New Stability and Exact Observability Conditions for Semilinear Wave Equations

# New Stability and Exact Observability Conditions for Semilinear Wave Equations

## Abstract

The problem of estimating the initial state of 1-D wave equations with globally Lipschitz nonlinearities from boundary measurements on a finite interval was solved recently by using the sequence of forward and backward observers, and deriving the upper bound for exact observability time in terms of Linear Matrix Inequalities (LMIs) [5]. In the present paper, we generalize this result to n-D wave equations on a hypercube. This extension includes new LMI-based exponential stability conditions for n-D wave equations, as well as an upper bound on the minimum exact observability time in terms of LMIs. For 1-D wave equations with locally Lipschitz nonlinearities, we find an estimate on the region of initial conditions that are guaranteed to be uniquely recovered from the measurements. The efficiency of the results is illustrated by numerical examples.

D
1

footnoteinfo]This work was supported by Israel Science Foundation (grant no. 1128/14).

a]Emilia Fridman, a]Maria Terushkin

istributed parameter systems; wave equation; Lyapunov method; LMIs; exact observability.

## 1 Introduction

Lyapunov-based solutions of various control problems for finite-dimensional systems can be formulated in the form of Linear Matrix Inequalities (LMIs) [3]. The LMI approach to distributed parameter systems is capable of utilizing nonlinearities and of providing the desired system performance (see e.g. [4, 7, 12]). For 1-D wave equations, several control problems were solved by using the direct Lyapunov method in terms of LMIs [8, 5]. However, there have not been yet LMI-based results for n-D wave equations, though the exponential stability of the n-D wave equations in bounded spatial domains has been studied in the literature via the direct Lyapunov method (see e.g. [18, 9, 1, 6]).

The problem of estimating the initial state of 1-D wave equations with globally Lipschitz nonlinearities from boundary measurements on a finite interval was solved recently by using the sequence of forward and backward observers, and deriving the upper bound for exact observability time in terms of LMIs [5]. In the present paper, we generalize this result to n-D wave equations on a hypercube. This extension includes new LMI-based exponential stability conditions for n-D wave equations. Their derivation is based on n-D extensions of the Wirtinger (Poincare) inequality [10] and of the Sobolev inequality with tight constants, which is crucial for the efficiency of the results. As in 1-D case, the continuous dependence of the reconstructed initial state on the measurements follows from the integral input-to-state stability of the corresponding error system, which is guaranteed by the LMIs for the exponential stability. Some preliminary results on global exact observability of multidimensional wave PDEs will be presented in [?].

Another objective of the present paper is to study regional exact observability for systems with locally Lipschitz in the state nonlinearities. Here we restrict our consideration to 1-D case, and find an estimate on the region of initial conditions that are guaranteed to be uniquely recovered from the measurements. Note that our result on the regional observability cannot be extended to multi-dimensional case (see Remark 4 below for explanation and for discussion on possible n-D extensions for different classes of nonlinearities). The efficiency of the results is illustrated by numerical examples.

The presented simple finite-dimensional LMI conditions complete the theoretical qualitative results of e.g. [15] (where exact observability of linear systems in a Hilbert space was studied via a sequence of forward and backward observers) and [2] (where local exact observability of abstract semilinear systems was considered).

Notation: denotes the -dimensional Euclidean space with the norm , is the space of real matrices. The notation with means that is symmetric and positive definite. For the symmetric matrix , and denote the minimum and the maximum eigenvalues of respectively. The symmetric elements of the symmetric matrix will be denoted by . Continuous functions (continuously differentiable) in all arguments, are referred to as of class (of class ). is the Hilbert space of square integrable , where , with the norm . For the scalar smooth function denote by () the corresponding partial derivatives. For define , . is the Sobolev space of absolutely continuous functions with the square integrable . is the Sobolev space of scalar functions with absolutely continuous and with .

