R E S E A R C H
Open Access
Variational formulation and optimal
control of fractional diffusion equations with
Caputo derivatives
Qing Tang
1*and Qingxia Ma
1,2*Correspondence:
1School of Mathematics and
Physics, China University of Geosciences, Lumo 388, Wuhan, 430074, China
Full list of author information is available at the end of the article
Abstract
In this paper we start by giving a new definition of weak Caputo derivative in the sense of distributions, and we give a variational formulation to a fractional diffusion equation with Caputo derivative. We prove the existence and uniqueness of the solution to this weak formulation and use it to obtain a result on optimal control.
MSC: 37L05; 47J30
Keywords: fractional diffusion equation; distributional weak Caputo derivative; fractional Sobolev space
1 Introduction
In this paper we study fractional diffusion equations with controls by the method of an ab-stract variational formulation. For a comprehensive treatment of the subject of fractional calculus and fractional differential equations we refer to Kilbaet al.[]. There has been a large and fast increasing literature on diffusion equations with time fractional derivatives [–]. An important obstacle to study solutions in fractional Sobolev spaces is that the Ca-puto derivative was not clearly defined when the first order derivative does not exist in the strong sense. In the recent work of Gorenfloet al.[], they gave a definition of the Caputo derivative in fractional Sobolev space. In this paper we attempt to give a new definition of the weak fractional Caputo derivative via distribution theory and an integration by parts formula. This definition makes it very natural to adopt the theory of operational differ-ential equations (Lions []) and gives an abstract variational formulation of the fractional diffusion equation.
We will study this weak formulation with the classic Faedo-Galerkin method. This is essentially a Hilbert space method []. We adopt it for fractional diffusion equations by using energy estimate inequalities and lemmas on weak convergence.
We note that the integration by parts technique has been developed and extensively used in the theory of fractional calculus of variations, of which we refer to the monograph of Malinowska and Torres [].
We obtain a solution in the functional setting of a fractional Sobolev space due to the following fact: theL-fractional derivative is the fractional power of the realization of a
derivative inLspace []. For a detailed analysis and characterization of the fractional
power of differential operator in the setting of Sobolev space, we refer to [, ].
Using the fractional integration by parts formula, we can also construct the adjoint sys-tem to our variational (weak) formulation. By a classic result of convex analysis we can characterize the optimal control of a system of partial differential equations and inequal-ities, which can be applied to concrete fractional diffusion equations.
Consider the following fractional diffusion equation: Letbe a bounded open subset ofRnwith sufficiently regular boundary, with cylinder defined as
Q=×],T[, =×],T[.
We will study the system with Dirichlet boundary condition:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
∂α
ty–
n i,j=
∂
∂xi(aij(x,t) ∂y
∂xj) =f inQ,t∈(,T], y= on,
y(x, ) = in,
(.)
and the system with distributed control:
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
∂α
ty(u) –
n
i,j=∂∂xi(aij(x,t) ∂y(u)
∂xj ) =f +u inQ,t∈(,T], y= on,
y(x, ) = in,
(.)
herey(t,x;u) andudenote the state and control, respectively,α∈(, ). We assume the conditions for the coefficient:
aij∈L∞(Q) and symmetric, ≤i,j≤n. (.)
Also∃θ> , such that∀ξ∈Rn,
n
i,j=
aij(x,t)ξiξj≥θ n
i=
ξi, a.e. inQ. (.)
Our approach has the following novelties:
. We define the notion of a weak Caputo derivative and prove that it has similar properties to the weak derivative defined in [] (norm equivalence with the inverse of Riemann-Liouville operator).
. The variational method allows us to consider fractional diffusion equation where the coefficientaij(x,t)of the diffusion operator depends on time.
. The variational method allows us to considerf taken in a larger space thanL(Q).
2 Preliminaries
His a separable Hilbert space.V is a dense subspace ofHwith continuous injection:
V →H;
as usual,His identified with its dualH. We denote the evolution triple{V,H,V}with embeddings:
V →H →V (.)
being continuous and dense.
