Low-regret control for a fractional wave equation with incomplete data

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Low-regret control for a fractional wave equation with incomplete data

Dumitru Baleanu, Claire Joseph, Gisèle Mophou

To cite this version:

Dumitru Baleanu, Claire Joseph, Gisèle Mophou. Low-regret control for a fractional wave equa-

tion with incomplete data. Advances in Difference Equations, Hindawi Publishin Corporation /

SpringerOpen, 2016, �10.1186/s13662-016-0970-8�. �hal-02548520�

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« doctopic: OriginalPaper numbering style: ContentOnly reference style: mathphys»

Baleanu et al.Advances in Difference Equations_#####################_

DOI 10.1186/s13662-016-0970-8

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R E S E A R C H Open Access

Low-regret control for a fractional wave equation with incomplete data

Dumitru Baleanu^1,2, Claire Joseph³and Gisèle Mophou^3,4*

*Correspondence:

gmophou@univ-ag.fr

3Université des Antilles et de la Guyane, Campus Fouillole, 97159, Pointe-à-Pitre, Guadeloupe (FWI), France

4Laboratoire MAINEGE, Université Ouaga 3S, 06 BP 10347, Ouagadougou 06, Burkina Faso Full list of author information is available at the end of the article

Abstract

We investigate in this manuscript an optimal control problem for a fractional wave equation involving the fractional Riemann-Liouville derivative and with missing initial condition. For this purpose, we use the concept of no-regret and low-regret controls.

Assuming that the missing datum belongs to a certain space we show the existence and the uniqueness of the low-regret control. Besides, its convergence to the no-regret control is discussed together with the optimality system describing the no-regret control.

Keywords: Riemann-Liouville fractional derivative; Caputo fractional derivative;

optimal control; no-regret control; low-regret control

1 Introduction

Let us considerN∈N∗anda bounded open subset ofR^Npossessing the boundary∂

of classC^. When the timeT> , we considerQ=×],T[ and6=∂×],T[ and we discuss the fractional wave equation:











D^α_RLy(x,t) –1y(x,t) =v(x,t), (x,t)∈Q, y(σ,t) = , (σ,t)∈6,

I^–αy(x, ⁺) =y^, x∈,

∂

∂tI^–αy(x, ⁺) =g, x∈,

()

such that / <α< ,y^∈H^()∩H_^(),I^–αy(x, ⁺) =lim_t→I^–αy(x,t) and _∂t^∂I^–αy(x,

⁺) =lim_t_→__∂t^∂I^–αy(x,t) where the fractional integralI^α of orderα and the fractional derivativeD^α_RL of orderαare within the Riemann-Liouville sense. The functiong is unknown and belongs toL^() and the controlv∈L^(Q).

Since the initial condition is unknown, the system () is a fractional wave equations with missing data. Such equations are used to model pollution phenomena. In this systemg represents the pollution term.

According to the data, we know that system () admits a unique solution y(v,g) = y(x,t;v,g) inL^((,T);H_^())⊂L^(Q) []. Hence, we can deﬁne the following functional:

J(v,g) =

y(v,g) –z_d



L^(Q)+Nkvk^_L(Q), ()

wherezd∈L^(Q) andN> .

©2016 Baleanu et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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In this manuscript, we discuss the optimal control problem, namely inf

v∈L^(Q)J(v,g), ∀g∈L^(). ()

If the functiongis given, namelyg=g∈L^(), then system () is completely determined and problem () becomes a classical optimal control problem []. Such a problem was studied by Mophou and Joseph [] with a cost function deﬁned with a ﬁnal observation.

Actually, the authors proved that one can approach the fractional integral of order  <

 –α< / of the state at ﬁnal time by a desired state by acting on a distributed control.

For more literature on fractional optimal control, we refer to [–] and the references therein.

Since the functiongis unknown, the optimal control problem () has no sense because L^() is of inﬁnite dimension. So, to solve this problem, we proceed as Lions [, ] for the control of partial diﬀerential equations with integer time derivatives and missing data.

This means that we use the notions of no-regret and low-regret controls. There are many works using these concepts in the literature. In [] for instance, Nakoulimaet al.utilized these concepts to control distributed linear systems possessing missing data. A generaliza- tion of this approach can be found in [] for some nonlinear distributed systems possessing incomplete data. Jacob and Omrane used the notion of no-regret control to control a linear population dynamics equation with missing initial data []. Recently, Mophou [] used these notions to control a fractional diﬀusion equation with unknown boundary condition. For more literature on such control we refer to [–] and the references therein.

