Applied Mathematics and Computation 262 (2015) 42–55
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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
A new proof for the existence and uniqueness of the discrete
evolutionary HJB equations
Salah Boulaaras a,b,∗, Mohamed Haiour c
a
Department of Mathematics, Colleague of Science and Arts, Al-Ras, Al-Qassim University, Kingdom Of Saudi Arabia
Laboratory of Fundamental and Applied Mathematics, As-Sania University, Oran, Algeria
c
Department of Mathematics, Faculty of Science, University of Annaba, Box. 12, Annaba 23000, Algeria
b
a r t i c l e
i n f o
Keywords:
QVIs
Finite elements
Theta scheme fixed point
HJB equations
a b s t r a c t
In this paper, we establish a new proof for the existence and uniqueness of the discrete
solution of evolutionary HJB equations which can be approximated by a weakly coupled
system of coercive elliptic quasi-variational inequalities (OVIs). For that some properties of
the presented algorithm (cf., e.g., Boulaaras and Haiour, 2013; Boulbrachene and Haiour, 2001)
using the theta-scheme with respect to the t-variable combined with a finite element spatial
approximation are proved.
© 2015 Published by Elsevier Inc.
1. Introduction
The aim of the present paper is to extend a previous work by Boulbrachene and Haiour (cf. [5]) for the stationary case and
Boulaaras and Haiour for the evolutionary case (cf. [3]) where an error estimate and an asymptotic behavior results for solutions
of the stationary and evolutionary HJB equations have been established. Here we use a new approach based on the previous our
algorithm which has been given for evolutionary free boundary problems, using the concepts of L -stability (cf. [3]) . Namely,
we consider the following Hamilton–Jacobi–Bellman equations: find u ∈ Rn such that
⎧
ut + max Ai u − f i = 0 in ⎪
⎪
1≤
j≤
M
⎪
⎨
ui = 0 on ⎪
⎪
⎪
⎩ i
u x, 0 = u0 in ,
(1)
where is a bounded open domain of RN ,N 1, with boundary sufficiently smooth and set in Rn × R, = × [0, T] with
T¨ < +∞, the f i , (i = 1, . . ., M) are given smooth positive functions, and the Ai , (i = 1, . . ., M) are second-order, uniformly elliptic
operators defined over (H1 ())M
Ai u = −
∗
N
N
∂ ui ∂ ui
∂ i
ajk (x)
+
bik (x)
+ ai0 (x) ui
∂
xj
∂
xk
∂
xk
j,k=1
k=1
Corresponding author. Tel.: +966557618327.
E-mail addresses: [email protected], [email protected] (S. Boulaaras), [email protected] (M. Haiour).
http://dx.doi.org/10.1016/j.amc.2015.03.095
0096-3003/© 2015 Published by Elsevier Inc.
(2)
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
43
¯ ))M , x ∈ ¯ , 1 ≤ k, j ≤ N are sufficiently smooth coefficients and satisfy
and whose coefficients aik,j (x), bik (x), ai0 (x) ∈ (L∞ () ∩ C 2 (
the following conditions
aijk (x) = aikj (x);
ai0 (x) ≥ β > 0,
β is a constant,
(3)
with
N
aijk (x)ξj ξk ≥ γ |ξ |2 ;
¯
ξ ∈ Rn , γ > 0, x ∈ (4)
j,k=1
and the bilinear forms associated with Ai , for u, v ∈ H01 a (u, v) =
i
⎛
⎝
N
aijk
j,k=1
⎞
N
∂ ui ∂ vi ∂ ui i
(x)
+
bi (x)
v + ai0 (x) ui vi ⎠ dx.
∂ xj ∂ xk j=1 k ∂ xj
(5)
f i is a regular function satisfying
f i ∈ (L2 (0, T, L∞ ) ∩ C 2 (0, T, H−1 ()))M , f i ≥ 0.
(6)
We shall also need the following norm
∀ W = (w1 , w2 , . . ., wM ) ∈
M
L∞ (),
i=1
W ∞ = max wi ∞ .
1≤ i≤M
Let (., .) be the scalar product in L2 ().
The stationary HJB equations are encountered in several applications; for example, in stochastic control, the solution of (1)
characterizes the infimum of the cost function associated to an optimally controlled stochastic switching process without cost
for switching and for the calculus of quasi-stationary state for the simulation of petroleum or gaseous deposit (cf., e.g., [1]). From
the mathematical analysis point of view, elliptic case of the problem (1) was intensively studied in the late 1980s (cf. [8,10–13]).
On the numerical and computational side (cf. [5–8]). However, as far as finite element approximation is concerned, only few
works are known in the literature (cf. [5,6,13]).
In (cf. [3]) we applied a new time-space discretization using the theta time scheme combined with a finite element approximation, we found (1) can be transformed into the following full-discrete HJB equation:
⎧
⎪
max Bi uhi,θ ,k − f i,θ ,k = 0
⎪
⎪
1≤ i≤ M
⎪
⎪
⎨
(7)
u = 0 on ∂ ⎪
⎪
⎪
⎪
⎪
⎩ u x, 0 = u0 in ,
1
= θnT ,
where f i, θ , k = θ f i (tk ) + (1 − θ )f i (tk − 1 ), Bi = Ai + μI and ui,hθ ,k = θ uih (tk ) + (1 − θ )uih (tk−1 ) such that Ai defined on (2), μ = θ t
respectively, with θ [0, 1].
