), INRIAin Rocquencourt (France), Ecole Centrale de Lille (France), ValenciennesUniversity (France), Leicester University (UK), Kent University (UK), CIN-VESTAV (Mexico), Zhejiang University (China), St. Petersburg IPM (Russia),Melbourne University (Australia), Supelec (France), KTH (Sweden).

Her research interests include time-delay systems, networked control sys-tems, distributed parameter systems, robust control, singular perturbations andnonlinear control. She has published two monographs and about 200 articlesin international scientific journals. In 2014 she was Nominated as a HighlyCited Researcher by Thomson ISI. She serves/served as Associate Editor inAutomatica, SIAM Journal on Control and Optimization and IMA Journalof Mathematical Control and Information. She is currently a member of theIFAC Council. She is IEEE Fellow since 2019.


Sampled-data control of 2D Kuramoto-Sivashinsky equation

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