Finite-Parameters Feedback Control for Stabilizing Damped Nonlinear Wave Equations

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by  Varga K. Kalantarov

FINITE-PARAMETERS FEEDBACK CONTROL FOR STABILIZING
DAMPED NONLINEAR WAVE EQUATIONS
VARGA K. KALANTAROV AND EDRISS S. TITI
5
1 Abstract. Inthispaperweintroduceafinite-parametersfeedbackcontrolalgorithmfor
0
stabilizing solutions of various classes of damped nonlinear wave equations. Specifically,
2
stabilizationthezerosteadystatesolutionofinitialboundaryvalueproblemsfornonlin-

n
earweaklyandstronglydampedwaveequations,nonlinearwaveequationwithnonlinear
a
damping term and some related nonlinear wave equations, introducing a feedback con-
J
trol terms that employ parameters, such as, finitely many Fourier modes, finitely many
3
volume elements and finitely many nodal observables and controllers. In addition, we

] alsoestablishthe stabilizationofthe zerosteadystate solutiontoinitial boundaryvalue
P
problem for the damped nonlinear wave equation with a controller acting in a proper
A
subdomain. Notably, the feedback controllers proposed here can be equally applied for
.
h stabilizing other solutions of the underlying equations.
t
a
m
[

1. Introduction
1
v
6 This paper is devoted to the study of finite-parameters feedback control for stabiliz-
5 ing solutions of initial boundary value problems for nonlinear damped wave equations.
5
Feedback control, stabilization and control of wave equations is a well established area
0
0 of control theory. Many interesting results were obtained, in the last dacades, on stabi-
.
1 lization of linear and nonlinear wave equations (see , e.g., [4], [17], [20],[23],[25],[27],[28]
0 and references therein). Most of the problems considered are problems of stabilization by
5
interior and boundary controllers involving linear or nonlinear damping terms. However,
1
: only few results are known on feedback stabilization of linear hyperbolic equation by em-
v
i ploying finite-dimensional controllers (see, e.g., [3] and references therein).
X
We study the problem of feedback stabilization of initial boundary value problem for
r
a damped nonlinear wave equation
∂2u−∆u+b∂ u−au+f(u) = −µw, x ∈ Ω,t > 0, (1.1)
t t
nonlinear wave equation with nonlinear damping term
∂2u−∆u+bg(∂ u)−au+f(u) = −µw, x ∈ Ω,t > 0, (1.2)
t t
and the strongly damped wave equation
∂2u−∆u−b∆∂ u−λu+f(u) = −µw, x ∈ Ω,t > 0. (1.3)
t t
Date: January 3, 2015.
1991 Mathematics Subject Classification. 35B40,35B41, 35Q35.
Key words and phrases. damped wave equation, strongly damped wave equation, feedback control,
stabilization, finite-parameters feedback control.
1
2 FEEDBACK CONTROL
Here and in what follows µ,a,b,ν are given positive parameters, w is a feedback control
input (different for different problems), f(·) : R → R is a given continuously differentiable
function such that f(0) = 0 and
s
f(s)s−F(s) ≥ 0,f′(s) ≥ 0, ∀s ∈ R, F(s) := f(τ)dτ, ∀s ∈ R. (1.4)
Z0
We show that the feedback controller proposed in [1] for nonlinear parabolic equations
can be used for stabilization of above mentioned wide class of nonlinear dissipative wave
equations. We also show stabilization of the initial boundary value problem for equation
(1.1), with the control input acting on some proper subdomain of Ω.
Our study, as well as the results obtained in [1], are inspired by the fact that dissipa-
tive dynamical systems generated by initial boundary value problems, such as the 2D
Navier - Stokes equations, nonlinear reaction-diffusion equation, Cahn-Hilliard equation,
damped nonlinear Schro¨dinger equation, damped nonlinear Klein- Gordonequation, non-
linear strongly damped wave equation and related equations and systems have a finite
dimensional asymptotic (in time) behavior (see, e.g., [2],[6]- [10],[12]-[14],[21],[22],[29] and
references therein). This property has also been implicitly used in the feedback control of
the Navier-Stokes equations overcoming the spillover phenomenon in [5]
Motivated by this observation, we specifically show
(1) stabilization of 1D damped wave equation (1.1), under Neuman boundary condi-
tions, when the feedback control input employs observables based on measurement
of finite volume elements,
(2) stabilization of equation (1.1), under homogeneous Dirichlet boundary condition,
with one feedback controller supported on some proper subdomain of Ω ⊂ Rn,
(3) stabilization of equation (1.1) and equation (1.3), under the Dirichlet boundary
condition, when the feedback control involves finitely many Fourier modes of the
solution, based on eigenfunctions of the Laplacian subject to homogeneous Dirich-
let boundary condition,
(4) stabilization of the 1D equation (1.3), when the feedback control incorporates
observables at finitely many nodal points.