On the other hand, throughout Evans and coworkers (cf. [10–14]) derived the HJB equation of the noncoercive case can be
approximated by a weakly coupled system coercive elliptic quasi-variational inequalities. Moreover, from (cf. [4]) we proved
that the theorem of the geometrical convergence result and the existence and uniqueness of the discrete solution using a new
algorithm based in Bensoussan’s algorithm remained true for the coercive case. Also, in (cf. [4]) the parabolic quasi variational
inequalities can be transformed into a system of strongly coercive quasi variational inequalities defined by the following full
, u2,k
, . . ., uhM,k ) ∈ Vhi,k solution of
discrete system: find (u1,k
h
h
⎧ i,θ ,k
i,θ ,k
i
i
⎪
≥ f i,θ ,k + μuhi,θ ,k−1 , vih − uhi,θ ,k ,
⎨ b uh , vh − uh
⎪
⎩
with
vih ∈ Vhi,k
(8)
uhi,θ ,k ≤ rh Muhi,θ ,k−1 ,
i = 1, 2...., M,
⎧ i,θ ,k
⎪
bi uh , vih − uhi,θ ,k = μ ui,hθ ,k , vih − uhi,θ ,k
⎪
⎪
⎪
⎪
⎪
⎪
⎨
+ai uhi,θ ,k , vih − uhi,θ ,k , vih , ui,hθ ,k ∈ Vhi ,
.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩ μ = 1 = n , k = 1, . . ., n, i = 1, . . ., M,
θ t θ T
(9)
where M is an operator defined by
Mui = L + ψ(ui ),
(10)
44
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
with L > 0 and ψ (ui ) is a continuous operator from L () into itself satisfying the following assumptions:
1. ψ ui ≤ ψ ũi whenever ui ≤ ũ,i a : e : in , i = 1, . . . , M.
2. ψ ui + γ ≤ ui + γ i , γ ≥ 0.
The class of the system of QVIs with coercive bilinear form includes at least two well-known important problems: the system of
variational inequality of feedback obstacle (VIs) (when ψ (ui ) = ui + 1 and u1 = uM + 1 , (cf. [2])), and the system of quasi-variational
inequality related to management of energy production (when ψ (ui ) is identically equal to inf uμ , μ > 0, (cf. [1])).
i
=μ
Knowing that the discrete spaces Vhi of finite element given by
Vhi
⎫
⎧
M
⎨ ui,θ ,k ∈ L2 0, T, H1 () ∩ C 0, T, H1 (
¯)
, such that ⎬
0
0
h
=
,
⎭
⎩ i,θ ,k
uh |Ki ∈ P1 , Ki ∈ τhi , and ui (., 0) = ui0 in , ui = 0 on ∂ (11)
where rh is the usual interpolation operator defined by
() v Ki ϕi (x)
m h
rh v =
(12)
i=1
and τ h denote the set of all those elements h > 0 is the mesh size and it is regular and quasi-uniform. Moreover, the usual basis
of affine functions ϕ l , l = {1, . . . , m(h)} defined by ϕ l (Ks ) = δ ls where Ks is a vertex of the considered triangulation. With the
satisfied discrete maximum principle assumption which it said the matrices Bi are M-matrices (see [9]), where Bi denote the
finite element matrices defined by
(Bi )ls = bi (ϕl , ϕs ) 1 ≤ i ≤ M, 1 ≤ l, s ≤ m(h).
(13)
The system (8) plays a fundamental role in solving evolutionary Hamilton–Jacobi–Bellman equations of stochastic control
theory problems (cf. e.g., [2]) arise in diverse fields [11] such as flow-through porous media, obstacle type problems, stochastic
impulse control problems, sandpile growth, phase-field model with temperature-dependent constraint, and critical-state model
in superconductivity where the critical current density depends on the magnetic field.
The structure of this paper is as follows. In Sections 2 and 3 we lay down some definitions and standard propositions needed
throughout the paper and we associate with the discrete a system of QVIs problem a fixed point mapping and we use that on
defining the discrete algorithm based on theta time scheme and we introduce a monotone iterative scheme based in Bensoussan’s
algorithm , and prove its some properties. We will see that such properties together with the subsolutions concepts will play
a crucial role in proving the existence and uniqueness of the problem which has been introduced in this paper, knowing that
the proof is based on the L -stability of the solution with respect to the right-hand side and its characterization as the least
upper bound of the subsolutions set (see also [5,6]). It is worth mentioning that this approach is entirely different from the one
developed for the evolutionary problem. Also, it is used for the first time for a system of QVIs. In Section 4, we have introduced
the main result deals with a new proof for the existence and uniqueness of the discrete evolutionary HJB equations which will
be carried out in four steps.. Finally, we have written the perspective that will be studied for completing this work.
2. Approximation of the evolutionary HJB equation by a system discrete coercive QVIs
2.1. The discrete coercive system of QVIs
2.1.1. Stability analysis for the discrete QVIs.
In (cf. [4]), it is possible to analyze stability taking advantage of the structure of eigenvalues of the bilinear form a(., .) defined
in (5) and we obtained, if θ ≥ 12 the θ -scheme is stable unconditionally. However, if 0 ≤ θ < 12 the θ -scheme is stable unless
2
.
1 − 2θ ρ (A)
t < Moreover, we (cf. [4]) deduced
p
p
2
2 i,p i,θ ,k
i,θ ,k
i
≤ C (p) ui0h +
t f
uh + t a uh , uh
2
k=1
2
i,θ ,k
+ εϕ
i,θ ,k 2
2
.
k=1
We shall first recall some results related to coercive quasi variational inequalities that are necessarily in proving some useful
qualitative properties.
