doi:10.1155/2011/385324
Research Article
Study of an Approximation Process of
Time Optimal Control for Fractional Evolution Systems in Banach Spaces
JinRong Wang
1and Yong Zhou
21Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
2Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China
Correspondence should be addressed to Yong Zhou,[email protected] Received 1 October 2010; Accepted 9 December 2010
Academic Editor: J. J. Trujillo
Copyrightq2011 J. Wang and Y. Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The paper is devoted to the study of an approximation process of time optimal control for fractional evolution systems in Banach spaces. We firstly convert time optimal control problem into Meyer problem. By virtue of the properties of the family of solution operators given by us, the existence of optimal controls for Meyer problem is proved. Secondly, we construct a sequence of Meyer problems to successive approximation of the original time optimal control problem. Finally, a new approximation process is established to find the solution of time optimal control problem. Our method is different from the standard method.
1. Introduction
It has been shown that the accurate modelling in dynamics of many engineering, physics, and economy systems can be obtained by using fractional differential equations. Numerous applications can be found in viscoelasticity, electrochemistry, control, porous media, electromagnetic, and so forth. There has been a great deal of interest in the solutions of fractional differential equations in analytical and numerical sense. One can see the monographs of Kilbas et al.1, Miller and Ross2, Podlubny3, and Lakshmikantham et al.
4. The fractional evolution equations in infinite dimensional spaces attract many authors including ussee, for instance,5–21and the references therein.
When the fractional differential equations describe the performance index and system dynamics, a classical optimal control problem reduces to a fractional optimal control problem. The optimal control of a fractional dynamics system is a fractional optimal control with system dynamics defined with partial fractional differential equations.
There has been very little work in the area of fractional optimal control problems18,22, especially the time optimal control for fractional evolution equations19. Recalling that the research on time optimal control problems dates back to the 1960s, many problems such as existence and necessary conditions for optimality and controllability have been discussed, for example, see23for the finite dimensional case and7,24–37for the infinite dimensional case. Since the cost functional for a time optimal control problem is the infimum of a number set, it is different with the Lagrange problem, the Bolza problem and the Meyer problem, which arise some new difficulties. As a result, we regard the time optimal control as another problem which is not the same as the above three problems.
Motivated by our previous work in18–21,38, we consider the time optimal control problemPof a fractional evolution system governed by
CDqtzt Azt ft, zt, Btvt, t∈0, τ, q∈0,1,
z0 z0∈X, v∈Vad, 1.1
where CDtqis the Caputo fractional derivative of orderq,A:DA → Xis the infinitesimal generator of a strongly continuous semigroup{Tt, t ≥ 0},Vadis the admissible control set andf:Iτ : 0, τ×X×X → Xwill be specified latter.
Let us mention, we do not study the time optimal control problemPof the above system by standard method used in our earlier work 19. In the present paper, we will construct a sequences of Meyer problemsPεn to successive approximation time optimal control problemP. Therefore, we need introduce the following new fractional evolution system
CDsqxs kqAxs kqfks, xs, Bksus, s∈0,1,
x0 z0 ∈X, w u, k∈W, 1.2
whose controls are taken from a product spaceWwill be specified latter.
By applying the family of solution operatorsTk andSk seeLemma 3.7associated with the family ofC0-semigroups with parameters and some probability density functions, the existence of optimal controls for Meyer problems Pεis proved. Then, we show that there exists a subsequence of Meyer problemsPεnwhose corresponding sequence of optimal controls{wεn} ∈Wconverges to a time optimal control of problemPin some sense. In other words, in a limiting process, the sequence{wεn} ∈Wcan be used to find the solution of time optimal control problemP. The existence of time optimal controls for problemPis proved by this constructive approach which provides a new method to solve the time optimal control.
The rest of the paper is organized as follows. In Section 2, some notations and preparation results are given. InSection 3, we formulate the time optimal control problemP and Meyer problemPε. InSection 4, the existence of optimal controls for Meyer problems Pεis proved. Finally, we display the Meyer approximation process of time optimal control and derive the main result of this paper.