In the sequel we will use the notations:
• (·,·) and k·k denote the inner product and the norm of L2(Ω),
and the following inequalities:
• Young’s inequality
ε
ab ≤ εa2 + b2, (1.5)
4
that is valid for all positive numbers a,b and ε
• the Poincar´e inequality
kuk2 ≤ λ−1k∇uk2, (1.6)
1
FEEDBACK CONTROL 3
which holds for each u ∈ H1(Ω).
0
2. Feedback control of damped nonlinear wave equations
In this section, we show that the initial boundary value problem for nonlinear damped
wave equation can be stabilized by employing finite volume elements feedback controller,
feedback controllers acting in a subdomain of Ω, or feedback controllers involving finitely
many Fourier modes.
1. Stabilization employing finite volume elements feedback control. We
consider the following feedback control problem
N
∂2u−ν∂2u+b∂ u−λu+f(u) = −µ u χ (x), x ∈ (0,L), t > 0,
t x t k Jk
 k=1 (2.1)
∂ u(0,t) = ∂ u(L,t) = 0, t > 0, P
 x x


u(x,0 = u (x), ∂ u(x,0) = u (x), x ∈ (0,L).
0 t 1

Here J := (k −1)L,kL , for k = 1,2,···N −1 and J = [N−1L,L],
 k N N N N
φ := 1 φ(x)dx, and χ (x) is the characteristic function of the interval J . In what
k |Jk| (cid:2) Jk(cid:1) k
J
k
follows weRwill need the following lemma
Lemma 2.1. (see [1]) Let φ ∈ H1(0,L). Then
N
kφ− φ χ (·)k ≤ hkφ k, (2.2)
k Jk x
k=1
X
and
N 2
h
kφk2 ≤ h φ2 + kφ k2, (2.3)
k 2π x
k=1 (cid:18) (cid:19)
X
where h := L.
N
By employing this Lemma, we proved the following theorem:
Theorem 2.2. Suppose that the nonlinear term f(·) satisfies the condition (1.4) and that
µ and N are large enough satisfying
δ b L2 δ b
µ ≥ 2 λ+ 0 and N2 > λ+ 0 , (2.4)
2 2νπ2 2
(cid:18) (cid:19) (cid:18) (cid:19)
where
b
δ = min{1,ν}. (2.5)
0
2
Then each solution of the problem (2.1) satisfies the following decay estimate:
k∂ u(t)k2 +k∂ u(t)k2 ≤ K(ku k2 +k∂ u k2)e−δ0t, (2.6)
t x 1 x 0
where K is some positive constant depending on b,λ and L.
4 FEEDBACK CONTROL
Proof. First observe that one can use standard tools of the theory of nonlinear wave
equations to show global existence and uniqueness of solution to problem (2.1) (see, e.g.,
[24]).
Taking the L2(0,L) inner product of (2.1) with ∂ u+εu , where ε > 0 is a parameter, to
t
be determined later, gives us the following relation:
d 1 ν 1 L 1 N
k∂ uk2 + k∂ uk2 − (εb−λ)kuk2+ F(u)dx+ hµ u2 +ε(u,∂ u)
dt 2 t 2 x 2 2 k t
" Z0 k=1 #
X
N
+(b−ε)k∂ uk2 +ενk∂ uk2 −ελkuk2+ε(f(u),u)+εµh u2 = 0. (2.7)
t x k
k=1
X
It follows from (2.7) that
d
Φ (t)+δΦ (t)+(b−ε)k∂ uk2 +ενk∂ uk2 −ελkuk2+ε(f(u),u)
ε ε t x
dt
N
δ δν δ
+εµh u2 − k∂ uk2 − k∂ uk2 − (εb−λ)kuk2 −δ(F(u),1)
k 2 t 2 x 2
k=1
X
N
δ
− hµ u2 −δε(u,∂ u) = 0. (2.8)
2 k t
k=1
X
Here
1 ν 1 L
Φ (t) := k∂ u(t)k2 + k∂ u(t)k2 + (εb−λ)ku(t)k2 + F(u(x,t))dx
ε t x
2 2 2
Z0
N
1
+ hµ u2(t)+ε(u(t),∂ u(t)).
