Optimal control of PDEsin a complex space setting;application to the Schrödinger equation

# Optimal control of PDEs in a complex space setting; application to the Schrödinger equation

## Abstract.

In this paper we discuss optimality conditions for abstract optimization problems over complex spaces. We then apply these results to optimal control problems with a semigroup structure. As an application we detail the case when the state equation is the Schrödinger one, with pointwise constraints on the “bilinear” control. We derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly.

###### Key words and phrases:
Optimal control, partial differential equations, optimization in complex Banach spaces, second-order optimality conditions, Goh-transform, semigroup theory, Schrödinger equation, bilinear control systems.
The second and third author were supported by the project ”Optimal control of partial differential equations using parameterizing manifolds, model reduction, and dynamic programming” funded by the Foundation Hadamard/Gaspard Monge Program for Optimization and Operations Research (PGMO).

July 17, 2018

Keywords:

## 1. Introduction

In this paper we derive no gap second order optimality conditions for optimal control problems in a complex Banach space setting with pointwise constraints on the control. This general framework includes, in particular, optimal control problems for the bilinear Schrödinger equation.

Let us consider , an open bounded set, , , and . The Schrödinger equation is given by

 (1) i˙Ψ(t,x)+ΔΨ(t,x)−u(t)B(x)Ψ(t,x)=0,Ψ(x,0)=Ψ0(x),

where , , and with the time-dependent electric field, the wave function, and the coefficient of the magnetic field. The system describes the probability of position of a quantum particle subject to the electric field ; that will be considered as the control throughout this paper. The wave function belongs to the unitary sphere in .

For and , the optimal control problem is given as

 (2) ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩minJ(u,Ψ):=12∫Ω|Ψ(T)−ΨdT|2dx+% 12∫Q|Ψ−Ψd|2dxdt+∫T0(α1u(t)+12α2u(t)2)dt, subject to (???) and u∈Uad,

with , and for , and desired running and final states and , resp. The control of the Schrödinger equation is an important question in quantum physics. For the optimal control of semigroups, the reader is referred to Li et al. [37, 38], Fattorini et al. [29, 28] and Goldberg and Tröltzsch [33]. In the context of optimal control of partial differential equations for systems in which the control enters linearly in both the state equation and cost function (we speak of control-linear problems), in a companion paper [3], we have extended the results of Bonnans [17] (about necessary and sufficient second order optimality conditions for a bilinear heat equation) to problems governed by general bilinear systems in a real Banach space setting, and presented applications to the heat and wave equation.

The contribution of this paper is the extension to a complex Banach space setting of the optimality conditions of a general class of optimization problems and of the framework developed in [3]. More precisely, we consider optimal control problems governed by a strongly continuous semigroup operator defined in a complex Banach space and derive necessary and sufficient optimality conditions. In particular (i) the study of strong solutions when , and (ii) the control-affine case, i.e. when , are addressed. The results are applied to the Schrödinger equation.

While the literature on optimal control of the heat equation is quite rich (see, e.g., the monograph by Tröltzsch [43]), much less is available for the optimal control of the Schrödinger equation. We list some references on optimal control of Schrödinger equation and related topics. In Ito and Kunisch [35] necessary optimality conditions are derived and an algorithm is presented to solve the unconstrained problem, in Baudouin et al. [7] regularity results for the Schrödinger equation with a singular potential are presented, further regularity results can be found in Baudouin et al. [8] and Boscain et al. [21] and in particular in Ball et al. [5]. For a minimum time problem and controllability problems for the Schrödinger equation see Beauchard et al. [12, 13, 11]. For second order analysis for control problems of control-affine ordinary differential systems see [2, 32]. About the case of optimal control of nonlinear Schrödinger equations of Gross-Pitaevskii type arising in the description of Bose-Einstein condensates, see Hintermüller et al. [34]; for sparse controls in quantum systems see Friesecke et al. [31].

The paper is organized as follows. In Section 2 necessary optimality conditions for general minimization problems in complex Banach spaces are formulated. In Section 3 the abstract control problem is introduced in a semigroup setting and some basic calculus rules are established. In Section 4 first order optimality conditions, in Section 5 sufficient second order optimality conditions are presented; sufficient second order optimality conditions for singular problems are presented in Section 6, again in a general semigroup setting. Section 7 presents the application, resp. the control of the Schrödinger equation and Section 8 a numerical tests supporting the possibility of existence of a singular arc.