2. Preliminaries
Throughout this paper, we denote byX a Banach space with the norm · . For eachτ <
∞, letIτ ≡ 0, τand CIτ, Xbe the Banach space of continuous functions from Iτ to X with the usual supremum norm. Let A : DA → X be the infinitesimal generator of a strongly continuous semigroup{Tt, t ≥ 0}. This means that there existsM > 0 such that supt∈I
τTt ≤M. We will also usefLpIτ,Rto denote theLpIτ, R norm offwhenever f∈LpIτ, R for somepwith 1< p <∞.
Let us recall the following definitions in1.
Definition 2.1. The fractional integral of orderγ with the lower limit zero for a functionf is defined as
Iγft 1 Γ
γ t
0
fs
t−s1−γds, t >0, γ >0, 2.1
provided the right side is pointwise defined on0,∞, whereΓ·is the gamma function.
Definition 2.2. Riemann-Liouville derivative of orderγwith the lower limit zero for a function f:0,∞ → Rcan be written as
LDγft 1
Γ
n−γ dn dtn
t
0
fs
t−sγ 1−nds, t >0, n−1< γ < n. 2.2 Definition 2.3. The Caputo derivative of orderγfor a functionf :0,∞ → Rcan be written as
CDγft LDγ
ft−n−1
k0
tk k!fk0
, t >0, n−1< γ < n. 2.3
Remark 2.4. iIfft∈Cn0,∞, then
CDγft 1
Γ n−γ
t
0
fns
t−sγ 1−ndsIn−γfnt, t >0, n−1< γ < n. 2.4 iiThe Caputo derivative of a constant is equal to zero.
iii If f is an abstract function with values in X, then integrals which appear in Definitions2.1and2.2are taken in Bochner’s sense.
Lemma 2.5see38, Lemma 3.1. If the assumption [A] holds, then
1for givenk∈0,T,kAis the infinitesimal generator ofC0-semigroup{Tkt, t≥0}onX, 2there exist constantsC≥1 andω∈−∞, ∞such that
Tkt ≤Ceωkt, ∀t≥0, 2.5
3ifkn → kεin0,T asn → ∞, then for arbitraryx∈Xandt≥0,
Tknt−→s Tkεt, asn−→ ∞ 2.6
uniformly inton some closed interval of0,T in the strong operator topology sense.
3. System Description and Problem Formulation
Consider the following fractional nonlinear controlled system
CDqtzt Azt ft, zt, Btvt, t∈0, τ,
z0 z0∈X, v∈Vad. 3.1
We make the following assumptions.
A:Ais the infinitesimal generator of aC0-semigroup{Tt, t≥0}onXwith domain DA.
F:f : Iτ ×X ×X → X is measurable intonIτ and for eachρ > 0, there exists a constantLρ> 0 such that for almost allt∈Iτ and allz1, z2, y1, y2 ∈X, satisfying z1,z2,y1,y2 ≤ρ, we have
f t, z1, y1
−f
t, z2, y2 ≤L ρ
z1−z2 y1−y2 . 3.2 For arbitraryt, z, y∈Iτ×X×X, there exists a positive constantM >0 such that
f
t, z, y ≤M
1 z y . 3.3
B: Let Ebe a separable reflexive Banach space, B ∈ L∞Iτ, LE, X,B∞ stands for the norm of operator B on Banach space L∞Iτ, LE, X. B : LpIτ, E → LpIτ, X1< p < ∞is strongly continuous.
U: Multivalued mapsV·:Iτ → 2E\ {Ø}has closed, convex and bounded values.
V·is graph measurable andV·⊆ΩwhereΩis a bounded set ofE.
Set
Vad{v·|Iτ −→Emeasurable, vt∈ Vta.e.}. 3.4 Obviously,Vad/Ø see39, Theorem 2.1andVad ⊂ LpIτ, E 1 < p < ∞ is bounded, closed and convex.