We introduce the vector valued spaceL(,T;V) such thaty∈L(,T;V) if
T
y(t)Vdt /
<∞.
In the same way we can defineL(,T;H),L(,T;V). We have the following.
LetE,Ebe two Banach spaces,L(E,E) denotes the space of continuous linear
map-ping ofE→E.A(t) is an unbounded operator inH, and continuous linear operator in
V itself, such that
A∈LL(,T;V),L,T;V, A(t)∈LV,V.
We denote (A(t)y(t),v) =a(t;y(t),v) for allv∈V, and give the following conditions: (A) a(t;y(t),v)≤μy(t)v, whereμis a constant independent oft∈[,T],y,v∈V. (A) t→a(t;y(t),v)is measurable in],T[for ally(t),v∈V.
(A) ∃θ> independent oft,Rea(t;v,v)≥θy,∀v∈V. Hencea(t;y(t),v)is coercive
uniformly int.
To avoid confusion, we clarify here that we use Dom orDto denote the domain of an operator, andRdenotes the range.
Definition . The left and right Caputo derivative are defined, for <t≤T, <α< , by
∂tαy(t) =CDα
ty(t) =
( –α)
t
(t–s)–α∂y ∂s(s)ds,
C tD
α
Ty(t) =
( –α)
T
t
(s–t)–α∂y ∂s(s)ds.
From this definition we see that for∂tαy(t) to be well defined,∂y/∂tmust be well defined. This is quite restrictive in applications, hence motivating our study of the weak Caputo derivative.
Next we define left and right Riemann-Liouville integrals, for ≤t≤T, <α≤:
Jαy(t) =
Itαy(t) =
(α)
t
(t–s)α–y(s)ds, J=I,
tIαTy(t) =
(α)
T
t
Wheny∈L(,T), the left Riemann-Liouville integral can also be defined via a
convolu-tion ((.), p., []):
Jαy(t) =gα∗y(t),
where
gα(t) = ⎧ ⎨ ⎩
(α)t
α–, t> ,
, t≤.
We denote byHα(,T) the fractional Sobolev space of orderαon (,T).
Hα(,T) =
y∈Hα(,T) :y() = .
For details of these definitions we refer to Section ., Chapter of [].
It has been verified in previous literature that the Riemann-Liouville integral operator is injective [], hence we can define a new operatorJ–αas the inverse operator ofJα. By definition,D(J–α) =R(Jα).
We use the following result regarding a solution to a fractional differential equation []:
∂tαy=f, f ∈L,
is given byu=Jαf with the range of this operatorJαbeing
RJα= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Hα(,T), ≤α< ,
Hα(,T), <α≤,
{y∈H(,T) :T
t–|y(t)|dt<∞}, α= .
(.)
In the literature of interpolation theory, one sometimes denotes (Remark ., [])
H
(,T),H(,T) =H
(,T) =
y∈H(,T) :ty∈L(,T).
With the above definitions we have the following result.
Lemma .(Theorem ., []) The normsJ–αy
L andyHα(,T)are equivalent for y∈
R(Jα).
We denote byHα(,T;V,V) the vector valued fractional Sobolev space.
Wα,p(,T;V,V) is the restriction on (,T) ofWα,p(–∞,∞;V,V) given by the Fourier
transform and
Hα,T;V,V=Wα,,T;V,V.
Lemma .((.), []) Denote by Cu(,T;H)the space of uniformly continuous functions
from(,T)into H.Supposeα> /p( <α< , <p≤ ∞).Then
Wα,p,T;V,V→Cu(,T;H) with compact embedding.
Lemma .((.), []) Letα,p and q satisfy:
ifα> /pthenp≤q≤ ∞,
ifα= /pthenp≤q<∞,
ifα< /pthenp≤q≤p∗,whereα– /p= –/p∗,that is,p∗=p/( –αp)( <α< , ≤p≤q≤ ∞).
Then
Wα,p,T;V,V→Lq(,T;H) with compact embedding. (.)
