QUANTUM CONTROL FORCHEMICAL REACTION OFPARTICLES AT SURFACE

The Chinese University of Hong KongQuan-Fang Wang

2018 ACS National Meeting & ExpositionNew Orleans, Ernest N. Morial Convention Center, Hall D

PHYS Poster Session, March 21, 19:00-21:00, 2018

ABSTRACTA long time ago, researchers and scientists are working at the quantumcontrol and surface science independently, the contribution of concernedfield are collected in theoretical, computational, numerical, and experi-mental results. Most of these published papers are fallen into physicalchemistry area. By comprising quantitative investigation, experimentsconclusion are occupied the majority numbers. In this work, chemicalreaction of particles at surface will be considered for the extension ofquantum control theory. Nowadays, theres is a lot of consideration on theno-reaction case at the surface, then things become easily to progress forthat final environment is physical experiments to be taken account intotarget control objective. The question arising in here is what would be theresults in the case of particles that chemical reaction at matter surface?The situation would be much more complicated because chemical reactionitself and the changes of both particles and surface layers.

Suppose a well-defined chemical reaction, which indicated that beforeand after reaction occurred, the original particles and original surface,produced (generated) particles and produced (generated) surface could bemeasured and calculated at chemical and physical sense. Then, quantumcontrol can be considerable and meaningful. At first, general system areproposed to try the control application in the reacted surface. Certainly,the chemical reaction what desired is benefit as well as ideal results in thefield of industry, engineering, medical science and so forth. At second,the control variables (factors) might be contained temperature, pressure,particle number, reagent, particle type, size, etc (cf. [3]) It would be fairlyclose work to chemical realms. Roughly to speak, chemist in laboratory todo experiments by using instruments and tools, it is totally in the scope ofcontrol, every adjustment and confinement would be thinkable as controlfactors. It will be happy to find the near pathway from quantum controlto chemical reaction. The rest problem is to select the atom or moleculesthat reacted at chosen surface (cf. [4]). The necessary comparing resultis to control free chemical reaction. It quite interesting to explore theconclusion and present as poster to all the colleague. Further collaborationis hoped to future works at lab experiments.

OUTLINE

1. Introduction

2. Serval Concepts(a) Well-defined chemical reaction at surface(b) Particles at surface(c) Surface control concepts(d) Realistic examples(e) Time-dependent Schrodinger equation

3. Mathematical Setting(a) Complex Hilbert spaces(b) Variational method(c) Solution space(d) Weak solution

4. Applying Control Theory(a) Optimal control for reacted particles at surface(b) Optimal system (Euler-Lagrange system)

5. Conclusion

6. Discussion

7. Acknowledgments

8. References and recommendation books

INTRODUCTIONChemical reaction at surface, it is a big topic at the surface science andit is also a core place occupied at the chemical application (cf. Somorjai[1]). As is well known that chemical reaction is majority part in chemistryfield, and it is almost composed the whole sub-branches and subjectsin chemistry. Without chemical reaction at laboratory, it is hard to saychemistry. Hence, what is the status if chemical reaction at matter surface(such as metallic surface, iron Fe, crystal or catalytic surface)? Furtherproblem is how to control chemical reaction at specified surface? In fact,these research is closely concerning the frontier field of surface science,and a lot of breakthrough had been made at past of decades, some ofthem had awarded the Nobel Prize (Ertl 2007, cf. [2]).

In this work, control of chemical reaction at matter surface (cf. [9],[10])is proposed at the theoretical aspect. To do such a investigation, first ofall, it needs to make serval concepts clearly.

Well-defined chemical reaction at surface what is a well-defined chemical re-action at surface, it indicates that a chemical reaction which mean-ingful to be considered at scientific or academic level. Then how todefine such a chemical reaction.A. before and after reaction incident, chemical reaction are measur-able, calculable.B. A chemical reaction benefit to industrial, engineering, and humansociety.C. Connected with existing surface science, not be isolated.D. No conflict interest with current chemical or physical concepts,theoretical and experimental results.E. Can be developed, can be repeated, can be a defined, can beremained in self subject.F. Theoretic results agree with computation and experiments.

Particles at surfaces Particles (molecules, atoms) at matter surface whichparticipated the chemical reaction before and after reaction, theirnumbers always not 100%. The interacted surface, that is interface(i.e. 65 %). Namely, we must lay some common sense at academiclevel for proceeding the study of control particles at surface.1). The interface in two (2D) or three dimensions (3D) before andafter reaction must be measurable, calculable and repeatable.2). Interface (always at molecule level) can be simulated, observedusing chemical instruments such as TEM, STM, LEED, and so on.3). Taken placed interface would be meaningful and make sense.It means that destroy experiments and no below-up explosion is exclusive.