## 2. Optimality conditions in complex spaces

### 2.1. Real and complex spaces

We consider complex Banach spaces which can be identified with the product of two identical real Banach spaces. That is, with a real Banach space we associate the complex Banach space with element represented as , with , in and , and the usual computing rules for complex variable, in particular, for with , real, we define . Define the real and imaginary parts of a by and , resp.

Let be a real Banach space and the corresponding complex one. We denote by (resp. ) the duality product (resp. antiduality product, which is linear w.r.t. the first argument, and antilinear w.r.t. the second). The dual (resp. antidual) of (resp. ), i.e. the set of linear (resp. antilinear) forms, is denoted by (resp. ).

### 2.2. Optimality conditions

We next adress the questions of optimality conditions analogous to the obtained in the case of real Banach spaces [19]. Consider the problem

 (3) Minu,xf(u,x);g(u,x)∈Kg;h(u,x)∈Kh.

Here and are real Banach space, and are complex Banach spaces, and , are nonempty, closed convex subsets of and resp. The mappings , , from to respectively, , , and are of class . As said before, the complex space can be identified to a pair of real Banach spaces, with dual . Let , . Setting and observe that (by linearity/antilinearity of ) that

 (4) ⟨x∗,x⟩¯¯¯¯X=⟨x∗1,x1⟩X+⟨x∗2,x2⟩X+i(⟨x∗2,x1⟩X−⟨x∗1,x2⟩X),

and therefore the ‘real’ duality product in given by satisfies

 (5) ⟨x∗,^x⟩X×X=R⟨x∗,x⟩¯¯¯¯X.

Let , be two complex spaces associated with the real Banach spaces and . The conjugate transpose of is the operator defined by

 (6) ⟨y∗,Ax⟩¯¯¯¯Y=⟨A∗y∗,x⟩¯¯¯¯X, for all (x,y∗) in ¯¯¯¯¯X×¯¯¯¯Y∗.

If , identifying the real Banach space with the space of real parts of the corresponding complex Banach space , we may define by

 (7) ⟨A∗y∗,u⟩¯U=⟨y∗,Au⟩¯¯¯¯Y.

Combining this relation with (5), we deduce that

 (8) R⟨y∗,Au⟩¯¯¯¯Y=R⟨A∗y∗,u⟩¯U=⟨RA∗y∗,u⟩U.

We deduce the following expression of normal cones, for :

 (9) NKg(y)={y∗∈¯¯¯¯Y∗;R⟨y∗,z−y⟩¯¯¯¯Y≤0, for all z∈Kg}.

For and the Lagrangian of the problem is defined as

 (10) L(u,x,λ,μ):=f(u,x)+R⟨λ,g(u,x)⟩¯¯¯¯Y+⟨μ,h(u,x)⟩W.
###### Lemma 2.1.

The partial derivatives of the Lagrangian are as follows:

 (11) ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩∂L∂u=∂f∂u+R(∂g∂u∗λ)+∂h∂u⊤μ,∂L∂xr=∂f∂xr+R(∂g∂x∗λ)+∂h∂xr⊤μ,∂L∂xi=∂f∂xi+I(∂g∂x∗λ)+∂h∂xi⊤μ.

In particular, we have that

 (12) ∂L∂xr+i∂L∂xi=∂f∂xr+i∂f∂xi+∂g∂x∗λ+(∂h∂xr+i∂h∂xi)⊤μ.
###### Proof.

We have that, skipping arguments:

 (13) ∂L∂uv=∂f∂uv+R(λ,∂g∂uv)¯¯¯¯Y+(μ,∂h∂uv)W=∂f∂uv+R(∂g∂u∗λ,v)U+(∂h∂u⊤μ,v)U=(∂f∂u+R(∂g∂u∗λ)+∂h∂u⊤μ,v)U

for all . We have used that setting and , then

 (14) (R(∂g∂u∗λ),v)U=(R(a⊤−ib⊤)(λr+iλi),v)U=(a⊤λr+b⊤λi),v)U=R(∂g∂u∗λ,v)U.