## 2 Observers and exponential stability of n-D wave equations

### 2.1 System under study and Luenberger type observer

Throughout the paper we denote by the n-D unit hypercube with the boundary . We use the partition of the boundary:

 ΓD={x=(x1,...,xn)T∈Γ: ∃p∈1,...,n s.t. xp=0}ΓN,p={x∈Γ:xp=1},ΓN=⋃p=1,…,nΓN,p.

Here subscripts D and N stand for Dirichlet and for Neumann boundary conditions respectively.

We consider the following boundary value problem for the scalar n-D wave equation:

 ztt(x,t)=Δz(x,t)+f(z,x,t)inΩ×(t0,∞),z(x,t)=0onΓD×(t0,+∞),∂∂νz(x,t)=0onΓN×(t0,∞), (2.1)

where is a function, denotes the outer unit normal vector to the point and is the normal derivative. Let be the known bound on the derivative of with respect to :

 |fz(z,x,t)|≤g1∀(z,x,t)∈\Blackboard Rn+2. (2.2)

Since is a unit hypercube, the boundary conditions on can be rewritten as

 zxp(x,t)∣∣xp=1=0∀xi∈[0,1], i≠p, p=1,…,n.

Consider the following initial conditions:

 z(x,t0)=z0(x), zt(x,t0)=z1(x),x∈Ω. (2.3)

The boundary measurements are given by

 y(x,t)=zt(x,t)onΓN×(t0,∞). (2.4)

Similar to [5], the boundary-value problem (2.1) can be represented as an abstract differential equation by defining the state and the operators

 A=[0IΔz0],F(ζ,t)=[0F1(ζ0,t)],

where is defined as so that it is continuous in for each . The differential equation is

 ˙ζ(t)=Aζ(t)+F(ζ(t),t),t≥t0 (2.5)

in the Hilbert space , where

 H1ΓD(Ω)={ζ0∈H1(Ω) ∣∣∣ ζ0|ΓD=0}

and . The operator has the dense domain

 D(A)={(ζ0,ζ1)T∈H1ΓD(Ω)×H1ΓD(Ω)∣∣∣ Δζ0∈L2(Ω)and ∂∂νζ0|ΓN=−bζ1|ΓN},

where . Here the boundary condition holds in a weak sense (as defined in Sect. 3.9 of [16]), i.e. the following relation holds:

 ⟨Δζ0,ϕ⟩L2(Ω)+⟨∇ζ0,∇ϕ⟩[L2(Ω)]n=−b⟨ζ0,ϕ⟩L2(ΓN)∀ϕ∈H1ΓD(Ω).

The operator is m-dissipative (see Proposition 3.9.2 of [16]) and hence it generates a strongly continuous semigroup. Due to (2.2), the following Lipschitz condition holds:

 ∥F1(ζ0,t)−F1(¯ζ0,t)∥L2≤g1∥ζ0−¯ζ0∥L2 (2.6)

where Then by Theorem 6.1.2 of [14], a unique continuous mild solution of (2.5) in initialized by

 ζ0(t0)=z0∈H1ΓD(Ω), ζ1(t0)=z1∈L2(Ω)

exists in . If , then this mild solution is in and it is a classical solution of (2.1) with (see Theorem 6.1.5 of [14]).

We suggest a Luenberger type observer of the form:

 ˆztt(x,t)=Δˆz(x,t)+f(ˆz,x,t),t≥t0,x∈Ω (2.7)

under the initial conditions and the boundary conditions

 ˆz(x,t)=0on ΓD×(t0,∞)∂∂νˆz(x,t)=k[y(x,t)−ˆzt(x,t)]on ΓN×(t0,∞) (2.8)

where is the injection gain.

The well-posedness of (2.7), (2.8) will be established by showing the well-posedness of the estimation error . Taking into account (2.1), (2.3) we obtain the following PDE for the estimation error :

 ett(x,t)=Δe(x,t)+ge(x,t)t≥t0,x∈Ω (2.9)

under the boundary conditions

 e(x,t)=0on ΓD×(t0,∞)∂∂νe(x,t)=−ket(x,t)on ΓN×(t0,∞). (2.10)

Here and

 g=g(z,e,x,t)=∫10fz(z+(θ−1)e,x,t)dθ.