In our paper, we show that the low-regret control problem associated to () admits a unique solution which converges toward the no-regret control. We provide the singular optimality system for the no-regret control.

Below we present the organization of our manuscript. In the following section, we show brieﬂy some results about fractional derivatives and preliminary results on the existence and uniqueness of solution to fractional wave equations. In Section , we investigate the no-regret and low-regret control problems corresponding to ().

2 Preliminaries

Below, we give brieﬂy some results about fractional calculus and some existence results about fractional wave equations.

Deﬁnition .[, ] Iff :R₊→Ris a continuous function onR₊, andα> , then the expression of the Riemann-Liouville fractional integral of orderαis

I^αf(t) =  Ŵ(α)

Z t



(t–s)^α–f(s)ds, t> .

Deﬁnition .[, ] The form of the left Riemann-Liouville fractional derivative of order ≤n–  <α<n,n∈Noff is given by

D^α_RLf(t) = 

Ŵ(n–α)· dⁿ dtⁿ

Z t



(t–s)^n––αf(s)ds, t> .

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Deﬁnition .[, ] The left Caputo fractional derivative of order ≤n–  <α<n, n∈Noff is given by

D^α_Cf(t) =  Ŵ(n–α)

Z t



(t–s)^n––αf⁽ⁿ⁾(s)ds, t> . ()

We mention that in the above two deﬁnitions we considerf :R₊→R.

Deﬁnition .[–] Letf :R₊→R, ≤n–  <α<n,n∈N. Then the right Caputo fractional derivative of orderαoff is

D^α_Cf(t) = (–)ⁿ Ŵ(n–α)

Z T t

(s–t)^n––αf⁽ⁿ⁾(s)ds,  <t<T. ()

In all above deﬁnitions we assume that the integrals exist.

Lemma .[] Let y∈C^∞(Q)andϕ∈C^∞(Q).Then we have Z

Q

D^α_RLy(x,t) –1y(x,t)

ϕ(x,t)dx dt

= Z

ϕ(x,T)∂

∂tI^–αy(x,T)dx– Z

ϕ(x, )∂

∂tI^–αy x, ⁺ dx –

Z

I^–αy(x,T)∂ϕ

∂t(x,T)dx+ Z

I^–αy(x, )∂ϕ

∂t(x, )dx +

Z

6

y(σ,s)∂ϕ

∂ν(σ,s)dσdt– Z

6

∂y

∂ν(σ,s)ϕ(σ,s)dσdt +

Z

Q

y(x,t) D_C^αϕ(x,t) –1ϕ(x,t)

dx dt. ()

In the following we give some results that will be use to prove the existence of the low- regret and no-regret controls.

Theorem .[] Let/ <α< ,y^∈H^()∩H_^(),y^∈L^()and v∈L^(Q).Then the problem











D^α_RLy(x,t) –1y(x,t) =v(x,t), (x,t)∈Q, y(σ,t) = , (σ,t)∈6,

I^–αy(x, ⁺) =y^, x∈,

∂

∂tI^–αy(x, ⁺) =y^, x∈

()

has a unique solution y∈L^((,T);H_^()).Moreover,the following estimates hold:

kykL^((,T);H^_())≤1 y^

H_^()+ y^

L^()+kvkL^(Q)

, () I^–αy

C([,T];H_^())≤5 y^

H_^()+ y^

L^()+kvkL^(Q)

, ()

∂

∂tI^–αy

C([,T];L^())

≤2 y^

H^()+ y^

L^()+kvkL^(Q)

, ()

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with

1=max

C s

T^α–

(α– ),C s

T^α–

(α– ),C s

T^α α(α– )

,

5=sup

C√

,C√

T^–α,C

sT^–α ( –α)

,

and

2=max √

CT^α–,√

C .

Consider the fractional wave equation involving the left Caputo fractional derivative of order / <α< :











D^α_Cy(x,t) –1y(x,t) =f, (x,t)∈Q, y(σ,t) = , (σ,t)∈6,

y(x, ) = , x∈,

∂y

∂t(x, ) = , x∈,

()

wheref ∈L^(Q).