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
45
Definition 1. ζhθ ,k = (ζh1,θ ,k , . . . , ζhM,θ ,k ) is said to be a subsolution for the system of QVIs (8) if
⎧ i,θ ,k
i
i,θ ,k
⎪
+ μζhi,θ ,k−1 , ϕs ,
⎨ b ζh , ϕs ≤ f
⎪
⎩
∀ϕs , s = 1, . . ., m h
ζhi,θ ,k ≤ rh Mζhi,θ ,k−1 .
Theorem 1. cf. 3] Under the discrete maximum principle, there exists a constant α > 0 such that:
bi ui,hθ , ui,hθ = a ui,hθ , ui,hθ + μ ui,hθ , ui,hθ ≥ α ui,hθ ( )
H1 where
⎛ 2
⎞
i
b
j ∞
γ i ⎟
+ a0 ⎠ ,
μ=⎜
+
⎝
∞
2γ
2
α=
γ
2
,
(14)
.
Remark 1. In this situation, the existence of a unique continuous solution to the stationary system can be handled in the spirit
of [15] or by adapting the algorithmic approach developed for the coercive and noncoercive problems using the Bensoussan’s
algorithm, cf. [5] just a brief description of it and skip over the proofs.
Notation 1. Let Xh be the set of discrete subsolutions. Then, we have the following theorem.
Theorem 2. Under the discrete maximum principle, the solution of the system of QVI (8) is the maximum element of Xh .
Proof. We denote by ϕ + = max (ϕ , 0), ϕ − = max ( − ϕ , 0).
Let wi,hθ ∈ Vhi be a solution of the following of the full discrete system of parabolic quai variational inequalities using the theta
time scheme combined with a finite element spatial approximation (cf. [3,4])
⎧ i,θ ,k
i,θ
i,θ ,k−1
i
i,θ
⎪
, ṽh − wi,hθ ,k ,
⎨ b wh , v̆h − wh ≥ f + λwh
⎪
⎩
∀ṽh ∈ Vhi
(15)
wih ≤ rh Mwhi,θ ,k−1 ,
where v̆h =
ṽ ≤ rh Mwhi,θ ,k−1 ,
m
(h)
s=1
ṽs ϕs .
Since ṽ is a trial function, we choose ṽh = wi,hθ ,k − vh and vh > 0. Thus
bi whi,θ ,k , ϕs ≤ f i,θ ,k + μwhi,θ ,k , ϕs ,
(16)
that is to say whi,θ ,k ∈ Xh .
On the other hand; let zhi,θ ,k be a subsolution, such that
whi,θ ,k ≤ zhi,θ ,k .
Then we have
⎧ i,θ ,k
i
i,θ ,k
⎪
+ λzhi,θ ,k−1 , ϕs
⎨ b zh , ϕs ≤ f
⎪
⎩
zhi,θ ,k ≤ rh Mwhi,θ ,k−1 .
+
Setting vh = zhi,θ ,k − whi,θ ,k
≥ 0 as a trial function. Yields
⎧ + + i,θ ,k
i,θ ,k
i,θ ,k
⎪
i
⎪
≤ f i,θ ,k + λzhi,θ ,k−1 , zhi,θ ,k − whi,θ ,k
⎨ b z h , z h − wh
⎪
⎪
⎩
zhi,θ ,k ≤ rh Mwhi,θ ,k−1
and since wi,hθ ,k is a subsolution too, we have
⎧ + + i,θ ,k
i,θ ,k
i,θ ,k
⎪
i
⎪
≤ f i,θ ,k + λzhi,θ ,k−1 , zhi,θ ,k − wi,hθ ,k
⎨ b wh , z h − wh
⎪
⎪
⎩
zhi,θ ,k ≤ rh Mwhi,θ ,k−1 .
(17)
46
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
Thus, we deduce
−bi
zhi,θ ,k − whi,θ ,k
+ + , zhi,θ ,k − wi,hθ ,k
≥ 0.
Under the coerciveness of the bilinear form by using Theorem 1, we can get
+
zhi,θ ,k − whi,θ ,k = 0,
therefore
zhi,θ ,k ≤ whi,θ ,k .
(18)
Thus, from (17) and (18) we obtain
zhi,θ ,k = whi,θ ,k .
3. A fixed-point mapping associated with the system of QVIs
Now, we shall give a proofs for proving the existence and uniqueness for the discrete QVIs (8) using the algorithm based on
theta time scheme combined with a finite element approximation which has already used in the previous researches regarding
the evolutionary free boundary problems (see [4]).
, . . ., ūhM,0 ), ūi,0
is solution of
For that, let us first introduce the initial vector Ūh0 = (ū1,0
h
h
bi ūi,0
, vh = f i , vh ,
h
where
∀vh ∈ Vhi ,
(19)
bi ūi,0
, vh = ai ūi,0
, vh + μ ūi,0
, vh .
h
h
h
Let H+ =
M
i=1
∞
L∞
+ , where L+ denotes the positive cone of L ().
Now, we consider the mapping
M
Th : H+ −→ Vh
W −→ TW = ξhi,θ ,k =
ξh1,θ ,k , . . . , ξhM,θ ,k
(20)
= ∂h (f i,θ ,k + μwi,θ ,k−1 , rh Mwi+1,θ ,k−1 ),
where ξhi,θ ,k is solution of the following problem
⎧ i,θ ,k
i,θ ,k
i
⎪
≥ f i,θ ,k + μwi,θ ,k−1 , vh − ξhi,θ ,k
⎨ b ξh , vh − ξh
⎪
⎩
(21)
ξhi,θ ,k ≤ rh Mwi+1,θ ,k−1 , i = 1, . . . , M, k = 1, . . . , n.