The initial conditions for the error are given by

 e(x,t0)=z1(x)−z(⋅,t0),et(x,t0)=z2(x)−zt(⋅,t0)

The boundary conditions on can be presented as

 exp(x,t)∣∣xp=1=−ket(x,t)∀xi∈[0,1], i≠p, p=1,…,n.

Let be a mild solution of (2.1). Then is continuous and, thus, the function defined as

 F2(ζ0,t)=f(z,x,t)−f(z−ζ0,x,t)

satisfies the Lipschitz condition (2.6), where is replaced by . By the above arguments, where in the definition of we have , the error system (2.9), (2.10) has a unique mild solution initialized by Therefore, there exists a unique mild solution to the observer system (2.7), (2.8) with the initial conditions . If then is a classical solution of 2.9), (2.10) with for . Hence, if and , there exists a unique classical solution to the observer system (2.7), (2.8) with for .

### 2.2 Lyapunov function and useful inequalities

We will derive further sufficient conditions for the exponential stability of the error wave equation (2.9) under the boundary conditions (2.10). Let

 E(t)=12∫Ω[|∇e|2+e2t]dx, (2.11)

be the energy of the system. Consider the following Lyapunov function for (2.9), (2.10):

 V(t)=E(t)+χ∫Ω[2(xT⋅∇e)+(n−1)e]etdx+χk(n−1)2∫ΓNe2dΓ

with some constant . Note that the above Lyapunov function without the last term was considered in [1, 6, 18]. The time derivative of this new term of cancels the same term with the opposite sign in the time derivative of (cf. (2.23) below) leading to LMI conditions for the exponential convergence of the error wave equation.

We will employ the following n-D extensions of the classical inequalities:

###### Lemma 1

Consider such that . Then the following n-D Wirtinger’s inequality holds:

 ∫Ω[4π2n|∇e|2−e2]dx≥0. (2.12)

Moreover,

 ∫ΓNe2dΓ≤∫Ω|∇e|2dx. (2.13)

Proof :  Since , by the classical 1-D Wirtinger’s inequality [10]

 ∫10e2dx1≤4π2∫10e2x1dx1.

Integrating the latter inequality in we obtain

 ∫Ωe2dx≤4π2∫Ωe2xpdx

with . Clearly the latter inequality holds for all , which after summation in yields (2.12).

Since we have by Sobolev’s inequality

 e2(x)∣∣x1=1≤∫10e2x1dx1 ∀xi∈[0,1], i≠1,

that after integration in leads to

 ∫ΓN,pe2dΓ≤∫Ωe2xpdx

with . The latter inequality holds leading after summation in to (2.13).

### 2.3 Exponential stability of n-D wave equation

In this section we derive LMI conditions for the exponential stability of the estimation error equation. We start with the conditions for the positivity of the Lyapunov function:

###### Lemma 2

Let there exist positive scalars and such that

 Φ0\lx@stackrelΔ=⎡⎢ ⎢ ⎢⎣12−λ04π2n√nχ0∗12n−12χ∗∗λ0⎤⎥ ⎥ ⎥⎦>0. (2.14)

Then the Lyapunov function is bounded as follows:

 αE(t)≤V(t)≤βE(t),α=2λmin(Φ0),β=2(1+2π2n)λmax(Φ1)+χk(n−1), (2.15)

where .