Theorem . Let f∈L^(Q).Then problem()has a unique solution y∈C([,T];H_^()).

Moreover,^∂y_∂t ∈C([,T];L^())and there exists C> in such a way that

kykC([,T];H_^())≤C rT^α–

α– kfkL^(Q) ()

and

∂y

∂t

C([,T];L^())

≤C rT^α–

α– kfkL^(Q). ()

Proof Below we proceed as was mentioned in [].

Corollary . Let/ <α< andφ∈L^(Q).Consider the fractional wave equation:











D_C^αψ(x,t) –1ψ(x,t) =φ, (x,t)∈Q, ψ(σ,t) = , (σ,t)∈6,

ψ(x,T) = , x∈,

∂ψ

∂t(x,T) = , x∈,

()

whereD_C^αis the right Caputo fractional derivative of order <α< .Then()has a unique solutionψ∈C([,T];H_^()).Moreover, ^∂ψ_∂t ∈C([,T];L^()),and there exists C> ful- filling

kψkC([,T];H_^())≤C rT^α–

α– kφkL^(Q) ()

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and

∂ψ

∂t

_C([,T];L_₍₎₎≤C rT^α–

α– kφkL^(Q). ()

Proof If we make the change of variablet→T–tin (), then we conclude thatψˆ(t) = ψ(T–t) veriﬁes











D^α_Cψˆ –1ψˆ =φˆ inQ, ˆ

ψ=  on6, ˆ

ψ() =  in,

∂ψˆ

∂t() =  in,

()

whereφˆ(t) =φ(T–t) andD^α_Cis the left Caputo fractional derivative of order / <α< .

BecauseT–t∈[,T] whent∈[,T], we say thatφˆ∈L^(Q) due to the fact thatφ∈L^(Q).

It then suﬃces to use Theorem . to conclude.

We also need some trace results.

Lemma .[] Let f ∈L^(Q)and y∈L^(Q)such that D^α_RLy–1y=f.Then:

(i) y_|∂and_∂ν^∂y_|_∂exist and belong toH^–((,T);H^–/(∂))and H^–((,T);H^–/(∂))respectively.

(ii) I^–αy∈C([,T];L^()).

(iii) _∂t^∂I^–αy∈C([,T];H^–()).

3 Existence and uniqueness of no-regret and low-regret controls

Below, we show the existence and the uniqueness of the no-regret control and the low- regret control problem for system ().

Lemma . Let v∈L^(Q)and g∈L^().Then we have J(v,g) =J(,g) +J(v, ) –J(, )

+  Z

Q

y(,g) –y(, )

y(v, ) –y(, )

dt dx. ()

Here J denotes the functional given by()and y(v,g) =y(x,t;v,g)∈L^(,T;H_^())⊂L^(Q) is the solution of ().

Proof Let us considery(v, ) =y(x,t;v, ),y(,g) =y(x,t; ,g), andy(, ) =y(x,t; , ) be the solutions of











D^α_RLy(v, ) –1y(v, ) =v inQ, y=  on6,

I^–αy(;v, ) =y^ in,

∂

∂tI^–αy(;v, ) =  in,

()

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









D^α_RLy(,g) –1y(,g) =  inQ, y=  on6,

I^–αy(; ,g) =y^ in,

∂

∂tI^–αy(; ,g) =g in,

()

and











D^α_RLy(, ) –1y(, ) =  inQ, y=  on6,

I^–αy(; , ) =y^ in,

∂

∂tI^–αy(; , ) =  in,

()

whereI^–αy(;v,g) =lim_t→⁺I^–αy(x,t;v,g) and _∂t^∂I^–αy(;v,g) =lim_t→⁺ ∂

∂tI^–αy(x,t;v,g).

Sincev∈L^(Q),y^∈H_^()∩H^() andg∈L^(), we see from Theorem . thaty(v, ), y(,g) andy(, ) belong toL^((,T);H_^()).