Mathematical setting reaction equation Usual density functional theory (DFT)can be used for time-dependent evolution in surveying of particles atsurface. In fact, two different Schrodinger equations must be es-tablished. Before molecule (atom)-surface reaction (equation one),and after molecule (atom)-surface reaction (equation two). Due totwo type of participating particles and the particles which composedsurface, there are two wave function needs to be established. Rig-orously, four status need to be taken account into control system:particles before reaction ψbp, after reaction ψap, surface particles beforereaction ψbs, after reaction ψas. Neglecting the nonsense chemical re-action at surface, for a well-defined particle-surface reaction, such asphotochemical surface reaction, control can be proceeded definitely.

Surface control concepts Chemical control means to control molecule size,experimental pressure, temperature, reaction particles type, reactedparticles number, etc. On the other hands, control at quantumphysics field is not limited to such kind of physical factors and param-eters adjustments and confinements. It is quite necessary to developquantum control concepts to chemical reaction at matter surface.The question is what the differences between them?

Changes of depended factor or condition parameters is physical con-trol. Quantum control indicate to input external force in the observedchemical reaction process. What and whether there is a proper exter-nal input as to a surface reaction? Yes, there is the third force (laser,atomic laser, electronic rays, other kinds of radiation rays). Theseforces can be constantly acting at the molecule or atom-surface re-action, and can change as well as control progress of whole chemicalreaction. This is the exact means of “quantum control”.

Realistic examples Ancient people already to control the chemical reactionat surface. For instance, as we are enclosing a new bottle of wine forselling, usual using the fire to hot plastic rubble at the bottle neck(glass, a kind of crystal matter), the role of fire is the third control.At that time, the plastic rubble changed their shape obviously, andglass surface seldom (or tiny) changed. All the process controllableand have a clear target.Another example, painting a car body, making a sculpture using tool,and many other chemical reaction taking place at surface, surfacecontrol would be necessary and useful certainly.

MATHEMATICAL SETTINGIn the setting of mathematics for control problems of chemical reaction atsurface, the variational theory will be utilized in the Hilbert spaces. Time-dependent Schrodinger equation is employed before and after reaction. Inthe few second reaction duration, chemical reaction taken placed. Thetime interval t ∈ [0, τ ]. In here, two measurements to be collected for twotypes of particles: before reaction, the first two equations in followingsystem (1); after reaction, the last two equations in following system(1). In the word of control theory, the first equation is initial state ofboth particle (molecule, atom) or surface particles (i.e. Fe (100)), thesecond equation is the final state of both generated (produced) particlesor generated (produced) surface. More precisely, there are target states,which is the desired states what needs to be achieved.Without lost of generality, consider spatial space at two dimensional case,that is x = (x1, x2) ∈ R2. Let Ω be open bounded set of R2 for reactiontime 0 ≤ t ≤ T . Suppose i number particles take part into reaction beforechemical surface reaction, and 1 ≤ i ≤ n. There is ı number particlesremained after the surface reaction, 0 ≤ ı ≤ n. For unified to use the wavefunctions ψbp, ψap, ψbs, ψas, denote ψip = ψbp, ψıp = ψap, ψis = ψbs and ψıs = ψisfor i and ı particles, and more simplify ψi = ψip, ψı = ψıp, φi = ψis andφı = ψıs. Thus, the time-depended Schrodinger reaction particles controlsystem governed by

i~∂ψi

∂t= −

~2

2m

∂2ψi

∂x2+ ψiu, ψi(x,0, u) = ψi0(x, u).

i~∂ψı

∂t′= −

~2

2m

∂2ψı

∂x2+ ψıu, ψı(x,0, u) = ψı0(x, u).

i~∂φi

∂t= −

~2

2M

∂2φi

∂x2+ φiu, φi(x,0, u) = φi0(x, u).

i~∂φı

∂t′= −

~2

2M

∂2φı

∂x2+ φıu, φı(x,0, u) = φı0(x, u).

(1)

In here, i is imaginary part of complex unit. the wave function ψi(x, t, u) isrepresents the probability of i-th particle distribution at the time t beforereaction, and ψı(x, t′, u) is represents the probability of ı-th particle distribu-tion after reaction at time t′ for spatial coordinate x. The same with φi, φıfor particles at surface. u(x, t) is time-space varying control input whichrepresent shaped laser pulse (or other third force) at laboratory. m, M aremass of particles and surface particle, respectively. If target final time τand τ ′ for before and after reaction, usually set t′ = t− τ .

Remark. For initial condition in system (1), to eliminate confusion with chem-ical experiments results, some interpreting must make clear. At the viewpointof chemistry, reaction rates, reaction area can be obtained and calculated for achemical reaction incident at surface. The visible area at a surface could not makesense in here. It is not concern boundary control. For topic of boundary control ofreaction area, these sort of consideration is not inclusive in our discussion. Equa-tion (1) can take i = ı = 1 for the case of one particle reacted at surface, and alsocan take many different particles number at theoretic argument. What we focusis state change of particles itself. The connection with a real chemical conceptreaction rate is i/ı. In precisely, if taking vector ψ = (ψ1, ψ2, · · · , ψn, φ1, φ2, · · · , φn)for a virtual wave function, it is also not a boundary issue.

