Hamilton-Jacobi equations for control problems of parabolic equations

April 2006, Vol. 12, 311–349 www.edpsciences.org/cocv

DOI: 10.1051/cocv:2006004

HAMILTON-JACOBI EQUATIONS FOR CONTROL PROBLEMS OF PARABOLIC EQUATIONS

Sophie Gombao

and Jean-Pierre Raymond

Abstract. We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers’ type in 2D. To deal with a control acting in a boundary condition a fractional power (−A)^β – where (A, D(A)) is an unbounded operator in a Hilbert spaceX – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power (−A)^β appears in another nonlinear term whose behavior is different from the one of the Hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets inX, but only in bounded subsets in a spaceY →X. To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets inY to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are H¨older continuous with respect to the time variable.

Mathematics Subject Classification. 49K20, 49L25.

Received November 16, 2004.

1. Introduction

In this paper we study the uniqueness and existence of viscosity solution of the Hamilton-Jacobi-Bellman equation

−∂v

∂t (t, x)−

Dxv(t, x)|Ax

X+

Dxv(t, x)|(−A)^βF(t,Λx)

X

+H

t, x,(−A)^βD_xv(t, x)

= 0 in (0, T)×X, (1.1)

v(T, x) =g(x) inX.

In this settingX is a real Hilbert equipped with the inner product (· | ·)_Xand the norm|·|_X,Ais an unbounded operator with domain D(A) inX, it is supposed to be self-adjoint and strictly dissipative inX, (−A)^β is the β-fractional power of (−A), and 0 ≤ β ≤ ¹₂, Λ is a bounded linear operator from D((−A)^α) into X₀ with 0 ≤ α < ¹₂, X₀ is another real Hilbert space equipped with the inner product (· | ·)_X0 and the norm |·|_X0,

Keywords and phrases. Hamilton-Jacobi-Bellman equation, boundary control, semilinear parabolic equations.

1 Laboratoire MIP, UMR CNRS 5640, Universit´e Paul Sabatier, 31062 Toulouse Cedex 4, France;[email protected];

[email protected]

c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/cocvor http://dx.doi.org/10.1051/cocv:2006004

F ∈ C([0, T]×X₀;X), the Hamiltonian functional H is continuous in [0, T]×X ×X, and the mappingg is Lipschitz continuous inX. Precise assumptions are stated in Section 2.

Equation (1.1) is related to optimal control problems of semilinear parabolic equations (including in particular the case where the control acts in a nonlinear boundary condition). More precisely, for allt∈[0, T] andx∈X, consider an optimal control problem of the form

(Pt,x) min

J(t, u, y)|u∈ M(t, T;U) and (y, u) is solution of equation (1.2) ,

where the cost functionalJ is defined by J(t, y, u) =

T t

L(r, y(r), u(r)) dr+g(y(T)), and the state equation is

y=Ay+ (−A)^β

Bu−F(·,Λy)

, y(t) =x. (1.2)

The control spaceM(t, T;U) is a set of bounded measurable functions with values inU, andU is a bounded subset in X_Γ,X_Γ is a Banach space,B∈ L(XΓ, X). In Section 3 we prove that equation (1.1) admits at most one viscosity solution. In Section 4, we prove that the value function of problem (Pt,x) is the unique viscosity solution of equation (1.1), whenH is defined by

H(t, x, p) = sup

u∈U

−(p|Bu)_X−L(t, x, u) .

Applications are discussed in Section 5. Before presenting what is new in the present paper, observe that by setting

H(t, x,(−A)^βDxv(t, x)) =H(t, x,(−A)^βDxv(t, x)) +

(−A)^βDxv(t, x)|F(t,Λx)

X, equation (1.1) can be written in the form

−∂v

∂t(t, x)−

D_xv(t, x)|Ax

X+H

t, x,(−A)^βD_xv(t, x)

= 0 in (0, T)×X. (1.3) Equation (1.3) seems to be simpler to handle than equation (1.1). However assumptions on F(t,Λx) and on H(t, x, p) are different and we cannot simplify the presentation of the paper by considering equation (1.3) (see e.g. the estimates involvingH andF in the proof of Th. 3.5).