2 k t
k=1
X
Due to condition (1.4) and the Cauchy-Schwarz inequality we have the following lower
estimate for Φ (t)
ε
N
1 ν bε λ 1
Φ (t) ≥ k∂ uk2 + k∂ uk2 +( − −ε2)kuk2+ hµ u2.
ε 4 t 2 x 2 2 2 k
k=1
X
By choosing ε ∈ (0, b] and by employing inequality (2.3), we get from the above in-
2
equality:
N 2
1 ν h
Φ (t) ≥ k∂ uk2 + k∂ uk2 −λ h u2 + k∂ uk2
ε 4 t 2 x k 2π x
" #
k=1 (cid:18) (cid:19)
X
1 N 1 ν L2 µ N
+ µh u2 = k∂ uk2 + −λ k∂ uk2 +h( −λ) u2, (2.9)
2 k 4 t 2 4π2N2 x 2 k
k=1 (cid:18) (cid:19) k=1
X X
Thanks to the condition (2.4) we also have
1
Φ (t) ≥ k∂ u(t)k2 +d k∂ u(t)k2, (2.10)
ε t 0 x
4
FEEDBACK CONTROL 5
where d = δ0bL2 , and δ is defined in (4.21).
0 4N2νπ2 0
Let δ ∈ (0,ε), to be chosen below. According to condition (1.4) f(u)u ≥ F(u),∀u ∈ R.
Therefore (2.8) implies
d b bν δ
Φ (t)+δΦ (t)+ k∂ uk2 +( − )k∂ uk2
ε ε t x
dt 2 2 2
δλ δb2 bλ b δ N
+ − − kuk2 +µh − u2 ≤ 0. (2.11)
2 4 2 2 2 k
(cid:18) (cid:19) (cid:18) (cid:19)k=1
X
We choose here δ = δ := b min{ν,1} and employ inequality (2.3) to obtain:
0 2
d b bν b δ b L2
Φ (t)+δ Φ (t)+ k∂ uk2 + − λ+ 0 k∂ uk2
dt ε 0 ε 2 t 4 2 2 4π2N2 x
(cid:20) (cid:18) (cid:19) (cid:21)
N
bh µ δ b
+ − λ+ 0 u2 ≤ 0.
2 2 2 k
(cid:20) (cid:18) (cid:19)(cid:21)k=1
X
Finally by using condition (2.4) we get:
d
Φ (t)+δ Φ (t) ≤ 0.
ε 0 ε
dt
Thus by Gronwalls inequality and thanks (2.10) we have
k∂ u(t)k2 +k∂ u(t)k2 ≤ K(ku k2 +k∂ u k2)e−δ0t, (2.12)
t x 1 x 0
where δ is defined in (4.21), and K is a positive constant, depending on b,λ and L. (cid:3)
0
2.1. Stabilization with feedback control on a subdomain. In this section we study
the problem of internal stabilization of initial boundary value problem for nonlinear
damped wave equation on a bounded domain. We show that the problem can be ex-
ponentially stabilized by a feedback controller acting on a strict subdomain. So, we
consider the following feedback control problem:
∂2u−∆u+b∂ u−au+|u|p−2u = −µχ (x)u, x ∈ Ω,t > 0, (2.13)
t t ω
u(x,t) = 0, x ∈ ∂Ω, t > 0, (2.14)
u(x,0) = u (x), ∂ u(x,0) = u (x), x ∈ Ω. (2.15)
0 t 1
Here a > 0,p ≥ 2 are given numbers, and µ > 0 is a parameter to be determined, Ω ⊂ Rn
is a bounded domain with smooth boundary ∂Ω, χ (x) is the characteristic function of
ω
the subdomain ω ⊂ Ω with smooth boundary and ω ⊂ Ω.