In order to use the theory of operational differential equations, we need to interpret the weak Caputo derivative in the sense of distributions, through fractional integration by parts in the formula (() in [])
T
∂tαy(t),ψ(t)dt=
T
y,CtDα
Tψ(t)
dt+tIT–αψ(t)·y(t)
T
. (.)
Fory() = ,ψ(T) = , we have
tIT–αψ(t)·y(t)
T = .
Hence we can proceed to the construction of a weak Caputo derivative in the sense of distributions (Schwartz []), note that if a distribution function is infinitely differentiable then its Caputo fractional derivative must also exist. It greatly simplifies the situation since we have the initial conditiony() =y= .
We denote byD(],T[) the space of infinitely differentiable functions in ],T[ with com-pact support. We call every continuous linear mapping ofD(],T[) intoEa vectorial dis-tribution over ],T[ with values in a Banach spaceE, and we denote
D],T[;E=LD],T[;E.
Definition . Define the test functionϕ∈D(],T[) for the functionysuch thaty() = , we call∂α
tya distributional weak Caputo derivative if it is a linear functional onD(],T[)
that sendsϕintoT(y,C
tDαTϕ(t)).
Our new definition of a weak Caputo derivative generalizes the (left) Caputo derivative (Definition .) since it is well defined even when∂y/∂sdoes not exist in the strong sense. It coincides with the Caputo derivative if∂y/∂sdoes exist.
Lemma . ∂α
t(y(·),v) =∂tαy(·),vinD(],T[),for y∈Hα(,T;V,V ),v∈V.Here( , )
denotes duality in H,,denotes a duality pairing of V and V.Moreover,the weak Caputo derivative∂α
Proof Denote the functionϕ∈D(],T[). For allt,ϕ(t) is a scalar. We can writev(t) =ϕ(t)v. Observey(t),v∈V⊂Hand the duality,is compatible with the identification ofHwith its dual. This implies
v,y(t)=v,y(t)=y(t),v.
From Definition . and (.) we obtain
T
∂tαy(t),vϕ(t)dt=
T
v,y(t)Ct Dα
Tϕ(t)dt
=
T
y(t),vCtDαTϕ(t)dt
=
T
∂tαy(t),vϕ(t)dt,
hence∂α
t(y(·),v) =∂tαy(·),vinD(],T[).
Since the Sobolev spaceH(,T) is dense inR(Jα), for eachy∈R(Jα) we can construct
an approximating sequenceφnsuch that
lim
n→∞φn=y, φn∈H (,T).
By the Hahn-Banach theorem we can uniquely extend the domain of linear operator y→∂α
tyfromH(,T) toR(Jα). From [] (Lemma .) we know
∂tαφn=J–αφn, φn∈H(,T),
hence we obtain
T
∂tαy,ϕdt=
T
J–αy,ϕdt, y∈RJα.
From Definition . and the fact thatϕ∈D(],T[)⊂L(,T) we obtain the weak Caputo derivative∂α
ty=J–αyinLfory∈R(Jα).
From Lemma . and Lemma . we obtain the following: suppose we have a sequence of approximating solutionsym∈R(Jα); if we have a priori estimates independent ofm, such
thatym(t)∈L(,T;V) and∂tαym(t)∈L(,T;V), then we haveym(t)∈Hα(,T;V,V).
Definition . We define the variational fractional equation (we also call it a fractional operational differential equation). Supposef ∈L(,T;V),
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂tα(y(t),v) +a(t;y(t),v) =f(t),v inD(],T[),t∈(,T],
∀v∈V, y= .
(.)
Here∂α
From Lemma . we see that the first equation of (.) is equivalent to
∂tαy+A(t)y=f in the sense ofL,T;V,t∈(,T].