Remark 2. In the feedback obstacle case (cf. [5]), the obstacle become
ξhM,θ ,k = ∂h ( f M,θ ,k + μwM,θ ,k−1 , rh (k + w1,θ ,k−1 )).
Proposition 1. Under the previous notations and assumptions, the solution h (., .) of (21) is increasing according the obstacle
rh Mwi, θ , k − 1 and the right hand side FM, θ , k = f i, θ , k + μwi, θ , k − 1 , i.e., If we have
F i,θ ,k ≤ Gi,θ ,k and Mui,θ ,k−1 ≤ Mwi,θ ,k−1 ,
then
∂h (F i,θ ,k , rh MuM,θ ,k−1 ) ≤ ∂h (Gi,θ ,k , rh MwM,θ ,k−1 ).
Proof. We assume Fi, θ , k Gi, θ , k and Mvi, θ , k − 1 Mwi, θ , k − 1 .
Setting U1 = h (Fi, θ , k , rh MvM, θ , k − 1 ) and U1 = h (Gi, θ , k , rh MwM, θ , k − 1 )
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
47
and we have from the proof of Theorem 2
⎧ i,θ ,k
i
i,θ ,k
⎪
, ϕs ,
⎨ b uh , ϕs ≤ F
⎪
⎩
hence
uhi,θ ,k ≤ rh Muhi,θ ,k−1 ,
⎧ i,θ ,k
i
i,θ ,k
⎪
, ϕs ≤ Gi,θ ,k , ϕs ,
⎨ b uh , ϕs ≤ F
⎪
⎩
uhi,θ ,k ≤ rh Muhi,θ ,k−1 ≤ rh Mwhi,θ ,k−1 ,
then, we deduce
⎧ i,θ ,k
i
i,θ ,k
⎪
, ϕs ,
⎨ b uh , ϕs ≤ G
⎪
⎩
uhi,θ ,k ≤ rh Mwhi,θ ,k−1 .
Thus, uhi,θ ,k is a subsolution for the solution wi,hθ ,k , that is to say ui,hθ ,k ≤ whi,θ ,k . Therefore,
∂h (F i,θ ,k , rh MuM,θ ,k−1 ) ≤ ∂h (Gi,θ ,k , rh MwM,θ ,k−1 )
Lemma 1. (cf. [5]). Let δ be a positive constant. Then we have
∂h (F i,θ ,k , rh Mui+1,θ ,k−1 + δ) = .∂h (F i,θ ,k , rh Mui+1,θ ,k−1 ) + δ .
Proof. The proof is similar to that in (cf. [5]) for a noncoercive case with a simple obstacle. Proposition 2. Under the previous notations and assumptions
∂h (F i,θ ,k + Gi,θ ,k , rh Mui,θ ,k−1 + rh Mwi,θ ,k−1 ) ≥ .∂h (F i,θ ,k , rh Mui,θ ,k−1 ) + ∂h (Gi,θ ,k , rh Mwi,θ ,k−1 ),
where h (Fi, θ , k , rh Mui, θ , k − 1 ) is a solution of the problem (21) with the obstacle MuM, θ , k − 1 and the right hand side Fi, θ , k and
h (Gi, θ , k , rh Mwi, θ , k − 1 ) is a solution of the problem (21) with the obstacle MwM, θ , k − 1 and the right hand side Gi, θ , k .
Proof. Setting
uhi,θ ,k = ∂h (F i,θ ,k , rh Mui,θ ,k−1 )
(22)
and
whi,θ ,k = ∂h (Gi,θ ,k , rh Mwi,θ ,k−1 ).
It is clear that (22) verify the following system of (QVIs)
⎧ i,θ ,k
i,θ ,k
i
⎪
≥ F i,θ ,k , vh − ui,hθ ,k , vh ∈ V h
⎨ b uh , vh − uh
⎪
⎩
(23)
uhi,θ ,k
≤
rh Muhi,θ ,k−1 .
By addition, It can be written
⎧ i,θ ,k
bi uh + whi,θ ,k , vh + wi,hθ ,k − ui,hθ ,k + whi,θ ,k
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪
⎨ ≥ F i,θ ,k + Gi,θ ,k , vh + wi,hθ ,k − uhi,θ ,k + whi,θ ,k ,
⎪
⎪
⎪
i,θ ,k
i,θ ,k−1
⎪
+ rh Mwhi,θ ,k−1 ,
⎪
⎪ vh + wh ≤ rh Muh
⎪
⎪
⎪
⎪
⎩ i,θ ,k
uh + whi,θ ,k ≤ rh Muhi,θ ,k−1 + rh Mwhi,θ ,k−1 .
It can be taken the trial function vh = ui,hθ ,k − ζh with ζ h 0, we find
⎧ i,θ ,k
i,θ ,k
i
i,θ ,k
⎪
+ Gi,θ ,k , ζh , ζh ≥ 0
⎨ b uh + wh , ζh ≤ F
⎪
⎩
uhi,θ ,k + whi,θ ,k ≤ rh Muhi,θ ,k−1 + rh Mwhi,θ ,k−1 .
Therefore,
uhi,θ ,k + whi,θ ,k = ∂h (F i,θ ,k , rh Mui,θ ,k−1 ) + ∂h (Gi,θ ,k , rh Mwi,θ ,k−1 )
48
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
is a subsolution for the obstacle
rh Muhi,θ ,k−1 + rh Mwhi,θ ,k−1
and the right hand side
F i,θ ,k + Gi,θ ,k .