During the eighties and the nineties several fundamental advances have been made in the study of Hamilton- Jacobi equation in infinite dimension. These equations were first studied by Barbu and Da Prato (seee.g. [2]), mainly in classes of convex functions. The method of viscosity solutions has been extended to infinite dimension by Crandall and Lions in a series of papers [10–14]. All these papers correspond to the case when β = 0.

Other contributions are due to Cannarsa and Frankowska [5], Ishii [20], Soner [27], Tataru [28,29], Crandall and Lions [15, 16], Cannarsa and Tessitore [6–9] in order to deal with boundary controls. In particular equations of the form (1.3) with 0 < β < ¹₂ are studied in [6], [8] to treat Neumann boundary controls. The case of Dirichlet controls is considered in [7, 9], it corresponds to the situation when ¹₂ < β <1 and has to be studied independently. More recently the case of the Navier-Stokes equations has been studied in [18, 26].

The main motivation of the present paper is to characterize the value function of control problems governed by semilinear parabolic equations, including the case of equations with a nonlinear boundary condition, or the case of nonlinearity of Burgers’ type in two dimension, and with cost functionals whose growth is quadratic or even higher than quadratic. For example we study the case of partial differential equations with nonlinear boundary conditions of the form:

∂y

∂t −∆y+y=f in ]t, T[×Ω, ∂y

∂n+h(y) =u on ]t, T[×Γ, y(t) =x in Ω, (1.4)

with cost functionals of the type

J(t, y, u) = T

t

L(r, y(r), u(r)) dr +g(y(T)),

where h is any regular nondecreasing function obeying h(0) = 0, and where L and g may be quadratic cost functionals. Many thermal processes lead to the kind of model corresponding to equation (1.4) (see [23]). The papers mentioned above do not include this model in their possible applications. If the initial conditionxbelong toX =L²(Ω), equation (1.4) is well posed and it admits a unique weak solution belonging toC([0, T];X) (the solution also belongs to L²(0, T;H¹(Ω))). We can write equation (1.4) in the form

y=Ay+ (−A)^β

Bu−F(·,Λy)

, y(t) =x, (1.5)

by defining Λ as the trace mapping on Γ:

Λ : y−→y|Γ.

In this example Λ is bounded from H^2α(Ω) =D((−A)^α) intoX₀ = L²(Γ) for all ¹₄ < α < ¹₂, D(A) = {y ∈ H²(Ω)| ^∂y_∂n = 0}, Ay = ∆y, and we have to take ¹₄ < β < ¹₂. For a parabolic equation with a nonlinearity of Burgers’ type we can take β = ¹₂. Let us denote by yt,x,u the solution to equation (1.5). To characterize the value functionv(t, x) of the problem

(Pt,x) min

J(t, u, y)|u∈ M(t, T;U) and (y, u) is solution of equation (1.5) ,

we have to study the dependence ofy_t,x,uand of Λy_t,x,uwith respect totand tox. Due to the nonlinear term in equation (1.5), we can prove continuity properties fory_·,x,uand Λy_·,x,uand Lipschitz properties fory_t,·,uand Λyt,·,uwhen the initial conditionxstays in bounded subsets inY, for a spaceY →X, but these properties are not true if we consider only bounded subsets inX. Therefore it is natural to study the properties of the value functionv(t, x) whenxremains in bounded subsets ofY, and to look for solutions to equation (1.4) in a space of the type C([0, T];Y) or at leastL^∞_w(0, T;Y) (the space of bounded and weakly measurable functions from (0, T) intoY).

Another difficulty comes from the cost functional. In the literature on Hamilton-Jacobi-Bellman equations, it is often assumed that the cost functionals either are bounded or satisfy a linear growth condition [6, 8, 18, 20].

Thus the case of quadratic cost functionals is not treated in these papers.