Based on our previous work21, Lemma 3.1 and Definition 3.1, we use the following definition of mild solutions for our problem.
Definition 3.1. By the mild solution of system3.1, we mean that the functionx ∈ CIτ, X which satisfies
zt Ttz0
t
0
t−θq−1St−θfθ, zθ, Bθvθdθ, t∈Iτ, 3.5
where
Tt ∞
0
ξqθTtqθdθ, St q ∞
0
θξqθTtqθdθ, ξqθ 1
qθ−1−1/q q
θ−1/q
≥0,
qθ 1
π ∞ n1
−1n−1θ−qn−1Γ nq 1
n! sin nπq
, θ∈0,∞,
3.6
ξqis a probability density function defined on0,∞, that is
ξqθ≥0, θ∈0,∞, ∞
0
ξqθdθ1. 3.7
Remark 3.2. iIt is not difficult to verify that forv∈0,1 ∞
0
θvξqθdθ ∞
0
θ−qv qθdθ Γ1 v Γ
1 qv. 3.8
ii For another suitable definition of mild solutions for fractional differential equations, the reader can refer to13.
Lemma 3.3see21, Lemmas 3.2-3.3. The operatorsTandShave the following properties.
iFor any fixedt≥0,TtandStare linear and bounded operators; that is, for anyx∈X,
Ttx ≤Mx, Stx ≤ qM Γ
1 qx. 3.9
ii{Tt, t≥0}and{St, t≥0}are strongly continuous.
We present the following existence and uniqueness of mild solutions for system3.1.
Theorem 3.4. Under the assumptions [A], [B], [F] and [U], for everyv∈Vadandpq > 1, system 3.1has a unique mild solutionz∈CIτ, Xwhich satisfies the following integral equation
zt Ttz0
t
0
t−θq−1St−θfθ, zθ, Bθvθdθ. 3.10
Proof. Consider the ball given byB{x ∈C0, T1, X| xt−x0 ≤ 1,0≤ t≤T1}, where T1would be chosen, andxt ≤1 x0ρ, 0≤t≤T1,B ⊆C0, T1, Xis a closed convex set. Define a mapHonBgiven by
Hzt Ttz0
t
0
t−θq−1St−θfθ, zθ, Bθvθdθ. 3.11
Note that by the properties of T and S, assumptions A,F,B, and U, by standard processsee19, Theorem 3.2, one can verify thatHis a contraction map onBwithT1>0.
This means that system3.1has a unique mild solution on0, T1. Again, using the singular version Gronwall inequality, we can obtain the a prior estimate of the mild solutions of system 3.1and present the global existence of the mild solutions.
Definition 3.5admissible trajectory. Take two pointsz0,z1in the state spaceX. Letz0be the initial state and letz1be the desired terminal state withz0/z1, denotezv≡ {zt, v∈X | t≥0}be the state trajectory corresponding to the controlv∈Vad. A trajectoryzvis said to be admissible ifz0, v z0andzt, v z1for some finitet >0.
SetV0 {v∈Vad|zvis an admissible trajectory} ⊂ Vad. For givenz0, z1 ∈Xand z0/z1, ifV0/Øi.e., there exists at least one control from the admissible class that takes the system from the given initial statez0to the desired target statez1in the finite time., we say the system3.1can be controlled.
Letτv ≡ inf{t ≥ 0 | zt, v z1} denote the transition time corresponding to the controlv∈V0/Ø and defineτ∗inf{τv≥0|v∈V0}.
Then, the time optimal control problem can be stated as follows.