For given proper physical parameter and constants, the equation (1) canbe solved analytically. By comprising two different Schrodinger equationand their solution, one can found:(a). how many number of particles (molecule, atom) reacted (i.e. 65 %)(b). how much energy change status: release and absorption of energy.(c). how large area reacted at surface.(d). who’s who particles before and after reaction. e.g. desorption ofH2 from silicon surface, Si(100). Also for instance, CO +H2O → H2 + CO2,H + CH4 → H2 + CH3.(e). With two different equations for two particles, what can we know;what changed; what is control force; control duration [0, τ ]; control forcetype (time-depended, spatial depended); quantitive calculation, as well assimulation can be done. Those issues of computational chemistry left toour future work.

APPLY CONTROL THEORY

Generally, with time-evolution of one wave function at one equation, wecan do control for particles at surface in previous papers. Two differentequations must be adopted to discuss the reaction with taken place atsurface (particles changed; surface changed; energy changed; control ob-jective achieved).

Theoretically to speak, control of surface reaction is involving optimalforce control and optimal time control.

Time optimal control Assume U is Hilbert space of control variable u(x, t),which is depending on spatial variable x and time t. Denote Q = Ω× [0, τ ]if needed. Suppose a desired achieve time τ ∈ [0, T ]. Use ϕ representseach ψi, ψı, φi, or φı for simplification. The controllability of time optimalcontrol means that, for a scalar function value ϕ, there exist a desiredcontrol input u(x, t) ∈ Uad and desired state ϕ(u, τ) ∈ H such that

ϕ(u, τ) = ϕ (2)

for appropriate τ , where Uad ⊂ U is admissible subset of u.

Time optimal control is to solve two fundamental problems:i). Find such a t∗ ∈ [0, T ] such that t∗ = inf

τ∈[0,T ]for the ϕ∗: ψ∗i , ψ

∗ı , φ

∗i , or φ∗ı to

satisfy (2), i.e. ϕ(u∗, t∗) = ϕ∗.ii). The properties of such a t∗.Such a time t∗ is called the optimal time to achieve optimal control u∗.

Cost function The measurement cost function for reaction particles oftime-depended Schrodinger systems (1) is given by quadratic form as

J(u, τ) =

∫ τ

0

∫Ω

|ψi(x, s)− ψif(x, s)|2dxds+

∫Ω

|ψı(x, τ)− ψıf(x, τ)|2dx

+

∫ τ

0

∫Ω

|φi(x, s)− φif(x, s)|2dxds+

∫Ω

|φı(x, τ)− φıf(x, τ)|2dx

+

∫ τ

0

∫Ω

|u(x, s)|2dxds+

∫Ω

|u(x, τ)|2dx+ infτ∈[0,T ]

,

u ∈ Ω× [0, τ ], τ ∈ [0, T ], (3)

where ψif(x) is target state for wave function of reaction i-th particles.ψı(x, τ) is observed state at last time τ . The same fashion to φif andφı. The criteria function (3) will also get the minimized external forceand minima of reaction time duration. Usually replace t∗ = inf

τ∈[0,T ]with∫ τ

0

|s− τ |2ds in (3).

Notice: Before chemical reaction, observed distributed wave function to do spa-tial distributed control (per molecule or per atom), and after chemical reaction,observed final wave function at time τ to do terminal control (final states ofremained molecules or atoms).

Successful optimal control: using minima of energy (control force input)and minimization time to approach target (sealed a bottle of wine). Suchkind of task nowadays become automatic mechanical works with machineat manufacturing factory.

As is well known, quantum optimal control and time optimal control is tosolve two fundamental problems (cf. [5],[6]):

(i). Find an element u∗(x, t) and optimal time t∗ such that

infu∈Uad,t∈[0,τ ]

J(u, t) = J(u∗, t∗), ∀x ∈ Ω.

(ii). Characterization of u∗ and t∗.Such a pairing (u∗, t∗) is called quantum optimal control for Schrodingersystems (1) subject to cost function (3), and t∗ is optimal time for particleschemical reaction duration at surface.

For solving optimal control and optimal time issue in the framework ofthe variational method, introduce the concepts of Hilbert spaces, solutionspace, weak solution.

Hilbert spaces Define two Hilbert spaces H = L2(Ω), V = H10(Ω) with usual

norm and inner product. Then (V,H) is a Gelfand triple spaces V → H → V ′,in which two embeddings are continuous, dense and compact (cf.[8]).