Definition . yis a (distributional) weak solution to system (.); it satisfies (.) with
y∈
3.1 Existence and uniqueness of solution to the variational formulation
Theorem . For y() = ,f ∈L(,T;V),conditions(A)to(A)are satisfied,suppose
Supposey(t)∈V⊂H, we know the following inequality (inequality (.), []):
T
Sinceyis a weak solution hence by Definition . and Lemma . we have
∂tαy(t) =
and from condition (A) we obtain
g–α(T)
Existence. We proceed by constructing a Galerkin approximation of the equation. Recall thatVis a separable Hilbert space. Let{Vm}be a family of finite dimensional spaces, and
denote
From this we derive a finite dimensional approximationym(t) =
We proceed to establish a prior estimate independent ofm; by the factf ∈L(I;V ), using again inequality (.) and condition (A) we obtain
g–α(T)
By the condition on the unbounded linear operator we obtain
{Aum}is a bounded sequence inL
By definitionV is a reflexive Banach space (naturallyV as well), then the unit balls of L(,T;V) andL(,T;V ) are weakly compact, hence we obtain
strongly inV. Define ⎧
⎨ ⎩
ψm=ϕ⊗vm (i.e.ψm(t) =ϕ(t)vm),
By the formula of the derivation of a tensor product [],
and Definition . we obtain
C
Consider the approximating equation (, denotes the duality ofV andV,, V denotes
the duality ofV andV)
Then we can pass to the limit and obtain
Forα∈(, ),y∈Hα(,T;V,V) is (apart from a set of measure zero) equal to a continuous function from [,T]→H.
Finally we have to verify the initial conditiony= forα∈(, ): sincey() =
T
∂tαy(t),ϕ(t)vdt=
T
y(t),vCtDαTϕ(t)dt
and
T
∂tαym(t),vm
ϕ(t)dt=
T
ym(t),vm
C
tDαTϕ(t)dt+ (ym,v)IT–αϕ().
Passing to the limit, we have asm→ ∞
T
∂tαym(t),vm
ϕ(t)dt=
T
f(t),vϕ(t)dt–
T
a(t;y,v)ϕ(t)dt, (.)
T
∂tαym(t),vm
ϕ(t)dt=
T
y(t),vCtDαTϕ(t)dt+ (y,v)IT–αϕ(). (.)
From (.), (.), and (.) we obtain
(y,v)IT–αϕ() = , ∀v∈V,
so thaty= .
3.2 Optimal control of variational formulation
Denote by U the Hilbert space of controls. We define the control operator B∈L(U; L(,T;V)). Consider the system
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
∂α
ty(u) +A(t)y(u) =f +Bu inD(],T[),t∈(,T],f ∈L(,T;V ),
y(u) = ,
y(u)∈L(,T;V).
(.)
Define observation Cy(u) such thatC∈L(L(,T;V);H).His a Hilbert space.N is
given asN∈L(U;U) and
(Nu,u)U≥νuU, ν> . (.)
Define the cost function:
J(u) =Cy(u) –zd
H+ (Nu,u)U. (.)
Uadis a closed convex subset ofU(set of admissible controls),zd∈H. The optimal control
problem is to findw∈Uadsuch that
J(w) =inf
u∈UJ(u).
From the fact that the affine mapu→y(u) is continuous and (.), by Theorem . of [], there exists a unique optimal controlw∈Uadfor (.) with cost function given as
Lemma .(Lemma ., []) w∈Uadis optimal if and only if
J(w)·(u–w)≥, ∀u∈Uad. (.)
From (.) and (.) we see that (.) is equivalent to
Cy(w) –zd,C
y(u) –y(w)H+Nw,u–wU≥, ∀u∈Uad. (.)
Denote by(andU) the canonical isomorphism ofHontoH (and ofU ontoU ). Then it reduces to
C∗y(w) –zd
,y(u) –y(w)+Nw,u–wU≥, ∀u∈Uad. (.)
We introduce the adjoint system:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
C
tDαTp(u) +A∗(t)y(u) =C∗(Cy(u) –zd), t∈(,T],
p(T;u) = , p(u)∈L(,T;V).
(.)
Theorem . The optimal control w∈Uis characterized by systems of partial differential systems and inequality:
⎧ ⎨ ⎩
∂tαy(w) +A(t)y(w) =f+Bw inD(],T[),t∈(,T], y(;w) = ,
(.)