Observing that J(v, ) =

Z

Q

y(v, ) –zd

dt dx+Nkvk^_L(Q), (a) J(,g) =

Z

Q

y(,g) –zd

dt dx, (b)

J(, ) = Z

Q

y(, ) –z_d

dt dx, (c)

and using the fact that

y(v,g) =y(v, ) +y(,g) –y(, ), we have

J(v,g) =

y(v,g) –z_d



L^(Q)+Nkvk^_L(Q)

=

y(v, ) +y(,g) –y(, ) –z_d



L^(Q)+Nkvk^_L(Q)

=J(v, ) +  Z

Q

y(v, ) –z_d

y(,g) –y(, ) dt dx +

y(,g) –y(, ) –zd

 L^(Q)

=J(v, ) +  Z

Q

y(v, ) –y(, )

y(,g) –y(, ) dt dx + 

Z

Q

y(, ) –zd

y(,g) –y(, )

dt dx+

y(,g) –y(, ) –zd

 L^(Q). Using

y(,g) –y(, ) –z_d



L^(Q)=J(,g) +J(, ) – 

Z

Q

y(, ) –zd

y(,g) –y(, )

dt dx– J(, ),

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we conclude that

J(v,g) =J(,g) +J(v, ) –J(, ) + 

Z

Z T



y(v, ) –y(, )

y(,g) –y(, )

dt dx.

Lemma . Let v∈L^(Q)and g∈L^().Then we have J(v,g) =J(,g) +J(v, ) –J(, ) + 

Z

gζ(x, ;v)dx, ()

whereζ(v) =ζ(x,t;v)∈C([,T];H_^())be solution of











D_C^αζ(v) –1ζ(v) =y(v, ) –y(, ) in Q, ζ=  on6,

ζ(x,T;v) =  in,

∂ζ

∂t(x,T;v) =  in.

()

Proof Sincey(v, ) –y(, )∈L^(Q), from Proposition ., we know that the system () admits a unique solutionζ(v)∈C([,T];H_^()). Also, there existsC>  such that

ζ(v)

C([,T];H_^())≤C rT^α–

α– 

y(v, ) –y(, )

L^(Q) () and

∂ζ

∂t(v) _C

([,T];L^())

≤C rT^α–

α– 

y(v, ) –y(, )

L^(Q). () Setz=y(g, ) –y(, ). Thenzveriﬁes











D^α_RLz–1z=  inQ, z=  on6,

I^–αz() =  in,

∂

∂tI^–αz() =g in.

()

Sinceg∈L^(), it follows from Theorem . thatz∈L^((,T);H_^()),I^–αz∈C([,T], H_^()), and _∂t^∂I^–αz∈C([,T],L^()). So, if we multiply the ﬁrst equation of () byz utilizing the fractional integration by parts provided by Lemma ., we conclude

Z

Q

y(v, ) –y(, ) z dt dx

= Z

Q

D^α_Cζ(v) –1ζ(v) z dt dx

= Z

ζ(x, ;v)∂

∂tI^–αz()dx. ()

Thus, replacingzby (y(,g) –y(, )), we obtain Z

Q

y(v, ) –y(, )

y(,g) –y(, ) dt dx=

Z

ζ(x, ;v)g dx,

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and () becomes

J(v,g) –J(,g) =J(v, ) –J(, ) +  Z

ζ(x, ;v)g dx.

Now we consider the no-regret control problem:

inf

v∈L^(Q)

sup

g∈L^()

J(v,g) –J(,g)

. ()

From (), this problem is equivalent to the following one:

inf

v∈L^(Q)

sup

g∈L^()

J(v, ) –J(, ) +  Z

ζ(x, ;v)g dx

. ()

As the spaceL^() is a vector space, the no-regret control exists only if sup

g∈L^()

Z

ζ(x, ;v)g dx

= . ()

This implies that the no-regret control belongs toUdeﬁned by U=

v∈L^(Q)

Z

ζ(x, ;v)g dx

= ,∀g∈L^()

.

As a result such control should be carefully investigated. So, we proceed by penalization.

For allγ > , we discuss the low-regret control problem:

inf

v∈L^(Q) sup

g∈L^()

J(v,g) –J(,g) –γkgk^_L()

. ()

According to (), the problem () is equivalent to the following problem:

inf

v∈L^(Q)

J(v, ) –J(, ) +  sup

g∈L^()

Z

ζ(x, ;v)g dx–γ

kgk^_L()

.