But, we know by Theorem 2 the solution
∂h F i,θ ,k + Gi,θ ,k , rh Mui,θ ,k−1 + rh Mwhi,θ ,k−1
is the greatest element in the subsolutions set. Then
∂h F i,θ ,k + Gi,θ ,k , rh Mui,θ ,k−1 + rh Mwhi,θ ,k−1
≥ ∂h (F i,θ ,k , rh Mui,θ ,k−1 ) + ∂h (Gi,θ ,k , rh Mwi,θ ,k−1 ).
(24)
Proposition 3. Under the previous notations and assumptions, it can be extended the result of Lemma 1 to the following equation
∂h (F i,θ ,k + δ a0 + μ, rh Mui,θ ,k−1 + δ) = .∂h (F i,θ ,k , rh Mui,θ ,k−1 ) + δ ,
where δ be a positive constant and μ defined in (9).
Proof. We can deduce the first inequality
∂h (F i,θ ,k + δ a0 + μ, rh Mui,θ ,k−1 + δ) ≥ ∂h (F i,θ ,k , rh Mui,θ ,k−1 ) + δ
from Proposition 2.
It remains to prove only the following inequality
∂h (F i,θ ,k + δ a0 + μ, rh Mui,θ ,k−1 + δ) ≤ ∂h (F i,θ ,k , rh Mui,θ ,k−1 ) + δ .
We consider the following inequalities system
⎧ i,θ ,k
i,θ ,k
i
⎪
≥ F i,θ ,k + Gi,θ ,k , vh − ρhi,θ ,k
⎨ b ρh , vh − ρh
⎪
⎩
(25)
vh , ρhi,θ ,k ≤ rh Muhi,θ ,k−1 + δ .
It can be verified that
F i,θ ,k + δ a0 + μ, vh − ρhi,θ ,k = F i,θ ,k , vh − ρhi,θ ,k + (δ a0 + μ, vh − ρ i,θ ,k )
= (F i,θ ,k , vh − ρ i,θ ,k ) +
(δ a0 + μ)(vh − ρ i,θ ,k )dx.
It can be easily shown from the elliptic operator (5) that
ai
δ , vh − ρhi,θ ,k =
δ a0 vh − ρhi,θ ,k dx, δ ≥ 0,
thus
bi
δ , vh − ρhi,θ ,k =
δ a0 + μ vh − ρhi,θ ,k dx.
Hence
F i,θ ,k + δ a0 + μ, vh − ρhi,θ ,k = F i,θ ,k , vh − ρhi,θ ,k + bi δ , vh − ρhi,θ ,k .
From (25) and (26), we have
⎧ i,θ ,k
i,θ ,k
i
⎪
≥ F i,θ ,k , vh − ρhi,θ ,k + bi δ , vh − ρhi,θ ,k ,
⎨ b ρh , vh − ρh
⎪
⎩
vh , ρhi,θ ,k ≤ rh Muhi,θ ,k−1 + δ , vh ∈ V h .
(26)
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
Then
49
⎧ i,θ ,k
i,θ ,k
i
⎪
≥ F i,θ ,k , vh − ρhi,θ ,k , vh ∈ V h
⎨ b ρh − δ , vh − ρh
(27)
⎪
⎩
vh , ρhi,θ ,k − δ ≤ rh Muhi,θ ,k−1 .
It can be taken vh = ρhi,θ ,k − ṽ h with ṽh ≥ 0 in (27). This gives
⎧ i,θ ,k
i
i,θ ,k
⎪
, ϕl ,
⎨ b ρh − δ , ϕl ≤ F
⎪
⎩
ϕl = 1, . . . , m h
ρhi,θ ,k − δ ≤ rh Muhi,θ ,k−1 .
Therefore, ρhi,θ ,k − δ is the subsolution for the obstacle rh Muhi,θ ,k−1 and the right hand side Fi, θ , k .
As we know that
∂h F i,θ ,k , rh Muhi+1,θ ,k−1
is the greatest element in the subsoltions set, then we have
ρhi,θ ,k − δ ≤ ∂h F i,θ ,k , rh Muhi,θ ,k−1 ,
ie.,
ρhi,θ ,k ≤ ∂h F i,θ ,k , rh Muhi,θ ,k−1 + δ .
Thus,
∂ F i,θ ,k , +a0 δ + μ, rh Muhi,θ ,k−1 + δ ≤ ∂h F i,θ ,k , rh Muhi,θ ,k−1 + δ .
(28)
From the first inequality which has been deduced by Proposition 2 and (28), we can say
∂ F i,θ ,k , +a0 δ + μ, rh Muhi,θ ,k−1 + δ = ∂h F i,θ ,k , rh Muhi,θ ,k−1 + δ .
(29)
3.1. Some properties of the mapping Th
Let Ūh0 = (ūi,0
, . . . , ūhM,0 )t be the finite element approximation of the discrete equation (19).
h
Proposition 4. Under the preceding notations and assumptions, the mapping Th satisfies: for V, W H+
1.Th V ≤ Th W whenever V ≤ W,
2.Th W ≥ 0,
(30)
3.Th W ≤ Ūh0 .
Proof. 1. Let V = (v1 , . . . , vM ), W = (w1 , . . . , wM ) in H+ such that vi wi , i = 1, . . . , M
Then since h is increasing in two cases (the solution related to simple obstacle ψ for noncoercive and coercive case of the
solution related to right hand side for coercive case, cf [5,6] , it follows that
∂h ( f i,θ ,k + μvi,θ ,k−1 , Mvi,θ ,k−1 ) ≤ ∂h ( f i,θ ,k + μwi,θ ,k , rh Mwi,θ ,k−1 ),
that is to say
Th V ≤ Th W.