ProblemProblemP. Take two pointsz0,z1in the state spaceX. Letz0be the initial state and letz1 be the desired terminal state with z0/z1. Suppose that there exists at least one control from the admissible class that takes the system from the given initial statez0to the desired target statez1in the finite time. The time optimal control problem is to find a control v∗∈V0such that
τv∗ τ∗inf{τv≥0|v∈V0}. 3.12
For fixedv∈Vad,Tτv>0. Now, we introduce the following linear transformation
tks, 0≤s≤1, k∈ 0,T
. 3.13
Through this transformation, system3.1can be replaced by
CDsqxs kqAxs kqfks, xs, Bksus, s∈0,1,
x0 z0 z0∈X, w u, k∈W, 3.14
wherex· zk·,u· vk·, and define
W
u, k|us vks, 0≤s≤1, v∈Vad, k∈ 0,T
. 3.15
ByTheorem 3.4, one can obtain the following existence result.
Theorem 3.6. Under the assumptions ofTheorem 3.4, for everyw ∈Wandpq >1, system3.14 has a unique mild solutionx∈C0,1, Xwhich satisfies the following integral equation
xs Tksz0
s
0
s−θq−1Sks−θkfkθ, xθ, Bkθuθdθ, 3.16
where
Tks ∞
0
ξqθTkqsqθdθ, Sks q ∞
0
θξqθTkqsqθdθ, 3.17
and{Tkqt, t≥0}is aC0-semigroup generated by the infinitesimal generatorkqA. By Lemmas2.5and3.3, it is not difficult to verify the following result.
Lemma 3.7. The family of solution operatorsTkandSkgiven by3.17has the following properties.
iFor anyx∈X,t≥0, there exists a constantCkq >0 such that
Tktx ≤Ckqx, Sktx ≤ qCkq
Γ
1 qx. 3.18
ii{Tkt, t≥0}and{Skt, t≥0}are also strongly continuous.
iiiIfknq → kqεin0,Tasn → ∞, then for arbitraryx∈Xandt≥0
Tkqnt−→ Ts kqεt, asn−→ ∞,
Skqnt−→ Ss kεqt, as n−→ ∞ 3.19
uniformly inton some closed interval of0,T in the strong operator topology sense.
For system3.14, we turn to consider the following Meyer problem.
Meyer ProblemPε
Minimize the cost functional given by
Jεw 1
2εxw1−z12 k 3.20
overW, wherexwis the mild solution of3.14corresponding to controlw, that is, find a controlwε uε, kεsuch that the cost functionalJεwattains its minimum onWatwε.
4. Existence of Optimal Controls for Meyer Problem P
εIn this section, we discuss the existence of optimal controls for Meyer problemPε. We show that Meyer problemPεhas a solutionwε uε, kεfor fixedε >0.
Theorem 4.1. Under the assumptions ofTheorem 3.6. Meyer problemPεhas a solution.
Proof. Let ε > 0 be fixed. Since Jεw ≥ 0, there exists inf{Jεw, w ∈ W}. Denote mε ≡ inf{Jεw, w∈W}and choose{wn} ⊆Wsuch thatJεwn → mεwherewn un, kn∈W Vad×0,T. By assumptionU, there exists a subsequence{un} ⊆ Vadsuch thatun →w uε
in Vad asn → ∞, andVad is closed and convex, thanks to Mazur Lemma, uε ∈ Vad. By assumptionB, we have
Bun−→s Buε, inLp0,1, X, asn−→ ∞. 4.1
Sinceknknqis bounded andknkqn>0, there also exists a subsequence{kn}{kqn}denoted by{kn}{knq}⊆0,Tagain, such that
kn
kqn
−→kε
kqε
, in
0,T
, asn−→ ∞. 4.2
Letxnandxεbe the mild solutions of system3.14corresponding town un, kn∈ Wandwε uε, kε∈W, respectively. Then, we have
xns Tnsz0
s
0
s−θq−1Sns−θknqFnθdθ,
xεs Tεsz0
s
0
s−θq−1Sεs−θkεqFεθdθ,
4.3
where
Tn·≡ ∞
0
ξqθTkqn·qθdθ, Sn·≡q
∞
0
θξqθTkqn·qθdθ, Fn·≡fkn·, xn·, Bkn·un·,
Tε·≡ ∞
0
ξqθTkqε·qθdθ, Sε·≡q
∞
0
θξqθTkqε·qθdθ, Fε·≡fkε·, xε·, Bkε·uε·.