Proof :  By Cauchy-Schwarz inequality we have

 |xT⋅∇e|≤|x||∇e|≤√n|∇e|, (2.16)

Then

 |χ∫Ω[2(xT⋅∇e)+(n−1)e]etdx|≤χ∫Ω[2√n|∇e||et|+(n−1)|e||et|]dx,

 V(t)≥12∫Ω[e2t+|∇e|2]dx−χ∫Ω[2√n|∇e||et|+(n−1)|e||et|]dx. (2.17)

Taking into account the n-D Wirtinger inequality (2.12), we further apply S-procedure [17] 2, where we subtract from the right-hand side of (2.17) the nonnegative term

 λ0∫Ω[4π2n|∇e|2−e2]dx (2.18)

with :

 V(t)≥12∫Ω{e2t+|∇e|2}dx−χ∫Ω[2√n|∇e||et|+(n−1)|e||et|]dx−λ0∫Ω[4π2n|∇e|2−e2]dx=∫ΩηTΦ0η,

where .

Similarly

 V(t) ≤ 12∫Ω[e2t+|∇e|2]dx (2.19) + χ∫Ω[2√n|∇e||et|+(n−1)|e||et|]dx + χk(n−1)2∫ΓNe2dΓ ≤ ηT1Φ1η1+χk(n−1)2∫ΓNe2dΓ

with .

Then (2.15) follows from

 λmin(Φ0)[2E(t)+∫Ωe2dx]≤V(t)≤λmax(Φ1)[2E(t)+∫Ωe2dx]+χk(n−1)2∫ΓNe2dΓ

and from the inequalities (2.12) and (2.13).

We are looking next for conditions that guarantee along the classical solutions of the wave equation initiated from . Then and, thus, (2.15) yields

 ∫Ω[|∇e|2(x,t)+e2t(x,t)]dx (2.20) ≤βαe−2δ(t−t0)∫Ω[|∇(z0(x)−^z(x,t0))|2 +(z1(x)−^zt(x,t0))2]dx.

Since is dense in the same estimate (2.20) remains true (by continuous extension) for any initial conditions . For such initial conditions we have mild solutions of (2.1), (2.3).

###### Theorem 1

Given and , assume that there exist positive constants and that satisfy the LMI (2.14) and the following LMIs:

 Ψ1\lx@stackrelΔ=−k+(1+k2n)χ≤0,Ψ2\lx@stackrelΔ=⎡⎢ ⎢⎣ψ22δ√nχ√ng1χ∗−χ+δ12g1+δ(n−1)χ∗∗−λ1+g1(n−1)χ⎤⎥ ⎥⎦≤0,ψ2=−χ+δ(1+χk(n−1))+λ14π2n. (2.21)

Then, under the condition (2.2), solutions of the boundary-value problem (2.9), (2.10) satisfy (2.20), where and are given by (2.15), i.e. the system governed by (2.9), (2.10) is exponentially stable with a decay rate .

Proof :  Differentiating in time we obtain

 ˙V(t)=˙E(t)+χddt[∫Ω[2(xT⋅∇e)+(n−1)e]etdx]+χk(n−1)∫ΓNeetdΓ

We have

 ˙E(t)=∫Ω((∇e)T∇(et)+etett)dx.

Applying Green’s formula to the first integral term, substituting and taking into account (2.2), we find

 ˙E(t)=∫Γet∂e∂νdΓ−∫ΩetΔedx+∫Ωet[Δe+ge]dx≤−k∫ΓNe2tdΓ+g1∫Ω|e||et|dx

Furthermore, we have

 ddt{∫Ω[2xT∇e+(n−1)e]etdx}=∫Ωddt[2xT∇e+(n−1)e]etdx+∫Ω[2xT∇e+(n−1)e][Δe+ge]dx.