For complex-valued function ψi, φi and ψı, φı, define complex solution spaceL2(Ω) and H1

0(Ω). For unified denote ψ = (ψi, φi, ψı, φı) if needed. Usenotations H = L2(Ω),V = H1

0(Ω). Further, V′ is complex conjugate space ofV. Actually, (V,H) is also a complex Gelfand triple spaces V → H → V′, inwhich two embeddings are continuous, dense and compact too.

Define inner product and norm

Definition 1. L2(Ω) For complex function ϕi = ϕi1 + iϕi2 ∈ L2(Ω), ϕi1, ϕi2 ∈L2(Ω), define the norm of complex space L2(Ω) as

‖ϕi‖L2(Ω) =(‖ϕi1‖2

L2(Ω) + ‖ϕi2‖2L2(Ω)

)12 .

If ϕi = ϕi1 + iϕi2 ∈ L2(Ω) and ϕi′ = ϕi′1 + iϕi′2 ∈ L2(Ω), then the inner productof complex space L2(Ω) can be defined by

(ϕi, ϕi′)L2(Ω) =(

(ϕi1, ϕi′1)L2(Ω) + (ϕi2, ϕi′2)L2(Ω)

)+i(

(ϕi2, ϕi′1)L2(Ω) − (ϕi1, ϕi′2)L2(Ω)

).

Definition 2. H10(Ω) For the complex space H1

0(Ω), define its norm as

‖ϕi‖H10(Ω) =

(‖ϕi1‖2

H10

(Ω) + ‖ϕi2‖2H1

0(Ω)

)12 .

If ϕi, ϕi′ ∈ H10(Ω), then inner product of complex space H1

0(Ω) is defined as

(ϕi, ϕi′)H10(Ω) = (ϕi1, ϕi′1)H1

0(Ω) + (ϕi2, ϕi′2)H1

0(Ω).

For simplification, then one can mathematically to get the definition of innerproduct and norm for combined function ψ = (ψi, φi, ψı, φı).

Definition 3. Solution space The complex space W (0, T ;V,V′) is calledsolution space, defined by

W (0, T ;V,V′) =ϕi|ϕi,∈ L2(0, T ;V), ϕ′i ∈ L2(0, T ;V′)

.

If ϕ1i , ϕ

2i ∈W (0, T ;V,V′), then its inner product can be defined by

(ϕ1i , ϕ

2i )W (0,T ;V,V′) = (ϕ1

i , ϕ1i )V + (ϕ′1i , ϕ

′1i )V′ + (ϕ2

i , ϕ2i )V + (ϕ′2i , ϕ

′2i )V′.

Then inner product induced norm of W (0, T ;V,V′) is defined by

‖ϕi‖W (0,T ;V,V′) =(‖ϕi‖2

L2(0,T ;V) + ‖ϕ′i‖2L2(0,T ;V′)

)12

Thus, W (0, T ;V,V′) is a complex Hilbert space equipped with inner productand norm. Further, C(0, T ;H) denotes complex continuous functions set.By the inclusive of space of continuity, then W (0, T ;V,V′) ⊂ C(0, T ;H).

Definition 4. Weak solution Let τ > 0, the pairing (ψi, φi, ψı, φı) is calledweak solution of (1) if ψi, φi, ψı, φı ∈W (0, T ;V,V′) and satisfy weak form:

∫ τ

0

∫Ω

i~∂ψi

∂tηtdxdt =

∫ τ

0

∫Ω

[~2

2m

∂ψi

∂x

∂η

∂x+ ψiuη]dxdt.∫ τ

0

∫Ω

i~∂φi

∂tξtdxdt =

∫ τ

0

∫Ω

[~2

2M

∂φi

∂x

∂ξ

∂x+ φiuξ]dxdt.∫ τ ′

0

∫Ω

i~∂ψı

∂t′ηt′dxdt

′ =

∫ τ

0

∫Ω

[~2

2m

∂ψı

∂x

∂η

∂x+ ψıuη]dxdt′.∫ τ ′

0

∫Ω

i~∂φı

∂t′ξt′dxdt

′ =

∫ τ ′

0

∫Ω

[~2

2M

∂φı

∂x

∂ξ

∂x+ φıuξ]dxdt

′.

(4)

for all η, ξ ∈ C1(0, T ;V) in the sense of distribution on [0, τ ] and such thatη(τ) = 0, ξ(τ) = 0 a.e. x ∈ Ω. In here, η, ξ is the complex conjugate functionof η, ξ functional, respectively. Set t′ = t − τ for computational approach,theoretically use the same t and τ .