⎧ ⎨ ⎩
C
tDαTp(w) +A∗(t)y(w) =C∗(Cy(w) –zd), t∈(,T],
p(T;w) = , (.)
–UB∗p(w) +Nw,u–wU≥, ∀u,w∈Uad, (.)
⎧ ⎨ ⎩
y(w)∈L(,T;V),
p(w)∈L(,T;V).
(.)
Here ( , )Udenotes the scalar product betweenU andU.
Proof FromC∗∈L(H;L(,T;V )) we see that (.) is equivalent to
T
C∗y(w) –zd
,y(u) –y(w)dt+Nw,u–wU≥, ∀u∈Uad. (.)
By Definition . and the definition of an adjoint operator we obtain
T
C
tD
α
Tp(w),y(u) –y(w)
dt=
T
p(w),∂tαy(u) –∂tαy(w)dt, (.)
T
A(t)∗p(w),y(u) –y(w)dt=
T
By adding (.) and (.), and using the first equation of (.), we obtain
T
C(t)∗Cy(w) –zd
,y(u) –y(w)dt
=
T
p(w),A(t) +∂tαy(u) –y(w)dt
=
T
p(w),Bu–Bwdt
=B∗p(w),u–wU
=–UB∗p(w),u–wU.
Hence (.) and (.) are equivalent, thus we obtained the characterization of the
opti-mal controlw.
3.3 System with distributed control
Theorem . Suppose f ∈L(,T;H–())andU=L(,T;L()),then system(.)has
a unique weak solution y such that
y∈ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Hα(,T;H
(),H–()), ≤α<,
Hα(,T;H(),H–()), <α≤, {u∈H(,T;H
(),H–()) :
T t
–|y(t)|dt<∞}, α= ,
(.)
and forα∈(, )we have y∈Cu(,T;L())apart from a set of measure zero.
Proof We introduce the spaces
V=H(), H=L(), V =H–().
Hence we have
A(t)y= –
n
i,j=
∂ ∂xi
aij(x,t)
∂y ∂xj
.
TakeDom(A(t)) =V=H() andA(t)∈L(H(),H–()). Forv∈V from Green’s formula we have
A(t)y(t)v dx=at;y(t),v, (.)
hence we obtain
at;y(t),v= –
n
i,j=
aij(x,t)
∂y ∂xj
∂v ∂xi
dx,
Denotef=f +u∈L(,T;H–()). From Definition ., (.) is equivalent to the
vari-From Theorem . we see that there exists a unique solutionyto (.) (and hence to sys-tem (.)) which satisfies (.). From Definition . this is also the unique weak solution to system (.). From Lemma . we obtain forα∈(
, ),y∈Cu(,T;L
()) apart from a
set of measure zero.
Finally we consider the optimal control of (.) with regard to the cost function:
J(u) =
Now we give interpretations of notations from Section . in this situation. The ob-servation space H=H =L(,T;L()). C is the injection of L(,T;H()) onto
L(,T;L()).B,,Uare identify mappings.
From (.) and (.) we can see thatA(t) is a self adjoint operator such that
A∗= –
From Theorem . we can obtain a characterization of optimal controlwof system (.) by simultaneously solving the following system of partial differential equations and in-equality:
In this work we considered fractional diffusion equation with Dirichlet boundary condi-tions with distributed control. It will be interesting also to consider a system with bound-ary controls or observations using the weak formulation given here.
Competing interests
Authors’ contributions
Qing Tang read and approved the final manuscript. Qingxia Ma contributed during the revision of this article, read and approved of it.
Author details
1School of Mathematics and Physics, China University of Geosciences, Lumo 388, Wuhan, 430074, China.2Wuhan
Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, China. Acknowledgements
Part of the research for this article was conducted during the Tang’s study in University of Santiago de Compostela for a Ph.D. degree in mathematics.
Received: 27 February 2015 Accepted: 5 August 2015
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