4.4
ByLemma 3.7, assumptionsF,B,U, and singular version Gronwall Lemma, it is easy to verify that there exists a constantρ >0 such that
xεC0,1,X≤ρ, xnC0,1,X≤ρ. 4.5
Further, there exists a constantMε>0 such that
FεC0,1,X≤Mε
1 ρ B∞max
t∈0,1{ut}
. 4.6
Denote
R1Tnsz0− Tεsz0, R2
s
0
s−θq−1Sns−θknqFnθdθ− s
0
s−θq−1Sns−θkqnFnεθdθ , R3
s
0
s−θq−1Sns−θkqnFnεθdθ− s
0
s−θq−1Sεs−θkqεFεθdθ ,
4.7
where
Fnεθ≡fknθ, xεθ, Bkεθuεθ. 4.8
By assumptionF,
R2≤ qCkqnkqn
Γ 1 q
s
0
s−θq−1Fnθ−Fnεθdθ
≤ qCknqkqnL ρ Γ
1 q s
0
s−θq−1xnθ−xεθdθ
qCkqnkqnL ρ Γ
1 q s
0
s−θq−1Bknθunθ−Bkεθuεθdθ
≤R21 R22 R23,
4.9
where
Mkqn ≡ qCkqnknqL ρ Γ
1 q , R21 ≡Mkqn
s
0
s−θq−1xnθ−xεθdθ,
R22 ≡Mkqn s
0
s−θq−1Bknθuεθ−Bkεθuεθdθ,
R23 ≡Mkqn s
0
s−θq−1Bknθunθ−Bknθuεθdθ,
R3≤ s
0
s−θq−1 kqnSns−θFnεθ−kqεSns−θFεθ dθ kqε
s
0
s−θq−1Sns−θFεθ− Sεs−θFεθdθ
≤R31 R32 R33,
4.10
where
R31≡Mkqnknq
s
0
s−θq−1Fεnθ−Fεθdθ,
R32≡Mknq s
0
s−θq−1kqn−kqεFεθdθ, R33≡kqεMε
1 ρ
s 0
s−θq−1Sns−θ− Sεs−θdθ.
4.11
Note that Lemma 3.7and 4.1, combining H ¨older inequality with Lebesgue domi- nated convergence theorem, one can verify R1 → 0, R23 → 0, R31 → 0 and R33 → 0 asn → ∞immediately. Sinceknknq → kεkεqasn → ∞,FεC0,1,X and uεtE are bounded,R22 → 0,R32 → 0 asn → ∞.
Then, we obtain that
xns−xεs ≤R1 R2 R3
≤σε Mkqn s
0
s−θq−1xnθ−xεθdθ, 4.12
where
σεR1 R22 R23 R31 R32 −→0, asn−→ ∞. 4.13 By singular version Gronwall Lemma again, we obtain
xn−→s xε, inC0,1, X, asn−→ ∞. 4.14 Thus, there exists a unique controlwε uε, kε∈Wsuch that
mε lim
n→ ∞Jεwn Jεwε≥mε. 4.15
This shows thatJεwattains its minimum atwε∈W, and hencexεis the solution of system 3.14corresponding to controlwε.
5. Meyer Approximation Process of Time Optimal Control
In this section, we display the Meyer approximation process of the time optimal control problemP.
For the sake of convenience, we subdivide the approximation process into several steps.
Step 1. ByTheorem 4.1, there exists awε uε, kε∈Wsuch thatJεwattains its minimum atwε∈W, that is,
Jεwε
1
2εxwε1−z12 kε inf
w∈WJεw. 5.1
By controllability of problemP,V0/Ø. Takev ∈ V0 and let τv τ < ∞then zvτ z1. Defineus v τs, 0≤s≤1 andw u, τ∈W. Thenx· zvτ·is the mild solution of system3.14corresponding to controlw u,τ ∈W. Of course, we have
x1 z1.