Theorem 5 Existence of weak solution For given ψi0, φi0, ψı0, φı0 ∈ V. Ifu(x,0) ∈ L2(Ω) is bounded, then there exist a unique weak solution (ψi, φi, ψı, φı)of time-depended Schrodinger quantum system (1) of reaction particles,and ψi, φi, ψı, φı ∈ W (0, T ;V,V′) ⊂ C(0, T ;H). Further, the estimation ofψi, φi, ψı, φı is obtained as

‖ψi‖2L2(0,T ;H), ‖ψı‖

2L2(0,T ;V), ‖φi‖

2L2(0,T ;H), ‖φı‖

2L2(0,T ;V)

≤ C(‖ψi0‖2V + ‖ψı0‖2

V + ‖φi0‖2V + ‖φı0‖2

V) (5)

where C is bounded complex constant, independent of initial functionsψi0, φi0, ψı0 and φı0.

Notice: usually assume ψi(x, τ) = ψı(x,0), φi(x, τ) = φı(x,0), t′ = t − τ for con-tinuous at computational and numerical approach.

Following the routing manipulation to straightforwardly get the proof ofthe existence of weak solution.

REACTED PARTICLES CONTROL

By the existence of weak solution to know, assume the admissible controlset Uad is closed convex and bounded subset of U (i.e. L2(0, T ; Ω)), then forarbitrary u(x, t) ∈ Uad, via the virtual of Theorem 5, there exists a uniqueweak solution pairing (ψi(u, t), φi(u, t), ψı(u, t), φı(u, t)) of Schordinger system(1) in complex solution space W (0, T ;V,V′).

It is means that there is a solution mapping (u, t) → (ψi, φi, ψı, φı) can bedefined from

U × [0, τ ]→W (0, T ;V,V′)×W (0, T ;V,V′)×W (0, T ;V,V′)×W (0, T ;V,V′).

The solutions ψi(u, t), φi(u, t), ψı(u, t), φı(u, t) is called the states of Schrodingersystem (1). The notations u∗, t∗ is called quantum optimal control and op-timal time of optimal state ψ∗i , φ

∗i , ψ

∗ı , φ

∗ı for control system (1).

PROOF OF CONTROL THEORY

Theorem 6 Suppose ψi0, φi0, ψı0, φı0 ∈ V, if Uad ⊂ U is bounded closed convexsubset, then there is at least one quantum optimal control u∗ ∈ Uad andoptimal time t∗ ∈ [0, τ ] of time-depended Schrodinger system (1) subjectto cost function (3) .Proof. If setting J = inf

u∈Uad,t∈[0,τ ]J(u, t), since Uad is non-empty admissible set,

by optimal time controllability and time interval bounded, there exist asequence un ⊂ Uad and τn ⊂ [0, τ ] can be selected, such that t∗ = inf

τ∈[0,T ]and

τn → t∗ as n→∞. Set ϕn = ϕ(un, τn), and ϕ∗ represents each ψ∗i , φ∗i , ψ

∗ı , φ

∗ı , then

by the boundedness of admissible set Uad that ϕn bounded in L2(0, T ;V),and ϕ′n bounded in L2(0, T ;V′). There is subsequence unk ⊂ un in Uad,and subsequence ϕnk ⊂ ϕn in L2(0, T ;V) can be chosen such that unk → u∗

in Uad, and

ϕnk → ϕ∗ in L2(0, T ;V), ϕ′nk → ϕ∗′ in L2(0, T ;V′), as nk →∞.

By ϕ(0) = ϕ0 and uniqueness of limitation that ϕ∗(0) = ϕ0(u∗) = ϕ(u∗,0).On the other hand, by the calculation

ϕ(unk, τn)− ϕ(u∗, t∗) = ϕ(unk, τn)− ϕ(unk, t∗) + ϕ(unk, t

∗)− ϕ(u∗, t∗)

to know that ϕ(unk, t∗)→ ϕ(u∗, t∗) weakly in space H. Further,

‖ϕ(unk, τn)− ϕ(unk, t∗)‖V′ = ‖

∫ τn

t∗

‖ϕ′(unk, τ)dτ‖V′

≤ (τn − t∗)12 (

∫ τn

t∗

‖ϕ′(unk, τ)‖2V′dτ)

12 ≤ C(τn − t∗)

12 . (6)

Hence, ϕ(unk, τn)→ ϕ(u∗, t∗) weakly convergence at V′. That is ϕ∗ = ϕ(u∗, t∗).

Set infunk∈Uad,t∈[0,τ ]

J(u) = limnk→∞

J(unk) = J. Clearly J(unk) is bounded in R+.