Let us denote by λ (µ,Ω) the first eigenvalue of the elliptic problem
1
−∆v +µχ (x)v = λv; x ∈ Ω, v = 0, x ∈ ∂Ω,
ω
and denote by λ (Ω ) the first eigenvalue of the problem
1 ω
−∆v = λv, x ∈ Ω ; v = 0, x ∈ ∂Ω ,
ω ω
where Ω := Ω\ω. We will need the following Lemma in the proof of the main result of
ω
this section:
6 FEEDBACK CONTROL
Lemma 2.3. (see, e.g.,[30]) For each d > 0 there exists a number µ (d) > 0 such that
0
the following inequality holds true
|∇v(x)|2 +µχ (x)v2(x) dx ≥ (λ (Ω )−d) v2(x)dx, ∀v ∈ H1(Ω), (2.16)
ω 1 ω 0
ZΩ ZΩ
(cid:0) (cid:1)
whenever µ > µ .
0
Theorem 2.4. Suppose that
3b2
λ (Ω ) ≥ 4a+ , and µ > µ , (2.17)
1 ω 0
2
where µ is the parameter stated in Lemma 2.3 , corresponding to d = 1λ (Ω ). Then
0 2 1 ω
the energy norm of each weak solution of the problem (2.13)-(2.15) tends to zero with an
exponential rate. More precisely, the following estimate holds true:
k∂ u(t)k2 +k∇u(t)k2 +ku(t)kp ≤ C e−2bt, (2.18)
t Lp(Ω) 0
where C is a positive constant depending on initial data.
0
Proof. Taking the L2(Ω) inner product of (2.13) with ∂ u+ bu we get
t 2
d b
E (t)+ k∂ uk2 +k∇uk2 −akuk2 +kukp +µ χ (x)u2(x)dx = 0, (2.19)
dt b 2 t Lp(Ω) ω
(cid:20) ZΩ (cid:21)
where
1 1 a 1 µ
E (t) := k∂ uk2 + k∇uk2 − kuk2+ kukp + χ (x)u2(x)dx
b 2 t 2 2 p Lp(Ω) 2 ω
ZΩ
b b2
+ (u,∂ u)+ kuk2.
t
2 4
By Cauchy-Schwarz and Young inequalities we have
b 1 b2
|(u,∂ u)| ≤ k∂ uk2 + kuk2. (2.20)
t t
2 4 4
By employing (2.20), we obtain the lower estimate for E (t):
b
1 1 1 1
E (t) ≥ k∂ uk2 + k∇uk2 + kukp + k∇uk2+2µkuk2 −2akuk2 . (2.21)
b 4 t 4 p Lp(Ω) 4 L2(ω)
h i
According to Lemma 2.3
1
k∇uk2 +µkuk2 ≥ λ (Ω )kuk2, for every µ > µ . (2.22)
L2(ω) 2 1 ω 0
Thus, thanks to condition (2.17), we have
1 1 1
E (t) ≥ k∂ uk2 + k∇uk2 + kukp . (2.23)
b 4 t 4 p Lp(Ω)
FEEDBACK CONTROL 7
Adding to the left-hand side of (2.19) the expression E (t)−δE (t) with some δ > 0 (to
b b
be chosen below), we get
d 1 1
E (t)+δE (t)+ (b−δ)k∂ uk2 + (b−δ)k∇uk2
b b t
dt 2 2
1 δb2 µ 1
+ (aδ −ab− )kuk2+ (b−δ)kuk2 − δb(u,∂ u) = 0
2 2 2 L2(ω) 2 t
We use here the inequality
1 b bδ2
δb|(u,∂ u)| ≤ k∂ uk2 + kuk2,
t t
2 4 4
then in the resulting inequality we choose δ = b, and get:
2
d b b 3
E (t)+ E (t)+ k∇uk2 +µkuk2 −(2a+ b2)kuk2 .
dt b 2 b 4 L2(ω) 4
(cid:20) (cid:21)
Finally, by using the condition (2.17), thanks to Lemma 2.3 we obtain
d b
E (t)+ E (t) ≤ 0.
b b
dt 2
Integrating the last inequality and taking into account (2.23), we arrive at the desired
estimate (2.18).