To overcome the two difficulties mentioned above, the one coming from the nonlinearity in the state equation and the other one due to the growth condition of the cost functional, we suggest to proceed as follows. First, we show that, for an initial condition inB_Y(M₀) (the ball inY centered at the origin and with radiusM₀), the solutiony of equation (1.2) satisfies y(·)∈B_Y(R_T) in (t, T) for someR_T =R(M₀, T) which can be explicitly estimated independently of t ∈ (0, T). Next, we associate with the mappings L(t, ·, u), g and F(t, Λ·), other mappingsL(t,·, u),gandF(t,Λ·) which are identical to the previous ones in the ballBY(RT), but which satisfies some global boundedness and Lipschitz properties. Let us consider the problem (Pt,x) – the one introduced at the beginning of the introduction – defined with L(t,·, u), g and F(t,Λ·). We are able to show that value function v(t, x) of problem (Pt,x) obeys v(t, x) = v(t, x) for t ∈ (0, T) and x∈ BY(RT). We show that v is the unique viscosity solution of the Hamilton-Jacobi equation (1.1). Thus v is not the viscosity solution to equation (1.1), but it is equal to the viscosity solution of equation (1.1) in bounded sets in (0, T)×Y.

Sections 2, 3 and 4 are devoted to the study of equation (1.2), equation (1.1), and the value function of problem (Pt,x). In these sections, only the mappingsL(t,·, u),g and F(t,Λ·) intervene. The assumptions are precisely stated in Section 2. The definition of the mappings L(t,·, u), g and F(t,Λ·) from L(t, ·, u), g and F(t,Λ·) is treated in examples of Section 5 by using projection operators. Three examples are considered. The first one is a control problem for the state equation (1.4), and the two others correspond to problems for a two dimensional scalar equation of Burgers’ type. The interest of the third example is to show that the method

using a projection operator in the cost functional and the state equation is flexible enough to involve different kind of projections adapted to the nonlinearity and to the functionals we have to deal with.

Let us finally mention that the definition of viscosity solutions that we take is not totally standard. Indeed we consider viscosity solutions which are H¨older continuous with respect to the time variable. This H¨older continuity condition, which is a new argument in the definition of viscosity solutions – see Definition 3.2 – plays a major role in the proof of uniqueness to estimate the nonlinear termF. A preliminary version of the present paper corresponds to a part of the Ph.D. thesis by the first author [17].

2. Preliminaries on the evolution equation

In this section we want to study properties of solutions of the evolution equation y =Ay+ (−A)^β

Bu−F(·,Λy)

in (t, T), y(t) =x, (2.1)

wheret∈[0, T).

2.1. Assumptions

Throughout the paper we make the following assumptions.

(i) The unbounded operatorA, with domainD(A) inX, is a closed and densely defined selfadjoint operator in X, such that (Ax|x)_X≤ −ω|x|²X for allx∈D(A), whereω >0.

(ii) B∈ L(X_Γ, X).

(iii) The linear operator Λ is bounded fromD((−A)^α) into X₀ for someα∈[0,¹₂[, that is:

|Λx|_X0 ≤C_α|(−A)^αx|_X for allx∈D((−A)^α). (2.2) The exponentβ ∈[0,¹₂] is given fixed.

(iv) F is a continuous mapping from [0, T]×X₀into X, which satisfies:

|F(t, x)−F(t, y)|X≤KF|x−y|X₀, and |F(t, x)|X ≤MF, (2.3) for allt∈[0, T], and allx,y∈X₀. Moreover, there existsη₁∈]0,1] such that:

|F(t, x)−F(s, x)|X ≤M₁,F 1 +|x|X₀

|t−s|^η1. (2.4)

In addition, we assume that eitherβ < ¹₂, orβ =¹₂ and D((−A)¹²)→X₀,

|F(t, x)|D((−A)^β0)≤M

β₀,|x|_D((−A)¹₂₎

for allt∈[0, T] and allx∈D((−A)¹²), (−A)^β0B∈ L(X_Γ, X), for some 0< β₀< β= ¹₂,

(2.5)

whereM

β₀,|x|_D((−A)¹₂₎

>0 only depends onβ₀ and|x|_D((−A)¹₂₎.