Since Uad × [0, τ ] is bounded closed and convex, choose a subsequenceunk, tk from unk, τn, and there exist (u∗, t∗) ∈ Uad × [0, τ ] such that

(unk, tk)→ (u∗, t∗) weakly in U × [0, τ ] as nk, k →∞, (7)

For ψi = ψi(u, t), φi = φi(u, t), ψı = ψı(u, t′), and φı = φı(u, t′), the estimate (5)in Theorem 5 imply that

‖ψi‖2W (0,T ;V,V′) ≤ C‖ψi0‖

2V, ‖φi‖2

W (0,T ;V,V′) ≤ C‖φi0‖2V (8)

‖ψı‖2W (0,T ;V,V′) ≤ C‖ψı0‖

2V, ‖φı‖2

W (0,T ;V,V′) ≤ C‖φı0‖2V. (9)

Because the time duration of chemical reaction [0, τ ] is bounded. Foru(x, t), the boundedness of Uad, (8) and (9) to know ψi(x, t), φi(x, t), ψı(x, t′),φı(x, t′) is bounded at W (0, T ;V,V′). Then, there exist a subsequence(ψi(unk, tk), φi(unk, tk), ψı(unk, t

′k), φı(unk, t

′k)) of (ψi(unk, τn), φi(unk, τn), ψı(unk, τ ′n),

φı(unk, τ ′n)), and a (ψi, φi, ψı, φı) belongs to W (0, T ;V,V′) such that

(ψi(unk, tk), φi(unk, tk), ψı(unk, t′k), φı(unk, t

′k))→ (ψi, φi, ψı, φı) weakly in W (0, T ;V,V′)

for ∀x ∈ Ω as nk →∞ and k →∞.

For simplification, denote ψnki = ψi(unk, tk), φnki = φi(unk, tk), ψ

nkı = ψı(unk, t

′k), φ

nkı =

φı(unk, t′k). Because the embedding V → H is compact, thus Aubin-Lions-

Temam compactness embedding Theorem (cf. [6]) to yield that

ψnki → ψi, φnki → φi, ψ

nkı → ψı, ψ

nkı → ψı strongly in L2(0, T ;H).

as nk → ∞ and k → ∞. Then for Schrodinger equation (1), the formulas(8) and (9) imply that

∂ψnki∂t→

∂ψi

∂t,∂φnki∂t→

∂φi

∂t,∂ψnkı

∂t′→

∂ψı

∂t′,∂φnkı

∂t′→

∂φı

∂t′weakly in L2(0, T ;V′),

∂ψnki∂x→

∂ψi

∂x,∂φnki∂x→

∂φi

∂x,∂ψnkı

∂x→

∂ψı

∂x,∂φnkı

∂x→

∂φı

∂xweakly in L2(0, T ;H),

as nk →∞ and k →∞. Recall definition of weak solutions for ψnki , φnki , ψ

nkı , φ

nkı

as

−∫ tk

0

∫Ω

i~∂ψnki∂t

ηtdxdt =

∫ tk

0

∫Ω

[~2

2m

∂ψnki∂x

∂η

∂x+ ψnki uη]dxdt.

−∫ tk

0

∫Ω

i~∂φnki∂t

ξtdxdt =

∫ tk

0

∫Ω

[~2

2M

∂φnki∂x

∂ξ

∂x+ φnki uξ]dxdt.

−∫ t′

k

0

∫Ω

i~∂ψnkı

∂t′ηt′dxdt

′ =

∫ tk

0

∫Ω

[~2

2m

∂ψnkı

∂x

∂η

∂x+ ψnkı uη]dxdt′.

−∫ t′

k

0

∫Ω

i~∂φnkı

∂t′ξt′dxdt

′ =

∫ t′k

0

∫Ω

[~2

2M

∂φnkı

∂x

∂ξ

∂x+ φnkı uξ]dxdt

′.

(10)

By the convergence of (7) and taking nk →∞, tk →∞ in (10) to get

−∫ tk

0

∫Ω

i~∂ψi

∂tηtdxdt =

∫ tk

0

∫Ω

[~2

2m

∂ψi

∂x

∂η

∂x+ ψiuη]dxdt.

−∫ tk

0

∫Ω

i~∂φi

∂tξtdxdt =

∫ tk

0

∫Ω

[~2

2M

∂φi

∂x

∂ξ

∂x+ φiuξ]dxdt.

−∫ t′

k

0

∫Ω

i~∂ψı

∂t′ηt′dxdt

′ =

∫ t′k

0

∫Ω

[~2

2m

∂ψı

∂x

∂η

∂x+ ψıuη]dxdt′.

−∫ t′

k

0

∫Ω

i~∂φı

∂t′ξt′dxdt

′ =

∫ t′k

0

∫Ω

[~2

2M

∂φı

∂x

∂ξ

∂x+ φıuξ]dxdt

′.

(11)

for all η, ξ ∈ C1(0, T ;V). Thus, by route manipulation to get that thelimit ψi(u, t), φi(u, t), ψı(u, t′), φı(u, t′) satisfy (11) for all ψi, φi, ψı, φı ∈ V in thesense of D′(0, T ), which is distribution on (0, τ). Via the convergency oftk, t′k ∈ tn to get t = t∗, t′ = t′∗. By the uniqueness of weak solution ofSchrodinger system (1) to know that

ψi(u, t) = ψi(u∗, t∗), φi(u, t) = φi(u

∗, t∗), ψı(u, t′) = ψı(u∗, t′∗), φı(u, t′) = φı(u

∗, t′∗).