2. This follows directly from the fact that f i, θ , k 0 and Mwi, θ , k − 1 0. Thus, we have Th W 0.
3. The fact that both of the solutions ξhi,θ ,k of (21) and ūi,0
of (19) belong to (Vh )M , we readily have
h
ξhi,θ ,k − ξhi,θ ,k + ūi,0
+
∈ (L∞ ())M .
+
Moreover, as ξ i + ūi,0
≥ 0, it follows that
ξhi,θ ,k − ξhi,θ ,k + ūi,0
+
≤ ξhi,θ ,k ≤ Mwi+1,θ ,k−1 .
+
Therefore, we can take vh = ξhi,θ ,k − ξhi,θ ,k + ūi,0
as a trial function in (21). This gives
bi
ξhi,θ ,k , − ξhi,θ ,k + ūi,0
+ + ≥ f i,θ ,k + μwi,θ ,k−1 , − ξhi,θ ,k + ūi,0
.
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S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
+
Also, for vh = ξhi,θ ,k + ūi,0
as trial function in (19), then it becomes
+ + bi ui,0 , ξhi,θ ,k + ūi,0
= f i , ξhi,θ ,k + ūi,0
,
∀vh ∈ V h ,
(31)
so, by addition, we obtain
−bi
ξhi,θ ,k + ūi,0
+ + , ξhi,θ ,k + ūi,0
≥ 0,
by Theorem 1, yields
ξhi,θ ,k + ūi,0
+
= 0.
Thus
ξhi,θ ,k ≤ ūi,0 , ∀i = 1, . . . , M.
Proposition 5. The mapping Th is concave on H+ , ie.,
Th
ηV + (1 − η)W ≥ ηTh V + (1 − η)Th W, ∀V, W ∈ H+ .
Proof. Let V, W H+ and Fi, θ , k = f i, θ , k − 1 + μvi, θ , k − 1 , Gi, θ , k − 1 = f i, θ , k + μwi, θ , k − 1 are the right hands sides of the inequalities
systems (21). It can be seen
Th V =
and
Th W =
∂h G1,θ ,k , rh Mwhi1,θ ,k−1 , . . . , ∂h Gi,θ ,k , rh Mwi,hθ ,k−1 , . . . , ∂h GM,θ ,k , rh MwhM,θ ,k−1
then we have
Th
∂h F 1,θ ,k , rh Mvhi1,θ ,k−1 , . . . , ∂h F i,θ ,k , rh Mvhi,θ ,k−1 , . . . , ∂h F M,θ ,k , rh MvhM,θ ,k−1
,
ηV + (1 − η)W = ∂h ηF 1,θ ,k + (1 − η)G1,θ ,k , rh ηMvh1,θ ,k−1 + rh (1 − η)Mwh1,θ ,k−1 , . . .,
∂h ηF i,θ ,k + (1 − η)Gi,θ ,k , rh ηMvhi,θ ,k−1 + rh (1 − η)Mwhi,θ ,k−1 , . . .,
θ ,k−1
∂h ηF M,θ ,k + (1 − η)GM,θ ,k , rh ηMvhM,θ ,k−1 + rh (1 − η)MwM,
h
Then, by using Proposition 2, we have
Th
ηV + (1 − η)W ≥
.
η.∂h F 1,θ ,k , rh Mvh1,θ ,k−1 + (1 − η).∂h G1,θ ,k , rh Mvh1,θ ,k−1
η.∂h F i,θ ,k , rh Mvhi,θ ,k−1 + (1 − η).∂h Gi,θ ,k , rh Mvhi,θ ,k−1
, . . .,
, . . .,
η∂h F M,θ ,k , rh MvhM,θ ,k−1 + (1 − η).∂h GM,θ ,k , rh MvhM,θ ,k−1
.
Then
Th (ηV + (1 − η)W ) ≥ ηTh (V ) + (1 − η)Th W,
which proves the concaveness of Th .
Proposition 6. Under the results of Propositions 2, 3 and using the properties of the operator Mu (cf. [16]) the mapping Th is Lipchitz
on H+ i.e.,
Th V − Th W ∞ ≤ V − W ∞ , V, W ∈ H+ .
Proof. We clearly have
i
i
Th V − Th W L∞ () = max Th V − Th W ∞
1≤i≤M
= max ∂h F i,θ ,k , rh Mvhi,θ ,k−1 − ∂h Gi,θ ,k , rh Mwi,hθ ,k−1 ,
∞
1≤i≤M
where (Th W)i and (Th V)i denote the ith components of the vectors W and V, respectively.
Setting
φ i,θ ,k = max rh Mvi,hθ ,k − rh Mwhi,θ ,k ,
∞
β
1 i,θ ,k
− Gi,θ ,k .
F
∞
+μ
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
51
We have
rh Mvhi,θ ,k ≤ rh Mwhi,θ ,k + rh Mvi,hθ ,k − rh Mwhi,θ ,k ∞
≤ rh Mwhi,θ ,k + φ i,θ ,k .
Moreover, we have
F i,θ ,k ≤ Gi,θ ,k + F i,θ ,k − Gi,θ ,k ∞
≤ Gi,θ ,k +
a0
β +μ
F i,θ ,k − Gi,k ∞
≤ Gi,θ ,k + a0 φ i,θ ,k
≤ Gi,θ ,k + a0 φ i,θ ,k + μ,
so, due to Proposition 3, it follows that
∂h F i,k , rh Mvhi,θ ,k ≤ ∂h Gi,θ ,k + a0 φ i,θ ,k + μ, rh Mwi,hθ ,k + φ i,k
≤
∂h Gi,k , rh Mwi,hθ ,k + φ i,k ,
hence
Th V ≤ Th W + φ i,k .