(cid:3)
3. Stabilization employing finitely many Fourier modes feedback controls.
In this section we consider the feedback control problem for damped nonlinear wave
equation based on finitely many Fourier modes, i.e. we consider the feedback system of
the following form:
N
∂2u−ν∆u+b∂ u−au+|u|p−2u = −µ (u,w )w , x ∈ Ω,t > 0, (2.24)
t t k k
k=1
X
u = 0, x ∈ ∂Ω, t > 0, (2.25)
u(x,0 = u (x), ∂ u(x,0) = u (x), x ∈ Ω, t > 0. (2.26)
0 t 1
Here ν > 0,a > 0,b > 0,µ > 0,p ≥ 2 are given numbers; w ,w ,...,w ,... is the set of or-
1 2 n
thonormal (in L2(Ω)) eigenfunctions of the Laplace operator −∆ under the homogeneous
Dirichlet boundary condition, corresponding to eigenvalues 0 < λ ≤ λ ··· ≤ λ ≤,···.
1 2 n
Theorem 2.5. Suppose that µ and N are large enough such that
ν ≥ (2a+3b2/4)λ−1 , and µ ≥ a+3b2/4. (2.27)
N+1
Then the following decay estimate holds true
k∂ u(t)k2 +k∇u(t)k2 + |u(x,t)|pdx ≤ E e−2bt, (2.28)
t 0
ZΩ
8 FEEDBACK CONTROL
where
1 ν b2 a 1 µ N b
E := ku k2+ k∇u k2+( − )ku k2+ |u (x)|pdx+ (u ,w )2+ (u ,u ).
0 1 0 0 0 0 k 0 1
2 2 4 2 p 2 2
ZΩ k=1
X
Proof. Multiplication of (2.24) by ∂ u+ bu and integration over Ω gives
t 2
N
d b
E (t)+ k∂ uk2 +νk∇uk2 −aku(t)k2 + |u|pdx+µ (u,w )2 = 0, (2.29)
b t k
dt 2
" ZΩ k=1 #
X
where
1 ν
E (t) := k∂ u(t)k2 + k∇u(t)k2
b t
2 2
b2 a 1 µ N b
+( − )ku(t)k2 + |u(x,t)|pdx+ (u(t),w )2 + (u(t),∂ u(t)).
k t
4 2 p 2 2
ZΩ k=1
X
Thanks to the Cauchy-Schwarz and Young inequalities we have
b 1 b2
|(u(t),∂ u(t))| ≤ k∂ uk2 + kuk2.
t t
2 4 4
Consequently,
N
1 ν a 1 µ
E (t) ≥ k∂ uk2 + k∇uk2− kuk2 + |u|pdx+ (u,w )2.
b t k
4 2 2 p 2
ZΩ k=1
X
Since
N N ∞
a µ 1 a
− kuk2 + (u,w )2 = (µ−a) (u,w )2 − (u,w )2,
k k k
2 2 2 2
k=1 k=1 k=N+1
X X X
by using the Poincar´e -like inequality
N
kφ− (φ,w )w k2 ≤ λ−1 k∇φk2, (2.30)
k k N+1
k=1
X
which is valid for each φ ∈ H1(Ω), we get
0
N
1 1 1 1
E (t) ≥ k∂ uk2 + ν −aλ−1 k∇uk2 + (µ−a) (u,w )2 + |u|pdx. (2.31)
b 4 t 2 N+1 2 k p
k=1 ZΩ
(cid:0) (cid:1) X
Thus,
1 ν 1
E (t) ≥ k∂ uk2 + k∇uk2 + |u|pdx. (2.32)
b t
4 4 p
ZΩ
Adding to the left-hand side of (2.29) the expression δE (t) − δE (t) (with δ to be
b b
chosen later), we obtain
FEEDBACK CONTROL 9
d 1 ν
E (t)+δE (t)+ (b−δ)k∂ uk2 + (b−δ)k∇uk2
b b t
dt 2 2
ba δa δb2 b δ
+ − + − ku2k+ − kukp
2 2 4 2 2 Lp(Ω)
(cid:18) (cid:19) (cid:18) (cid:19)
N
µ 1
+ (b−δ) (u,w )2 − bδ(u,∂ u) = 0.