The approximate solution is convergent as nk →∞ and k →∞

ψi(unk, tk)→ ψi(u∗, t∗), ψıf(unk, τ)→ ψıf(u∗, τ)

strongly in L2(0, τ ;L2(Ω)),L2(Ω) respectively.

φi(unk, tk)→ φi(u∗, t∗), φıf(unk, τ)→ φıf(u∗, τ)

strongly in L2(0, τ ;L2(Ω)),L2(Ω) respectively.

Since the lower semi-continuous of weak topology for norms ‖ · ‖L2(0,τ ;L2(Ω))

and ‖ · ‖L2(Ω) at the spaces L2(0, τ ;H) and H = L2(Ω), hence that

lim infnk→∞,k→∞

‖ψi(unk, tk)− ψif(unk, tk)‖2L2(0,tk;H) ≥ ‖ψi(u

∗, t∗)− ψif(u∗, t∗)‖2L2(0,tk;H),

lim infnk→∞

‖ψı(unk, τ)− ψıf(unk, τ)‖2H ≥ ‖ψi(u

∗, τ)− ψif(u∗, τ)‖2H,

lim infnk→∞,k→∞

‖φi(unk, tk)− φif(unk, tk)‖2L2(0,tk;H) ≥ ‖φı(u

∗, t∗)− φıf(u∗, t∗)‖2L2(0,tk;H),

lim infnk→∞

‖φı(unk, τ)− φıf(unk, τ)‖2H ≥ ‖φı(u

∗, τ)− φıf(u∗, τ)‖2H,

Similarly, for running cost term in criteria function (3), weak convergences(7) imply

lim infnk→∞

(unk, unk)U ≥ (u∗, u∗)U .

In particular, for optimal time term in criteria function (3), from bound-

edness of reaction duration [0, τ ] and (6) to get limk→∞

(tk − t∗)12 → 0. Since

weakly lower semi-continuous of J and Lebegue convergence to up-limittk of integration to obtain J = lim inf

nk→∞,k→∞J(unk, tk) ≥ J(u∗, t∗). Resultantly,

J(u∗, t∗) = inf(u,t)∈Uad×[0,τ ]

J(u, t). Hence, u∗ is quantum optimal control and t∗ is

optimal reaction time subject to cost function (3). It proved Theorem 6.

Remark Roughly, one can use symbol ϕ to represent each ψi, φi, ψı, φı or use vectorψ = (ψi, φi, ψı, φı) for arguments.

Notice. It is possible to do distributed control after chemical reaction for someexperiments, but it would be incredible to do terminal control before chemicalreaction, since it must be medium-states in process, and difficult to be observed.

OPTIMIZATION PROBLEM

Theorem 7 Suppose ψi0, φi0, ψı0, φı0 ∈ V, if Uad ⊂ U is bounded closed convexspace, then optimal reaction control u∗ and optimal reaction time t∗ forcost function (3) is characterized by following equations and inequality,called Euler-Lagrange optimal system:

−i~∂ψi

∂t=

~2

2m

∂2ψi

∂x2+ u∗ψi in Q, ψi(0) = ψi0 in Ω.

−i~∂φi

∂t=

~2

2M

∂2φi

∂x2+ u∗φi in Q, φi(0) = φi0 in Ω.

−i~∂ψı

∂t=

~2

2m

∂2ψı

∂x2+ u∗ψı in Q, ψı(0) = ψı0 in Ω.

−i~∂φı

∂t=

~2

2M

∂2φı

∂x2+ u∗φı in Q, φı(0) = φı0 in Ω.

(12)

−i~∂pi

∂t=

~2

2m

∂2pi

∂x2+(ψi(u

∗, t∗)− ψif(u∗, t∗))

in Q, pi0 = 0 in Ω.

−i~∂qi

∂t=

~2

2M

∂2q

∂x2+(φi(u

∗, t∗)− φif(u∗, t∗))

in Q, qi0 = 0 in Ω.

−i~∂pı

∂t=

~2

2m

∂pı

∂x2in Q, ipı(t

∗) = ψı(u∗, t∗)− ψıf(u∗, t∗) in Ω.

−i~∂qı

∂t=

~2

2M

∂qı

∂x2in Q, iqı(t

∗) = φı(u∗, t∗)− φıf(u∗, t∗) in Ω.

(13)

∫ t∗

0

(pi(u

∗) + qi(u∗), u− u∗

)H

+(pi(u

∗) + qi(u∗), t− t∗

)Hdt

+(pı(u

∗, t∗) + qı(u∗, t∗), u− u∗

)H

+ (pı(u∗, t∗) + qı(u

∗, t∗), t− t∗)H+(u∗, u− u∗)U + (t∗, t− t∗) ≥ 0. (14)

In here, ψi, φi, ψı, φı is weak solutions of states system (12), pi, qi, pı, qı ∈W (0, T ;V,V′) is weak solutions of adjoint system (13). As is well knownthat inequality (14) is necessary optimality condition of (u∗, t∗).