Then Green’s formula leads to (see (11.35) of [13])

 ddt{∫Ω[xT∇e+(n−1)e]etdx}=2∫ΓNxT∇e ∂e∂νdΓ−∫ΓN(xTν)|∇e|2dΓ+(n−2)∫Ω|∇e|2dx+∫ΓN(xTν)e2tdΓ−n∫Ωe2tdx+(n−1)∫Ωe2tdx+(n−1)∫ΓNe∂e∂νdΓ−(n−1)∫Ω|∇e|2dx+∫Ω[2xT∇e+(n−1)e]gedx (2.22)

Noting that on and taking into account the boundary conditions we obtain

 ddt{∫Ω[2xT∇e+(n−1)e]etdx}=−∫Ω{e2t+|∇e|2+[2xT∇e+(n−1)e]ge}dx−∫ΓN[|∇e|2+2kxT∇eet]dΓ+∫ΓN[e2t−k(n−1)eet]dΓ (2.23)

By inequalities (2.16) and (2.2) we have

 ∫Ω[2xT∇e+(n−1)e]gedx≤∫Ω[2|xT∇e||g||e|dx+(n−1)g1e2]dx≤∫Ω[2√ng1|∇e||e|+(n−1)g1e2]dx.

Further due to (2.16)

 −∫ΓN2kxT∇eetdΓ≤2k∫ΓN|xT∇e||et|dΓ≤2k√n∫ΓN|∇e||et|dΓ.

Then by completion of squares we find

 −∫ΓN[|∇e|2+2kxT∇eet]dΓ≤∫ΓN[k2ne2t−[|∇e|−k√n|et|]2]dΓ≤k2n∫ΓNe2tdΓ.

Summarizing we obtain

 ˙V(t)≤[χ(1+k2n)−k]∫ΓNe2tdΓ−∫Ω[χ[e2t+|∇e|2]−[2√nχg1|∇e||e|+(n−1)χg1e2+g1|e||et|]]dx. (2.24)

Therefore, employing (2.19) we arrive at

 ˙V(t)+2δV(t)≤∫ΓN[Ψ1e2t+δχk(n−1)e2]dΓ−(χ−δ)∫Ω[e2t+|∇e|2]dx+∫Ω[2√nχg1|∇e||e|+(n−1)χg1e2++[g1+2δχ(n−1)]|e||et|+4δχ√n|∇e||et|]dx. (2.25)

By taking into account Wirtinger’s inequality (2.12), we add to (2.25) the nonnegative term (2.18), where is replaced by . Denote . Then after employing the bound (2.13) we arrive at

 ddtV(t)+2δV(t)≤Ψ1∫ΓNe2tdΓ+∫ΩηT2Ψ2η2dx≤0

if the LMIs (2.21) are feasible.

###### Remark 1

For the term of leads to in (cf. (2.22)).

## 3 Exact observability of n-D wave equation

Our next objective is to recover (if possible) the unique initial state (2.3) of the solution to (2.1)-(2.3) from the measurements on the finite time interval

 y(x,t)=zt(x,t) on ΓN×[t0,t0+T], T>0. (3.1)
###### Definition 1

[5] The system (2.1), (2.3) with the measurements (3.1) is called exactly observable in time , if

(i) for any initial condition it is possible to find a sequence from the measurements (3.1) such that (i.e. it is possible to recover the unique initial state as );

(ii) there exists a constant such that for any initial conditions and leading to the measurements and and to the corresponding sequences and , the following holds:

 ∥limm→∞[zm0,zm1]T−limm→∞[¯zm0,¯zm1]T∥2H≤C∫t0+Tt0∫ΓN|y(x,t)−¯y(x,t)|2dΓdt. (3.2)

The time is called the observability time.

The system is called regionally exactly observable if the above conditions hold for all with for some .

Note that (3.2) means the continuous in the measurements recovery of the initial state. In this section we will derive LMI sufficient conditions for n-D wave equations with globally Lipschitz in the first argument , where (2.2) holds globally in . In Section 4, we will present LMI-based conditions for the regional observability for 1-D wave equation, where (2.2) holds locally in .

### 3.1 Iterative forward and backward observer design

In order to recover the initial state of the solution to (2.1) from the measurements (3.1) we use the iterative procedure as in [15]. Define the sequences of forward and backward observers with the injection gain :

 z(m)tt(x,t)=Δz(m)(x,t)+f(z(m)(x,t),x,t),z(m)(x,t)=0,x∈ΓD,∂∂νz(m)(x,t)=k[y(x,t)−