Similarly, interchanging the roles of w and w̃ we also get
Th W ≤ Th V + φ i,k ,
knowing that by (cf. [16]) M is Lipchiz, then we can easily deduce
Th V − Th W ∞ ≤ max rh Mvi,hθ ,k − rh Mwi,hθ ,k ,
∞
≤ max 1,
μ
μ+β
≤ V − W ∞ .
i,θ ,k
i,θ ,k vh − wh β
1 i,θ ,k
− Gi,θ ,k F
∞
+μ
∞
4. The main result
4.1. A discrete iterative scheme
Starting from Û 0 solution of (19) (respectively, Ǔ 0 = 0, . . . , 0 , we define
Û θ ,1 = T Û 0, ,
Û θ ,k = T Û θ ,k−1, , k = 2, 3, . . .,
(32)
respectively,
Ǔ θ ,1 = T Ǔ 0 ,
Ǔ θ ,k = T Ǔ θ ,k−1 ,
θ ∈ [0, 1], k = 2, 3, . . .,
(33)
Let
kh = {W ∈ H+ such that 0 ≤ W ≤ Û 0 },
where Ûhθ ,k = (û1,k
, . . . , ûhM,k ), the solution of the problem (23).
h
For k = 1, It can be calculated Û θ ,1 by Euler time scheme combined with a finite element approximation method.
L
Lemma 2. For 0 ≤ λ ≤ inf
, 1 , where L is defined in (10) , then, we have
Û0 ∞
Th (0) ≥ λÛ 0 ∞ .
(34)
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S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
Proof. From (33), Th (0) = Ǔ θ ,1 = (ǔ1,1 , . . . , ǔ1,M ), where ǔ1,1 is a solution of the following system of quasi variational inequalities
⎧ i,θ ,1
i,1
i
i,θ ,1
⎪
+ μǔi,0
, vh − ǔhi,,θ ,1 , vh ∈ V h
⎨ b ǔh , vh − ǔh ≥ f
h
⎪
⎩
(35)
≤ rh M ûi,0
.
ǔi,1
h
h
It can be taken the following trials functions
−
vh = ǔhi,θ ,1 − λûi,0
+ ǔi,hθ ,1
h
in the system of QVIs (34), and
−
− ǔhi,θ ,1 − λûi,0
h
in the problem (19).
Using the fact Fi, 1 0, we get by addition (19) with (34)
− − i,θ ,1
i,θ ,0
i,θ ,1
i,θ ,0
i,θ ,1
i,θ ,1
bi ǔhi,θ ,1 − λûi,0
,
−
λ
−
λ
F
,
−
λ
û
û
≥
F
ǔ
ǔ
h
h
h
h
h
− ≥ 1 − λ F i,θ ,1 , ǔhi,θ ,1 − λûi,0
≥ 0,
h
where F i,θ ,1 = f i,θ ,1 + μǔi,0
.
h
Thus, by using Theorem 1
−
ǔhi,θ ,1 − λûi,0
= 0,
h
i.e.,
ǔhi,θ ,1 ≥ λûi,0
, i = 1, . . . , M.
h
Then
Th 0 ≥ λ Û 0 ,
∞
which completes the proof.
Proposition 7. Let ω [0, 1] such that
W − V ≤ ωW,
∀W, V ∈ k,
(36)
then, under Propositions 4 and 5 , the following holds
Th V − Th W ≤ ω 1 − λ Th V.
(37)
Proof. By (36), we have
(1 − ω)W ≤ V,
thus, using the fact that Th is increasing and concave, it follows that
(1 − ω)Th V + γ Th (0) ≤ Th ((1 − ω)V + ω.0)
≤ Th W.
Finally, using Lemma 2 we get (37). Making use of properties of Proposition 4 and 7, we have the following main result.
Theorem 3. The sequences (Û θ ,k,n+1 ) and (Ǔ θ ,k,n+1 ) well defined in k and converge, respectively, from above and below, to the unique
solution of system of inequalities (23).
Proof. The proof will be carried out in four steps.
Step 1. The sequence (Û θ ,k,n+1 ) stays in k and is monotone decreasing.
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
53
From (32), (23) it is easy to see that ∀k ≥ 1, Û θ ,k = (ûθ ,1 , . . . , ûθ ,M )t is solution to
⎧ i,θ ,k
i,θ ,k
i
⎪
≥ F i,θ ,k−1 , vh − ûi,hθ ,k , vh ∈ V h
⎨ b ûh , vh − ûh
⎪
⎩
Since
(38)
ûhi,θ ,k
Fi, 0
rh Mûhi,θ ,k−1 .
≤
and û0 , combining comparison results in variational inequalities with a simple induction, it follows that
Û θ ,k ≥ 0.
(39)
Furthermore, by Proposition 4
0 ≤ Û 1 = Th (Û 0 ) ≤ Û 0 ,
thus we can deduce
Û θ ,1 = θ Û 1 + (1 − θ )Û 0 ≥ 0.
(40)
For k 2, we know by Proposition 4 that Th increasing. Thus, inductively
0 ≤ Û θ ,k+1 = Th (Û θ ,k ) ≤ Û θ ,k ≤, · · ·, ≤ Û θ ,1 ≤ Û 0 .
(41)
Step 2. Û θ ,k converges to the solution of the system (23).
From (39), (41) it is clear that
lim ûi,θ ,k = ūi,θ , x ∈ , ūi,θ ∈ k.