CONCLUSION

In this work, control problem of chemical reacted particles at surface hadbeen considered for theoretical results. Optimal control theory had beenapplied to time-depended Schrodinger quantum system. The optimal sys-tem is deduced contain optimal states system; optimal adjoint system;and necessary optimal condition. To the chemical reaction at surfacephenomena, control theory can be worked at matter surface (metal, crys-tal, catalysis etc). Fortunately, It needs to evident at experiments methodusing mutated optical technique (cf. [7]).

What’s new: optimal time control is required to minimize the chemicalreaction time duration. Scientifically, we did not hope the effect of waterdrop dig the stone at a long long time.

Mathematical difficulties: the precise equations structure to a variety ofchemical reaction at amount number of surfaces, would be incredible tomeet each one of the satisfaction. For unified and simplification, not lostof generality to use time-depended Schrodinger equation.

Chemical difficulties: For different chemical reactions at surfaces occurredat chemistry and physics lab, it needs to fit particular situation to calculate.

DISCUSSION

Future tend work is to quantitive study detailed particles at a specifiedsurface, computational approach and numerical simulation could be workedat the control of chemical reaction at surface.

The extension would not be limited to 2D surface, can also be developedto 3D surface.

The application of reaction at surface have huge space to the field ofindustrial, engineering, computer science, and technology.

At past time, using chemical experimental results directly to do realisticapplication in the world, now with theoretic results, one can try theoreticaltest at computer firstly, and find best solution to apply in industrial andengineering, meanwhile, also can do adaptive control or feedback controlto achieve a large scale complex ideal control purpose. One can avoidtry-false test, and using modern computer technique and subject softwareto execute real control problem occurred at society in real world.

Control theory would be worked at the particles-surface as their are reactedeach other.

At academic level, it is a closely contact for surface science and quantumcontrol theory. An enjoyable journey to both fields.

Molecules-surface reaction would be smoothly controlled by third forceinput. Atomic-surface reaction needs to do much more delicate work inthe future investigation.

ACKNOWLEDGMENTS

The author really appreciate 255th ACS National Meetings & Exposition,New Orleans, March 18∼22, 2018 for poster 365.

THANKS FOR YOUR ATTENTION

REFERENCES

[1], G. A. Somorjai, Y. Li, Impact of surface chemistry, PNAS Vol. 108, no.3,917-924, 2011.

[2].KLINGL VETENSKAPSAKADEMIEN, Chemical Processes on Solid Surface,Scientific Background on the Nobel Prize in Chemistry 2007.

[3]. M. K. Weldon, C.M. Friend, Probing surface reaction mechanisms usingchemical and vibrational methods: Alkyl oxidation and reactivity of alcohols ontransitions metal surfaces, Chemical Review 96,1391-1411,1996.

[4]. T.V. Albu, Computational Chemistry of Polyatomic reaction kinetics anddynamics: The quest for an accurate CH5 potential energy surface, ChemicalReview 107, 5101-5132, 2007.

[5]. R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods forScience and Technology, Vol. 5, Evolution Problems I, Berlin-Heidelberg-NewYork. Springer-Verlag,1992.

[6] J. L. Lions, Optimal Control of Systems Governed by Partial DifferentialEquations, Berlin-Heidelberg-New York, Springer-Verla, 1971.

[7] S. A. Rice, Optical Control of Molecular Dynamics, New York, Wiley, 2000.

[8] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,Second Edition, Applied Mathematica Sciences Vol. 68, Berlin-Heidelberg-NewYork, Springer-Verlag, 1997.

[9] Q. F. Wang, Quantum control at matter surface, ACS National Meeting &Exposition 2009, Poster.

[10] Q. F. Wang, Quantum surface control of Bose-Einstein-Condensate, 251thACS National Meeting & Exposition, Poster, 2016.

RECOMMENDED BOOKS

Quan-Fang Wang, Optimal Control for Nonlinear Parabolic DistributedParameter Systems: with numerical analysis, Lambert Academic Pub-lishing, Germany, Feb. 1, 2011.

Quan-Fang Wang, Practical Application of Optimal Control Theory:computational approach, Lambert Academic Publishing, Germany,Nov. 11. 2011.

Quan-Fang Wang, Optimal Control for Cahn-Hilliard Issues: basics,concepts and tutorials, Lambert Academic Publishing, Germany, Mar.19. 2014.

Quan-Fang Wang, Identification in Inverse Problems: Parabolic par-tial differential equation, Lambert Academic Publishing, Germany,Jun. 8. 2015.