(v) The control ubelongs toM(t, T;U), the space of measurable functions from (t, T) intoU, where U is a nonempty, bounded and closed subset ofX_Γ,such that

|u|X_Γ ≤MU for all u∈U. (2.6)

We now state assumptions needed in Section 3 to study equation (1.1).

(vi) The mappingg∈C(X) is Lipschitz continuous and bounded inX,i.e.:

|g(x)−g(y)| ≤K_g|x−y|_X and |g(x)| ≤M_g, for allx, y∈X.

(vii) The Hamiltonian functionalH satisfies

|H(t, x, p)−H(s, y, q)| ≤K_H(|t−s|^η2+|x−y|_X+|p−q|_X). (2.7) In Section 4, we make the following additional assumption.

(viii) The Hamiltonian functionalH : [0, T]×X×X →Ris defined by:

H(t, x, p) = sup

u∈U[−(p|Bu)_X−L(t, x, u)], (2.8) where the functionalL∈C([0, T]×X×U) satisfies:

|L(t, x, u)−L(s, y, u)| ≤KL(|t−s|^η2+|x−y|X) and |L(t, x, u)| ≤ML, for allt, s∈[0, T],allx, y∈X,and allu∈U, with 0< η₂≤1.

Observe that ifH is defined by (2.8) and ifLsatisfies the estimate stated in (viii), then

|H(t, x, p)−H(s, y, q)| ≤K_H(|t−s|^η2+|x−y|_X+|p−q|_X), withKH= max (KL,BMU).Thus assumption (vii) is automatically satisfied in that case.

Due to assumption (i), (A, D(A)) is the infinitesimal generator of a strongly continuous analytic semigroup of contractions onX which satisfies

e^tA_L(X)≤e^−ωt. (2.9)

Moreover (see [19], Th. 1.4.3, Chap. 1 and [3], Prop. 5.1, Chap. 1), for all δ ≥0,there exists a constant M_δ such that, for allt >0:

(−A)^δe^At

L(X)≤Mδt^−δ. (2.10)

If 0< δ≤1,andx∈D((−A)^δ),we have:

e^At−Ix

X≤ 1

δM_1−δt^δ(−A)^δx

X. (2.11)

Besides, for allδ < γ and allx∈D((−A)^γ), one has:

(−A)^δx

X ≤M_δ,γ|(−A)^γx|_X^γδ |x|¹⁻_X ^γδ. (2.12) With Young’s inequality the last estimate implies that, for all δ ∈

0,¹₂

, and all σ >0, there exists a con- stantC_δ,σsuch that:

(−A)^δx

X ≤σ(−A)¹²x

X+C_δ,σ|x|_X forx∈D((−A)¹²). (2.13)

2.2. Properties and regularities of mild solutions of equation (2.1)

Theorem 2.1. For all x∈X and allu∈ M(t, T;U), equation (2.1) admits a unique mild solution y_t,x,u in L¹(t, T;D((−A)^α)), it obeys:

yt,x,u(s) = e^(s−t)Ax+ (−A)^β s

t e^(s−r)A

Bu(r)−F(r,Λyt,x,u(r))

dr, (2.14)

for alls∈[t, T]. Moreover yt,x,u belongs toC([t, T];X)and satisfies the estimate yt,x,uC([t,T];X)≤C(1 +|x|X+uL^∞(t,T;U)).

Proof. Lett₁∈(t, T] be such thatCαKFMα+β(t₁−t)^1−(α+β)

1−(α+β) ≤1/2. Let us setE=L¹(t, t₁;D((−A)^α)), and let us show that the mapping

y−→(Ψy) (s) = e^(s−t)Ax+ (−A)^β s

t e^(s−r)A[Bu(r)−F(r,Λy(r))] dr, is a contraction inE. First we have:

t₁

t |(−A)^α(Ψy) (s)|ds≤ t₁

t

(−A)^αe^(s−t)Axds+ t₁

t

s t

(−A)^α+βe^(s−r)A[Bu(r)−F(r,Λy(r))]drds

≤Mα|x|X

(t₁−t)^1−α

1−α +Mα+β

t₁ t

s t

1

(s−r)^α+β [BMU+MF] drds

≤M_α|x|_X (t₁−t)^1−α

1−α +M_α+β[BM_U+M_F] (t₁−t)^2−(α+β) [1−(α+β)] [2−(α+β)]· Thus, ify∈E, Ψybelongs toE. Moreover ify₁, y₂∈E, we can write

t₁

t |(−A)^α(Ψy₁) (s)−(−A)^α(Ψy₂) (s)|_Xds

≤ t₁

t

s t

(−A)^α+βe^(s−r)A[F(r,Λy₁(r))−F(r,Λy₂(r))]