(42)
k−→∞
Moreover, from (39) we have
F i,θ ,k−1 ≥ 0 and rh Mûi,θ ,k−1 ≥ 0.
Then, we can take vh = 0 as a trial function in (38), which yields
2
α ûhi,θ ,k Vh
≤ bi ûhi,θ ,k , ûi,hθ ,k ≤ F i,k−1 , ûi,hθ ,k
≤ F i,k−1 L2 () ûi,hθ ,k Vh
≤
Hence
f i,k−1 L2 () + μ ûhi,θ ,k 2
L
i,θ ,k ûh ()
Vh
.
α ûhi,θ ,k h ≤ Cf + μ ûi,hθ ,k h ,
V
V
or more simply
i,θ ,k ûh ≤ Cf,α ,μ ≤ C,
Vh
where C is a constant independent of k and we choose t such that
Hence,
ûi,hθ ,k
lim û
k−→∞
stays bounded in
i,θ ,k
Vh H1 ()
i,θ
= ū weakly in H
1
1
θ t
< α.
and consequently we can complete (42) by
.
(43)
Step 3. Ū θ = (ū1,θ , . . . , ūM,θ ) coincides with the solution of system (19). Indeed, since
ûhi,θ ,k ≤ rh Mûhi,θ ,k−1 ,
then from (43) implies
ūi,hθ ≤ rh Mūi,hθ .
Now, let vh ≤ rh Mūi,hθ , then vh ≤ rh Mûhi,θ ,k−1 , ∀k, We can, therefore, take vh as a trial function for the system (38). Consequently,
combining (42), (43) thus we have
lim bi ûhi,θ ,k , ûhi,θ ,k ≤ lim bi ûhi,θ ,k , vh − F i,k−1 , vh − ûi,hθ ,k , vh ∈ V h .
k−→∞
k−→∞
The continuous system of bi (vh , vh ) is a weak lower semi-continuity, then
lim bi ûhi,θ ,k , ûhi,θ ,k ≤ bi ūi,hθ , vh − F i,k−1 , vh − ūi,hθ , vh ∈ V h ,
k−→∞
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S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
but
0 ≤ bi ûhi,θ ,k − ūi,hθ , ûhi,θ ,k − ūi,hθ ≤ bi ûi,hθ ,k , ûi,hθ ,k − bi ûhi,θ ,k , ūi,hθ
−bi ūhi,θ ,k , ûhi,θ ,k + bi ūi,hθ , ūi,hθ .
(44)
Hence,
bi ûhi,θ ,k , ûhi,θ ,k ≥ bi ûi,hθ ,k , ūi,hθ + bi ūi,hθ , ûhi,θ ,k − bi ūi,hθ ,k , ūi,hθ
(45)
and passing to the limit in problem (45), we obtain
bi ūi,hθ , ūi,hθ ≤ lim bi ûi,hθ ,k , ûi,hθ ,k ≤ bi ūi,hθ , vh − F i,k−1 , vh − ūi,hθ .
k−→∞
Yields
⎧ i,θ
i,θ
i
i,k−1
⎪
, vh − ūi,hθ , vh ∈ V h
⎨ b ūh , vh − ūh ≥ F
⎪
⎩
ūi,hθ ≤ rh Mūi,hθ .
Thus Ūhθ is the solution of system (38)
Step 4. The monotone property of the sequence (Ǔ θ ,k ) can be shown similarly to that of sequence (Û θ ,k ).
Let us prove its convergence to the solution of system (38). Indeed, we use (32) and (33) with
V = Û 0 , Ṽ = Ǔ 0 ,
γ = 1,
therefore,
Th Û 0 − Th Ǔ 0 ≤ 1 − λ Th Û 0 ,
so
Û θ ,1 − Ǔ θ ,1 ≤ 1 − λ Û θ ,1
and applying (33) again, this yields
2
Û θ ,2 − Ǔ θ ,2 ≤ 1 − λ Û θ ,2
and quite generally
k
Û θ ,k − Ǔ θ ,k ≤ 1 − λ Û θ ,k ,
or
k
Û θ ,k − Ǔ θ ,k ≤ 1 − λ Û 0
≤ (1 − λ)k Û 0 ∞ .
We can prove that sequence Ǔ θ ,k −→ U θ similarly to that of sequence (Û θ ,k ) in Step 3.
k−→∞
Since (1 − λ)k −→ 0, and after passing to the limit, we get
Û θ ≤ Ǔ θ .
Similarly, interchanging the roles of Û θ ,k and Ǔ θ ,k we also get
Ǔ θ ≤ Û θ .
Finally, we deduce the solution of (38) is unique, i.e.,
Ǔ θ = Û θ = U
Remark 3. From the above proposition, one can observe that the solution of system (23) or (38) is a fixed point of Th , i.e.,
Th U = U.
Conclusion 1. In this paper, we have introduced a new proof for the existence and uniqueness of the discrete evolutionary HJB
equation which carried out in four steps. Moreover, it based on some properties of the discrete iterative algorithm using the
theta-scheme with respect to the t-variable combined with a finite element spatial approximation and which has been used for
proving the asymptotic behavior in uniform norm in the previous works (cf., e.g., [3,4]). A next manuscript, the convergence
discrete iterative schemes for the sequences (32) and (33) will be provided and we see that this result will play a major role in
the finite element error analysis section.
S. Boulaaras, M. Haiour / Applied Mathematics and Computation 262 (2015) 42–55
55
Acknowledgments
The author wish to thank deeply the anonymous referees for his/her useful remarks and his/her careful reading of the proofs
presented in this paper.
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