k t
2 2
k=1
X
We choose here δ = b, and obtain
2
d b νb
E (t)+δE (t)+ k∂ uk2 + k∇uk2
b b t
dt 4 4
ba b3 µb N b2
− + kuk2+ (u,w )2 − (u,∂ u) = 0. (2.33)
k t
4 8 4 4
(cid:18) (cid:19) k=1
X
Employing the inequality
b2 b b3
|(u,∂ u)| ≤ k∂ uk2 + kuk2,
t t
4 4 16
and inequality (2.30), we obtain from (2.34):
d b
E (t)+δE (t)+ ν −(a+3b2/4)λ−1 k∇uk2
dt b b 4 N+1
(cid:2) (cid:3) N
b
+ µ−(a+3b2/4) (u,w )2. (2.34)
k
4
k=1
(cid:2) (cid:3)X
d b δ b δν
E (t)+δE (t)+ − k∂ uk2 + (ν −aλ−1 )− k∇uk2+
dt b b 2 2 t 2 N+1 2
(cid:18) (cid:19) (cid:18) (cid:19)
N
b δµ b δ
(µ−a)− (u,w )2 + − |u|pdx
k
2 2 2 p
(cid:18) (cid:19)k=1 (cid:18) (cid:19)ZΩ
X
b2 a δb
−δ − kuk2+ (u,∂ u) ≤ 0. (2.35)
t
4 2 2
(cid:18) (cid:19)
By using the inequalities
δb b δ2b
|(u,∂ u)| ≤ k∂ uk2+ kuk2,
t t
2 4 4
and (2.30) in (2.35) we get
d b δ b δ2b δν
E (t)+δE (t)+ − k∂ uk2 + (ν −aλ−1 )− λ−1 − k∇uk2+
dt b b 4 2 t 2 N+1 4 N+1 2
(cid:18) (cid:19) (cid:18) (cid:19)
b δµ N b δ b2 a δ2b
(µ−a)− (u,w )2 + − |u|pdx−δ − − kuk2 ≤ 0.
k
2 2 2 p 4 2 4
(cid:18) (cid:19)k=1 (cid:18) (cid:19)ZΩ (cid:18) (cid:19)
X
10 FEEDBACK CONTROL
By choosing δ = b we obtain :
2
d b b b2 3 b µ 3 b2 N
E (t)+ E (t)+ ν −( + a)λ−1 k∇uk2 + − a− (u,w )2 ≤ 0.
dt b 2 b 2 8 2 N+1 2 2 2 8 k
(cid:20) (cid:21) (cid:20) (cid:21)k=1
X
Taking into account conditions (2.27) we deduce from the last inequality the inequality
d b
E (t)+ E (t) ≤ 0.
b b
dt 2
(cid:3)
Integrating the last inequality we get the desired estimate (2.28) thanks to (2.32).
Remark 2.6. We would like to note that estimate (2.28) allows us to prove existence of
a weak solution to the problem (2.24), (2.25) such that (see [24])
u ∈ L∞ R+;H1(Ω)∩Lp(Ω) , u ∈ L∞ R+;L2(Ω)
0
Note that there areno restr(cid:0)ictions onthe spatia(cid:1)l dimension(cid:0)of the dom(cid:1)ain Ω or the growth
of nonlinearity.
3. Nonlinear Wave Equation with Nonlinear Damping term:Stabilization
with finitely many Fourier modes
In this section we consider the initial boundary value problem for a nonlinear wave
equation with nonlinear damping term with a feedback controller involving finitely many
Fourier modes:
N
∂2u−ν∆u+b|∂ u|m−2∂ u−au+|u|p−2u = −µ (u,w )w , x ∈ Ω,t > 0, (3.1)
t t t k k
k=1
X
u = 0, x ∈ ∂Ω, t > 0, (3.2)
u(x,0) = u (x), ∂ u(x,0) = u (x), x ∈ Ω, (3.3)
0 t 1
where ν > 0,b > 0,a > 0,p ≥ m > 2 are given parameters.
We show stabilization of solutions of this problem to the zero with a polynomial rate.
Our main result is the following theorem:
Theorem 3.1. Suppose that µ ad N are large enough such that
ν > 2aλ−1 , and µ > a. (3.4)
N+1
Then for each solution of the problem (3.1)-(3.3) following estimate holds true
k∂tu(t)k2 +k∇u(t)k2 + |u(x,t)|pdx ≤ Ct−mm−1, (3.5)
ZΩ
where C is a positive constant depending on initial data.
Proof. Taking the inner product of equation (3.1) in L2(Ω) with ∂ u we get
t
d
E(t)+bk∂ u(t)km = 0, (3.6)
t
dt