Xdrds

≤ t₁

t

s t

M_α+β

(s−r)^α+βKF|Λy₁(r)−Λy₂(r)|X₀drds

=KF

t₁

t |Λy₁(r)−Λy₂(r)|X₀

t₁ r

M_α+β

(s−r)^α+βdsdr

≤CαKFMα+β

t₁

t |(−A)^αy₁(r)−(−A)^αy₂(r)|_X(t₁−r)^1−(α+β) 1−(α+β) dr

≤C_αK_FM_α+β(t₁−t)^1−(α+β) 1−(α+β)

t₁

t |(−A)^αy₁(r)−(−A)^αy₂(r)|_Xdr.

Thus Ψ is a contraction in E, and it admits a unique fixed point in E, which is the unique solution y in E to equation (2.1). In addition y belongs to C([t, t₁];X) and formula (2.14) is satisfied for all s ∈ [t, t₁]. We can repeat this process on the interval [t₁,2t₁], and step by step, we prove that equation (2.1) admits a unique solution inL¹(t, T;D((−A)^α)), which belongs toC([t, T];X) and satisfies formula (2.14).

Proposition 2.2. Assume that x∈D((−A)^α). Then the solution yt,x,u of (2.1) satisfies:

|Λyt,x,u(s)−Λx|_X0 →0 uniformly with respect tou∈ M(t, T;U) when st. (2.15) Proof. Letxbe inD((−A)^α). With inequality (2.2) we have

|Λ(yt,x,u(s)−x)|_X0 ≤C_α|(−A)^α(y_t,x,u(s)−x)|_X. Due to (2.14), we can write

|(−A)^α(yt,x,u(s)−x)|_X ≤(−A)^α

e^(s−t)Ax−x

X (2.16)

+

(−A)^β+α s

t e^(s−r)A[Bu(r)−F(r,Λyt,x,u(r))] dr X

.

We can estimate the two terms in the right hand side of (2.16) as follows:

(−A)^α

e^(s−t)Ax−x_X =

e^(s−t)A−I

(−A)^αx_X, (2.17)

and

(−A)^β+α s

t e^(s−r)A[Bu(r)−F(r,Λyt,x,u(r))] dr

X ≤Mα+β(s−t)^1−(α+β)

1−(α+β) (MF +BMU). (2.18) The two terms (2.17) and (2.18) go to 0 uniformly with respect to u ∈ M(t, T;U) when s t, because

(α+β)<1.

Proposition 2.3. Let yt,x,u be the weak solution of (2.1). There exists a constant C₁(β), independent of u, such that y_t,x,u(s)−e^(s−t)Ax

X ≤C₁(β) (s−t)^1−β for allx∈X. (2.19) Proof. Letyt,x,ube the weak solution of equation (2.1). With the integral formulation (2.14), we have

yt,x,u(s)−e^(s−t)Ax

X ≤ (−A)^β

s

t e^(s−r)A[Bu(r)−F(r,Λy(r))] dr X

.

From (2.3) and (2.10) it follows that (−A)^β

s

t e^(s−r)A[Bu(r)−F(r,Λy(r))] dr X ≤s

t M_β

(s−r)^β[BMU+MF] dr

≤M_β[BM_U+M_F](s−t)^1−β 1−β ·

The proof is complete.