Using the Legendre-Fenchel transform, we conclude

γ sup

g∈L^()

Z



γζ(x, ;v)g dx–  γ γ

kgk^_L()

=  γ

ζ(·, ;v)

 L^(), and problem () becomes: For anyγ > , ﬁndu^γ∈L^(Q) such that

Jγ u^γ

= inf

v∈L^(Q)Jγ(v), ()

where

Jγ(v) =J(v, ) –J(, ) +  γ

ζ(·, ;v)



L^(). ()

Proposition . Letγ > .Then()has a unique solution u^γ,called a low-regret control.

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Proof We recall that

Jγ(v) =J(v, ) –J(, ) +  γ

ζ(·, ;v)



L^()≥–J(, ).

Thus, we can say thatinf_v∈L(Q)Jγ exists. Let (v_n)∈L^(Q) be a minimizing sequence such that

n→+∞lim Jγ(vn) = inf

v∈L^(Q)Jγ(v). ()

Thenyn=y(x,t;vn, ) is a solution of () andynsatisﬁes

D^α_RLy_n(x,t) –1y_n(x,t) =v_n(x,t) inQ, (a)

yn(x,t) =  on6, (b)

I^–αyn(x, ) =y^ in, (c)

∂

∂tI^–αyn(x, ) =  in. (d)

It follows from () that there existsC(γ) >  independent ofnsuch that

≤J(vn, ) +  γ

ζ(·, ;vn)



L^()≤C(γ) +J(, ) =C(γ).

From the deﬁnition ofJ(vn, ) we obtain

kv_nkL^(Q)≤C(γ), (a)

ζ(·, ;v_n)

L^()≤√γC(γ). (b)

Therefore, from Theorem ., we know that there exists a constantCindependent ofn such that

kynkL^((,T);H_^())≤C(γ), (a)

I^–αyn

L^((,T);H_^())≤C(γ), (b)

∂

∂tI^–αyn

L^((,T);L^())≤C(γ). (c)

Moreover, from (a) and (a), we have D^α_RLy_n–1y_n

L^(Q)≤C(γ). () Consequently, there existu^γ ∈L^(Q),y^γ ∈L^((,T);H_^()),δ∈L^(Q), η∈L^((,T);

H_^()),θ∈L^((,T);L^()) and we can extract subsequences of (vn) and (yn) (still called (vn) and (yn)) such that:

vn⇀u^γ weakly inL^(Q), (a)

D^α_RLyn–1yn⇀ δ weakly inL^(Q), (b)

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yn⇀y^γ weakly inL^ (,T);H_^()

, (c)

I^–αyn⇀ η weakly inL^ [,T],H_^()

, (d)

∂

∂tI^–αyn⇀ θ weakly inL^ (,T);L^()

. (e)

The remaining part of the proof contains three steps.

Step: We show that (u^γ,y^γ) fulﬁlls ().

SetD(Q), the set ofC^∞function onQwith compact support and denote byD′(Q) its dual. Multiplying (a) byϕ∈D(Q) and using Lemma ., (a), and (c), we prove as in [] that

D^α_RLyn–1yn⇀D^α_RLy^γ –1y^γ weakly inD^′(Q).

From (b) and the uniqueness of the limit, we conclude

D^α_RLy^γ –1y^γ=δ∈L^(Q). ()

Hence,

D^α_RLyn–1yn⇀D^α_RLy^γ –1y^γ weakly inL^(Q). () Then passing to the limit in (a) and using () and (a), we obtain

D^α_RLy^γ(x,t) –1y^γ(x,t) =u^γ(x,t), (x,t)∈Q. () On the other hand, we have

Z

Q

I^–αyn(x,t)ϕ(x,t)dt dx

= Z

Z T



yn(x,s) 

Ŵ( –α) Z T

s

(t–s)^–αϕ(x,t)dt

ds dx, ∀ϕ∈D(Q).

Thus using (c) and (d), while passing to the limit, we get Z

Q

ηϕ(x,t)dt dx= Z

Z T



y^γ(x,s) 

Ŵ( –α) Z T

s

(t–s)^–αϕ(x,t)dt

ds dx

= Z

Q

I^–αy^γ(x,t)ϕ(x,t)dt dx, ∀ϕ∈D(Q).

This implies that I^–αy^γ(x,t) =η inQ.