Now, for :

 (15) ∂L∂xrzr=∂f∂xrzr+R(λ,∂g∂xzr)¯¯¯¯Y+(μ,∂h∂xrzr)W=∂f∂xrzr+R(∂g∂x∗λ,zr)¯¯¯¯X+(∂h∂xr⊤μ,zr)¯¯¯¯X=(∂f∂xr+R(∂g∂x∗λ)+∂h∂xr⊤μ,zr)¯¯¯¯X

and for all :

 (16) ∂L∂xizi=∂f∂xizi+R(λ,∂g∂xiizi)¯¯¯¯Y+(μ,∂h∂xizi)W=∂f∂xizi−R(i∂g∂xi∗λ,zi)¯¯¯¯X+(∂h∂xi⊤μ,zi)¯¯¯¯X=∂f∂xizi+I(∂g∂x∗λ,zi)¯¯¯¯X+(∂h∂xi⊤μ,zi)¯¯¯¯X=(∂f∂xi+I(∂g∂x∗λ)+∂h∂xi⊤μ,zi)¯¯¯¯X.

The result follows. ∎

###### Remark 2.2.

Not surprisingly, we obtain the same optimality system as if we had represented the constraint as an element of the product of real spaces. The advantage of the complex setting is to allow more compact formulas.

## 3. The abstract control problem in a semigroup setting

Given a complex Banach space , we consider optimal control problems for equations of type

 (17) ˙Ψ+AΨ=f+u(B1+B2Ψ);t∈(0,T);Ψ(0)=Ψ0,

where

 (18) Ψ0∈¯¯¯¯¯H;f∈L1(0,T;¯¯¯¯¯H);B1∈¯¯¯¯¯H;u∈L1(0,T);B2∈L(¯¯¯¯¯H),

and is the generator of a strongly continuous semigroup on , in the sense that, denoting by the semigroup generated by , we have that

 (19) dom(A):={y∈¯¯¯¯¯H;limt↓0y−e−tAytexists}

is dense and for , is equal to the above limit. Then is closed. Note that we choose to define and not its opposite as the generator of the semigroup. We have then

 (20) ∥e−tA∥L(¯¯¯¯H)≤cAeλAt,t>0,

for some positive and . For the semigroup theory in a complex space setting we refer to Dunford and Schwartz [27, Ch. VIII]. The solution of (17) in the semigroup sense is the function such that, for all :

 (21) Ψ(t)=e−tAΨ0+∫t0e−(t−s)A(f(s)+u(s)(B1+B2Ψ(s)))ds.

This fixed-point equation (21) is well-posed in the sense that it has a unique solution in , see [3]. We recall that the conjugate transpose of has domain

 (22) dom(A∗):={φ∈¯¯¯¯¯H∗;for some c>0:|⟨φ,Ay⟩|≤c∥y∥,for all y∈dom(A)},

with antiduality product . Thus, has a unique extension to a linear continuous form over , which by the definition is . This allows to define weak solutions, extending to the complex setting the definition in [6]:

###### Definition 3.1.

We say that is a weak solution of (17) if and, for any , the function is absolutely continuous over and satisfies

 (23) ddt⟨ϕ,Ψ(t)⟩+⟨A∗ϕ,Ψ(t)⟩=⟨ϕ,f+u(t)(B1+B2Ψ(t))⟩,for a.a. t∈[0,T].

We recall the following result, obvious extension to the complex setting of the corresponding result in [6]:

###### Theorem 3.2.

Let be the generator of a strongly continuous semigroup. Then there is a unique weak solution of (23) that coincides with the semigroup solution.

So in the sequel we can use any of the two equivalent formulations (21) or (23). The control and state spaces are, respectively,

 (24) U:=L1(0,T);Y:=C(0,T;¯¯¯¯¯H).

For we set . Let be given and solution of (17). The linearized state equation at , to be understood in the semigroup sense, is

 (25) ˙z(t)+Az(t)=^u(t)B2z(t)+v(t)(B1+B2^Ψ(t));z(0)=0,

where It is easily checked that given the equation (25) has a unique solution denoted by , and that the mapping from to is of class , with .

The results above may allow to prove higher regularity.