Proposition 2.4. We assume thatβ =¹₂ and that the corresponding additional conditions of assumption(iv) is satisfied. There exists a constant C₁

β₀,|x|_D((−A)¹₂₎

, independent of u, such that y_t,x,u(s)−e^(s−t)Ax

X≤C₁

β₀,|x|_D((−A)¹₂₎

(s−t)^12+β0 for all x∈D((−A)¹²). (2.20)

Proof. Assume thatx ∈D((−A)¹²). Let yt,x,u be the weak solution of equation (2.1). By using the integral formulation (2.14), we have

y_t,x,u(s)−e^(s−t)Ax

X ≤

(−A)^β−β0 s

t e^(s−r)A

(−A)^β0Bu(r)−(−A)^β0F(r,Λy(r)) dr

X

.

With (2.5) and (2.10) we have (−A)^β−β0

s

t e^(s−r)A

(−A)^β0Bu(r)−(−A)^β0F(r,Λy(r)) dr

X

≤ s

t

M_β−β0 (s−r)^β−β0

(−A)^β0BMU+M

β₀,|x|_D((−A)¹₂₎ dr

≤Mβ−β₀

(−A)^β0BMU+M

β₀,|x|_D((−A)¹₂₎(s−t)^1−β+β0 1−β+β₀ ·

The proof is complete.

To prove the other propositions, we need the following theorem.

Theorem 2.5 ([1] Th. 3.3.1, Chap. 2). Let δ andγ be in[0,1[and >0. If the mapping t∈J ⊂R⁺→t^δu(t),

belongs toL^∞_loc(J;R), and if there exist two positive constantsa, bsuch that u(t)≤a t^−δ+b

t 0

(t−τ)^−γu(τ) dτ, for a.e. t∈J^∗=J\ {0}, then there exits a positive constant c:=c(δ, γ, )independent ofaandb such that

u(t)≤a t^−δ

1 +c b t^1−γe^(1+)k(γ,b)t

for a.e. t∈J^∗, wherek(γ, b) := (Γ (1−γ)b)^1/(1−γ).

Proposition 2.6. Letxandx₀be inX,u∈ M(t, T;U), and lety_t,x,uandy_t,x0,ube the corresponding solutions to equation (2.1). Then, for allθ∈[0,1−α[, there exists a constant C₂(α, β, θ)such that:

|Λyt,x,u(r)−Λyt,x₀,u(r)|_X0 ≤C₂(α, β, θ) (r−t)^α+θ

(−A)^−θ(x−x₀)

X, (2.21)

for allr∈(t, T]. (The constantC₂(α, β, θ)is explicitly given in (2.24).)

Proof. Using the integral formulation (2.14) foryt,x,uandyt,x₀,u, and (2.2), we obtain

|Λyt,x,u(r)−Λyt,x₀,u(r)|_X0 ≤Cα(−A)^α+θe^(r−t)A(−A)^−θ(x−x₀)

X

+Cα

(−A)^β+α r

t e^(r−s)A[F(s,Λyt,x,u(s))−F(s,Λyt,x₀,u(s))] ds X

. (2.22) Setting θ= 0 in this estimate, we first obtain

|Λyt,x,u(r)−Λy_t,x0,u(r)|_X0 ≤ C_αM_α

(r−t)^α|x−x₀|_X+2C_αM_α+βM_F

1−(α+β) (r−t)^1−(α+β).

Multiplying both sides by (r−t)^α+θ we have (r−t)^α+θ|Λyt,x,u(r)−Λyt,x₀,u(r)|_X0

≤(T−t)^θCαMα|x−x₀|X+2C_αM_α+βM_F

1−(α+β) (T−t)^1+θ−β∈L^∞(t, T;R). (2.23) Next with (2.22) we write

|Λyt,x,u(r)−Λy_t,x0,u(r)|_X0

≤ CαMα+θ

(r−t)^α+θ

(−A)^−θ(x−x₀)

X+ r

t

CαMβ+α

(r−s)^β+αKF|Λyt,x,u(s)−Λyt,x₀,u(s)|_X0ds.