Thus, (d) becomes

I^–αyn⇀I^–αy^γ weakly inL^ [,T],H_^()

. ()

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In view of (), we have

∂

∂tI^–αyn⇀ ∂

∂tI^–αy^γ weakly inD^′(Q), and as we have (e), we obtain

∂

∂tI^–αyn⇀ ∂

∂tI^–αy^γ =θ weakly inL^(Q). ()

Sincey^γ ∈L^(Q) andD^α_RLy^γ–1y^γ ∈L^(Q), in view of Lemma ., we know thaty^γ_|_∂and

∂y^γ

∂v|∂exist and belong toH^–((,T);H^–/(∂)) andH^–((,T);H^–/(∂)), respectively.

Moreover, we haveI^–αy^γ ∈C([,T];L^()) and_∂t^∂I^–αy^γ∈C([,T];H^–()).

Now multiplying (a) by a function ϕ ∈C^∞(Q) such that ϕ_|∂ =  and ϕ(x,T) =

∂ϕ

∂t(x,T) =  in, and integrating by parts overQ, we obtain Z

Q

v_n(x,t)ϕ(x,t)dx dt= Z

Q

D^α_RLy_n(x,t) –1y_n(x,t)

ϕ(x,t)dx dt

= Z

y^∂ϕ

∂t(x, )dx +

Z

Q

yn(x,t) D_C^αϕ(x,t) –1ϕ(x,t) dx dt

because we have (c) and (d). Thus, using (a) and (c) while passing to the limit in the latter identity, we get

Z

Q

u^γ(x,t)ϕ(x,t)dx dt= Z

y^∂ϕ

∂t(x, )dx +

Z

Q

y^γ(x,t) D_C^αϕ(x,t) –1ϕ(x,t) dx dt,

∀ϕ∈C^∞(Q) such thatϕ_|∂= ,ϕ(T) =^∂ϕ_∂t(T) =  in, which, according to Lemma ., can be rewritten as

Z

Q

u^γ(x,t)ϕ(x,t)dx dt= Z

y^∂ϕ

∂t(x, )dx +

Z

Q

D^α_RLy^γ(x,t) –1y^γ(x,t)

ϕ(x,t)dx dt +

ϕ(x, ), ∂

∂tI^–αy^γ(x, )

H_^(),H^–()

– Z

∂ϕ

∂t(x, ),I^–αy^γ(x, )dx –

y^γ(σ,t),∂ϕ

∂ν(σ,t)

H^–((,T);H^–/(∂)),H_^((,T);H^/(∂))

,

∀ϕ∈C^∞(Q) such thatϕ_|∂= ,ϕ(T) =^∂ϕ_∂t(T) =  in.

Using (), we obtain

 = Z

y^∂ϕ

∂t(x, )dx

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+

ϕ(x, ), ∂

∂tI^–αy^γ(x, )

H_^(),H^–()

– Z

∂ϕ

∂t(x, ),I^–αy^γ(x, )dx –

y^γ(σ,t),∂ϕ

∂ν(σ,t)

H^–((,T);H^–/(∂)),H_^((,T);H^/(∂))

, ()

∀ϕ∈C^∞(Q) such thatϕ_|_∂= ,ϕ(T) =^∂ϕ_∂t(T) =  in.

Choosing successively in ()ϕ such thatϕ(x, ) =^∂ϕ_∂t(x, ) =  andϕ(x, ) = , we deduce that

y^γ(x,t) = , (x,t)∈6, ()

I^–αy^γ(x, ) =y^, x∈, ()

and then

∂

∂tI^–αy^γ(x, ) = , x∈. ()

In view of (), (), (), and (), we see thaty^γ =y^γ(x,t;u^γ, ) is a solution of ().

Step: We showζ_n=ζ(x,t;vn) converges toζ^γ =ζ(x,t;u^γ).

In view of (),ζ_n=ζ(x,t;vn) veriﬁes











D_C^αζ_n–1ζ_n=y(vn, ) –y(, ) inQ, ζn=  on6,

ζ_n(T) =  in,

∂

∂tζ_n(T) =  in.

()

Setzn=y(v_n, ) –y(, ). In view of () and (),znveriﬁes











D^α_RLz_n–1z_n=v_n inQ, zn=  on6,

I^–αzn() =  in,

∂

∂tI^–αzn() =  in.

It follows, from Theorem . and (a), that kznkL^((,T);H^_())=

yn(vn, ) –y(, )

L^((,T);H^_())≤C(γ).