For anyε >0, submittingw toJε, we have
Jεw τ≥Jεwε
1
2εxwε1−z12 kε. 5.2
This inequality implies that
0≤kε≤τ,
xwε1−z12≤2ετ, hold for all ε >0.
5.3
We can choose a subsequence{εn}such thatεn → 0 asn → ∞and
kqεn −→kq0, in 0,T
, kεn −→k0, in
0,T ,
xwεn1≡xεn1−→z1, inX, asn−→ ∞, uεn
−→w u0, inVad, wεn uεn, kεn∈W.
5.4
SinceVadis closed and convex, thanks to Mazur Lemma again,u0∈Vad. Further, by assumptionB, we obtain
kqεn −→kq0, in 0,T
, kεn −→k0, in
0,T ,
xwεn1≡xεn1−→z1, inX, asn−→ ∞, Buεn
−→s Bu0, inLp0,1, X.
5.5
Step 2. Letxεnandx0be the mild solutions of system3.14corresponding towεn uεn, kεn∈ Wandw0 u0, k0∈W, respectively. Then, we have
xεns Tεnsz0
s
0
s−θq−1Sεns−θkqεnFεnθdθ,
x0s T0sz0
s
0
s−θq−1S0s−θkq0F0θdθ,
5.6
where
Tεn·≡ ∞
0
ξqθTkqεn·qθdθ, Sεn·≡q
∞
0
θξqθTkqεn·qθdθ, Fεn·≡fkεn·, xεn·, Bkεn·uεn·,
T0·≡ ∞
0
ξqθTk0q·qθdθ, S0·≡q
∞
0
θξqθTk0q·qθdθ, F0·≡f
k0·, x0·, B k0·
u0· .
5.7
Recalling 5.5 and the process in Theorem 4.1, after some calculation, using the singular version Gronwall Lemma again, we also obtain
xεn
−→s x0, inC0,1, X, asn−→ ∞. 5.8
Step 3. It follows from Steps1and2, xεn1−z1 ≤
2εnτ−→0, asn−→ ∞, xεn1−x01 −→0, asn−→ ∞,
x01−z1 ≤ xεn1−z1 xεn1−x01 −→0, asn−→ ∞,
5.9
thatx01 z1. It is very clear thatk0/0 unlessz0z1. This implies thatk0>0.
Definev0· u0·/k0. In fact,z0· x0·/k0is the mild solution of system3.1 corresponding to control v0 ∈ V0, then z0k0 x01 z1 and τv0 k0 > 0. By the definition ofτ∗inf{τv≥0|v∈V0}, we havek0≥τ∗.
For anyv∈V0,
τv≥Jεwε
1
2εxwε1−z12 kε. 5.10
Thus,τv≥kε. Further,τv≥kεnfor allεn>0.
Sincek0is the limit ofkεn asn → ∞,τv≥τv0 k0for allv∈V0. Hence,k0 ≤τ∗. Thus, 0< τv0 k0 τ∗. This implies thatv0is an optimal control of ProblemPandk0>0 is just optimal time.
Remark 5.1. Under the above assumptions, there exists a sequence of Meyer problemsPεn whose corresponding sequence of optimal controls{wεn} ∈Wcan successive approximation the time optimal control problemPin some sense. In other words, by limiting process, the sequence of the optimal controls{wεn} ∈W can be used to find the solution of time optimal control problemP.
As a result, we obtain the existence result of time optimal control for system 3.1 directly.
Theorem 5.2. Under the assumptions ofTheorem 4.1. The time optimal control problemPhas a solution, that is, there exists an optimal controlv∗∈V0⊂Vadsuch that
τv∗ τ∗inf{τv≥0|v∈V0}. 5.11
Acknowledgments
This research was supported by National Natural Science Foundation of China No.
10971173, Tianyuan Special Funds of National Natural Science Foundation of ChinaNo.
11026102, and National Natural Science Foundation of Guizhou Province2010, No. 2142.
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