Since the function r → (r−t)^α+θ|Λyt,x,u(r)−Λyt,x₀,u(r)|_X0 belongs to L^∞(t, T), we can use Theorem 2.5 with for exampleε= 1, and we obtain (2.21) by setting

C₂(α, β, θ) =C_αM_α+θ

1 +C_β+α,α+θ(c C_αM_βK_F)T^1−β+αe^{2k(β+α,KFCαMβ+α)T}

(2.24)

wherec is the constant appearing in Theorem 2.5.

Proposition 2.7. Letxandx₀be inX,u∈ M(t, T;U), and lety_t,x,uandy_t,x0,ube the corresponding solutions to equation (2.1). Then, for all θ∈[0,1−α[, there exists a constant C₃(α, β, θ) such that

|y_t,x,u(r)−y_t,x0,u(r)|_X ≤

M_θ

(r−t)^θ+ C₃(α, β, θ)

1−(α+θ)(r−t)^{1−(α+β+θ)} (−A)^−θ(x−x₀)

X. (2.25) (The constantC₃(α, β, θ)is explicitly given in (2.27).)

Proof. With Proposition 2.6, we have:

|yt,x,u(r)−yt,x₀,u(r)|_X≤(−A)^θe^(r−t)A(−A)^−θ(x−x₀)

X+ r

t

M_β

(r−s)^βKF|Λyt,x,u(s)−Λyt,x₀,u(s)|_X0ds

≤ M_θ (r−t)^θ

(−A)^−θ(x−x₀)

X+M_βK_FC₂(α, β, θ)(−A)^−θ(x−x₀)

X

r t

1 (r−s)^β

1 (s−t)^α+θds

≤

M_θ

(r−t)^θ +M_βK_FC₂(α, β, θ) r

t

1 (r−s)^β

1

(s−t)^α+θds (−A)^−θ(x−x₀)

X. By using the integral formula of the beta function we have:

r t

1

(r−s)^β(s−t)^α+θds= Γ (1−β) Γ (1−(α+θ))

Γ (2−(α+θ+β)) (r−t)^{1−(α+θ+β)}. (2.26) By setting

C₃(α, β, θ) =Γ (1−β) Γ (1−(α+θ))

Γ (2−(α+θ+β)) MβKFC₂(α, β, θ), (2.27) and

C₄(α, β, θ, t;r) =

Mθ

(r−t)^θ + C₃(α, β, θ)

1−(α+θ)(r−t)^{1−(α+β+θ)}

, (2.28)

we obtain

|yt,x,u(r)−yt,x₀,u(r)|_X ≤C₄(α, β, θ, t;r)(−A)^−θ(x−x₀)

X.

Proposition 2.8. Let xbe in X,s, t∈[0, T), and u∈ M(min (t, s), T;U). Let us denote byyt,x,uandys,x,u

the solutions of equation (2.1) respectively corresponding to the initial data(t, x)and(s, x). Then there exist a constant C₅(α, β)and a continuous mapping¯a(t, s, x)(independent of α) such that, for allr∈]max (s, t), T], we have:

|Λyt,x,u(r)−Λys,x,u(r)|_X0≤ C₅(α, β)

(r−max (t, s))^α¯a(t, s, x). (2.29) The function s → ¯a(t, s, x) goes to 0 when s goes to t, for all fixed x ∈ X. (The constant C₅(α, β) and the mappinga¯ are explicitly defined in (2.34)and(2.35).)

Proposition 2.9. With the same assumptions and notation as in the previous proposition, there exists a constant C₆(α, β) such that, for allr∈[max (s, t), T], we have:

|yt,x,u(r)−ys,x,u(r)|_X≤C₆(α, β) ¯a(t, s, x). (2.30) (The constantC₆(α, β)is explicitly defined in(2.36).)