Hence, from Corollary ., we deduce that

kζ_nkC([,T];H_^())≤C(γ), ()

∂

∂tζ_n

_C([,T];L_₍₎₎≤C(γ). ()

Since the embedding of C([,T];H_^()) into L^((,T);H_^()) and the embedding of C([,T];L^()) into L^(Q) are continuous, we can conclude that there exists ζ^γ ∈

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L^((,T);H_^()) such that

ζ_n⇀ ζ^γ weakly inL^ (,T);H_^()

. ()

Therefore,

∂

∂tζ_n⇀ ∂

∂tζ^γ weakly inD^′(Q) and, consequently,

∂

∂tζ_n⇀ ∂

∂tζ^γ weakly inL^(Q). ()

Sinceζ^γ ∈L^((,T);H_^()) and _∂t^∂ζ^γ ∈L^(Q), we see that ζ^γ() and ζ^γ(T) belongs to L^(). In view of (), we have

ζ^γ(T) =  in ()

and in view of ()and (), we set

∂

∂tζ^γ(T) =  in. ()

From (b), we deduce that there existsρ∈L^() such that

ζ(·, ;vn)⇀ ρ weakly inL^(). ()

Multiplying the ﬁrst equation of () byφ∈D(Q) then, using the integration by parts given by Lemma ., we obtain

Z

Q

y(x,t;vn, ) –y(x,t; , )

φ(x,t)dt dx

= Z

Q

D^α_RLφ(x,t) –1φ(x,t)

ζn(x,t)dt dx.

Hence, using (c) and () while passing to the limit in the latter identity, we have Z

Q

y x,t;u^γ, 

–y(x,t; , )

φ(x,t)dt dx

= Z

Q

ζ^γ(x,t)dt dx, ∀φ∈D(Q), () which by using again Lemma . gives

Z

Q

y x,t;u^γ, 

–y(x,t; , )

φ(x,t)dt dx

= Z

Q

D^α_Cζ^γ(x,t) –1ζ^γ(x,t)

φ(x,t)dt dx, ∀φ∈D(Q).

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«ADE  layout: Onecolumn v.. ﬁle: ade.tex (Aurimas) class: bmc-onecol-v v.// Prn://; : p. /»

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This implies that D^α

Cζ^γ–1ζ^γ=y u^γ, 

–y(, ) inQ. ()

Now, if we multiply the ﬁrst equation of () byφ∈C^∞(Q) withφ_|_∂=  andI^–αφ() =  inand integrating by parts overQ, we obtain

Z

Q

y_n(x,t) –y(x,t; , )

φ(x,t)dt dx

= Z

Q

D^α_Cζ_n(x,t) –1ζ_n(x,t)

φ(x,t)dt dx

= Z

Q

ζ_n(x,t)dt dx+ Z

ζ(x, ,v_n)∂

∂tI^–αφ()dx.

Using (c), (), and () while passing the latter identity to the limit, we obtain Z

Q

y^γ(x,t) –y(x,t; , )

φ(x,t)dt dx

= Z

Q

ζ^γ(x,t)dt dx+ Z

ρ∂

∂tI^–αφ()dx,

∀φ∈C^∞(Q) such thatφ_|_∂= ,I^–αφ() =  in, () which by using again Lemma ., (), (), and () gives

– Z

6

∂

∂νφ(σ,t)ζ^γ(σ,t)dσdt+ Z

ρ(x)∂

∂tI^–αφ()dx

= Z

ζ^γ()∂

∂tI^–αφ()dx

∀φ∈C^∞(Q) such thatφ_|_∂= ,I^–αφ() =  in.

Hence, choosingφ∈C^∞(Q), such thatφ_|_∂= ,I^–αφ() =_∂t^∂I^–αφ() = , we get

ζ^γ =  on6, ()

and then

ζ^γ() =ρ in. ()

In view of (), (), (), and (), we see thatζ^γ=ζ(u^γ) is a solution of











D_C^αζ^γ –1ζ^γ =y(u^γ, ) –y(, ) inQ, ζ^γ=  on6,

ζ^γ(T) =  in

∂ζ^γ

∂t (T) =  in.

()

Moreover, using (), equation () becomes ζ(·, ;vn)⇀ ζ^γ() =ζ ·, ;u^γ

weakly inL^(). ()