Proof of Proposition 2.8. Consider the case wheres < t. The case t < scan be treated in a similar way. Let be r > t > s, with estimate (2.2) and with (2.14), we have

|Λyt,x,u(r)−Λys,x,u(r)|_X0 ≤Cα(−A)^αe^(r−t)A

e^|t−s|Ax−x_X (2.31)

+C_α t

s (−A)^α+βe^(r−σ)A[Bu(σ)−F(σ,Λy_s,x,u(σ))] dσ X

(2.32) +Cα

r

t (−A)^α+βe^(r−σ)A[F(σ,Λyt,x,u(σ))−F(σ,Λys,x,u(σ))] dσ X

. (2.33) Now we can write

(2.31)≤ C_αM_α (r−t)^α

e^|t−s|A−I x

X. Since (r−t)^α≤(r−σ)^α for allσ∈(s, t), we have

(2.32)≤CαMα+β(MF+BMU) t

s

dσ

(r−σ)^α+β ≤CαMα+β(MF+BMU) (r−t)^α

t s

dσ (r−σ)^β· Similarly, for allσ∈(s, t), we have (r−σ)^β≥(t−σ)^β. Then

t s

dσ (r−σ)^β ≤

t s

dσ

(t−σ)^β =(t−s)^1−β 1−β , and therefore we obtain

(2.32)≤C_αM_α+β(M_F+BM_U)

(r−t)^α(1−β) |t−s|^1−β. The last term can be estimated as follows

(2.33)≤r t

C_αM_α+βK_F

(r−σ)^α+β |Λyt,x,u(σ)−Λys,x,u(σ)|_X0dσ

≤ CαMα+βKF

(r−t)^α r

t

1

(r−σ)^β|Λyt,x,u(σ)−Λys,x,u(σ)|_X0dσ.

From the estimates obtained for (2.31), (2.32), (2.33), we deduce that the function r → (r−t)^α|Λyt,x,u(r)−Λys,x,u(r)|_X0 belongs toL^∞(t, T). Applying Theorem 2.5, we obtain

|Λyt,x,u(r)−Λy_s,x,u(r)|_X0 ≤ C₅(α, β)

(r−t)^α ¯a(t, s, x), with

C₅(α, β) = 2Cαmax Mα

1−α,^Mα+β(M₁₋^F+_β ^BMU) 1 +c CαMβ+αKFT^(1−β)e^CT

, (2.34)

wherec andC are given in Theorem 2.5, and

¯a(t, s, x) =e^|t−s|Ax−x

X+|t−s|^1−β. (2.35)

The functions→¯a(t, s, x) goes to 0 when sgoes tot,for allxfixed inX.

Proof of Proposition 2.9. Consider the case where 0≤s < t.We have:

|yt,x,u(r)−ys,x,u(r)|_X

≤e^|t−s|Ax−x

X+Mβ(MF+BMU)

1−β (t−s)^1−β+C₅(α, β)KF¯a(t, s, x) r

t

Mβ

(r−σ)^β 1 (σ−t)^αdσ.

From (2.26) with θ= 0, it yields:

r t

1 (r−σ)^β

1

(σ−t)^αdσ≤Γ(1−β)Γ(1−α)

Γ(2−(α+β)) T^1−(α+β)≤4T^1−(α+β). Hence

|yt,x,u(r)−ys,x,u(r)|_X ≤

1 + M_β(M_F+BM_U)

1−β +C₅(α, β)KF4T^1−(α+β)

a¯(t, s, x). (2.36)

The proof is complete.

Proposition 2.10. Letxandx₀ be inX,t∈[0, T), andu∈ M(t, T;U). Let us denote byyt,x,uandyt,x₀,uthe solutions of equation (2.1) respectively corresponding to the initial data(t, x)and(t, x₀).Then, for allr∈[t, T] and all s∈]t, T], we have:

|Λyt,x,u(s)−Λy_t,x0,u(s)|_X0 ≤ C₇(α, β)

(s−t)^α |x−x₀|_X, (2.37) and

|yt,x,u(r)−yt,x₀,u(r)|_X≤C₈(α, β)|x−x₀|_X. (2.38) Proof. The functionw=yt,x,u−yt,x₀,usatisfies

|Λw(s)|X₀ ≤ M_α

(s−t)^α|x−x₀|X+

(−A)^β+α s

t e^(s−r)A[F(r,Λyt,x,u(r))−F(r,Λy,x₀,u(r))] dr X

≤ Mα

(s−t)^α|x−x₀|X+Mα+βKF

s t

1

(s−r)^α+β|Λw(r)|X₀dr.