###### Definition 3.3 (Restriction property).

Let be a Banach space, with norm denoted by with continuous inclusion in . Assume that the restriction of to has image in , and that it is a continuous semigroup over this space. We let denote its associated generator, and the associated semigroup. By (19) we have that

 (26) dom(A′):={y∈E;limt↓0e−tAy−ytexists}

so that , and is the restriction of to . We have that

 (27) ∥e−tA′∥L(E)≤cA′eλA′t

for some constants and . Assume that , and denote by the restriction of to , which is supposed to have image in and to be continuous in the topology of , that is,

 (28) B1∈E;B′2∈L(E).

In this case we say that has the restriction property.

### 3.1. Dual semigroup

Since is a reflexive Banach space it is known, e.g. [40, Ch. 1, Cor. 10.6], that generates another strongly continuous semigroup called the dual (backward) semigroup on , denoted by , which satisfies

 (29) (e−tA)∗=e−tA∗.

The reference [40] above assumes a real setting, but the arguments have an immediate extension to the complex one. Let be solution of the forward-backward system

 (30) {(i)˙z+Az=az+b,(ii)−˙p+A∗p=a∗p+g,

where

 (31) ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩b∈L1(0,T;¯¯¯¯¯H),g∈L1(0,T;¯¯¯¯¯H∗),a∈L∞(0,T;L(¯¯¯¯¯H)),

and for a.a. , is the conjugate transpose operator of , element of .

The solutions of (30) in the semigroup sense are , , and for a.a. :

 (32)

The following integration by parts (IBP) lemma follows:

###### Lemma 3.4.

Let satisfy (30)-(31). Then,

 (33) ⟨p(T),z(T)⟩+∫T0⟨g(t),z(t)⟩dt=⟨p(0),z(0)⟩+∫T0⟨p(t),b(t)⟩dt.
###### Proof.

This is an obvious extension of [3, Lemma 2.9] to the complex setting. ∎

## 4. First order optimality conditions

### 4.1. The optimal control problem

Let and be continuous quadratic forms over , with associated symmetric and continuous operators and in , such that and , where the operators and are self-adjoint, i.e.,

 (34) ⟨Qx,y⟩=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⟨Qy,x⟩for all x, y % in ¯¯¯¯¯H.

Observe that the derivative of at in direction is

 (35) Dq(y)x=2R⟨Qy,x⟩.

Similar relations for hold.

###### Remark 4.1.

The bilinear form associated with the quadratic form is

 (36) 12(q(x+y)−q(x)−q(y))=R⟨Qx,y⟩.

Then

 (37) I⟨Qx,y⟩=R(−i⟨Qx,y⟩)=R⟨Qx,iy⟩=12(q(x+iy)−q(x)−q(iy)).

Given

 (38) Ψd∈L∞(0,T;¯¯¯¯¯H);ΨdT∈¯¯¯¯¯H,

we introduce the cost function, where and , assuming that if :

 (39) J(u,Ψ):=∫T0(α1u(t)+12α2u(t)2)dt+12∫T0q(Ψ(t)−Ψd(t))dt+% 12qT(Ψ(T)−ΨdT)

The costate equation is

 (40) −˙p+A∗p=Q(Ψ−Ψd)+uB∗2p;p(T)=QT(Ψ(T)−ΨdT).

We take the solution in the (backward) semigroup sense:

 (41)

The reduced cost is

 (42) F(u):=J(u,Ψ[u]).

The set of feasible controls is

 (43) Uad:={u∈U;um≤u(t)≤uMa.e.% on [0,T]},

with given real constants. The optimal control problem is

Given , let denote the solution in the semigroup sense of

 (44) ˙y(t)+Ay(t)=f(t),t∈(0,T),y(0)=y0.

The compactness hypothesis is

 (45) {For given y0∈H, the mapping f↦B2y[y0,f]is compact from L2(0,T;¯¯¯¯¯H) to L2(0,T;¯¯¯¯¯H).
###### Theorem 4.2.

Let (45) hold. Then problem (P) has a nonempty set of solutions.

###### Proof.

Similar to [3, Th. 2.15]. ∎

We set

 (46) Λ(t):=α1+α2^u(t)+R⟨p(t),B1+B2^Ψ(t)⟩.
###### Theorem 4.3.

The mapping is of class from to and we have that

 (47) DF(u)v=∫T0Λ(t)v(t)dt,for all v∈U.
###### Proof.

That and are of class follows from classical arguments based on the implicit function theorem, as in [3]. This also implies that, setting and :

 (48) DF(u)v=∫T0(α1+α2u(t))v(t))dt+∫T0R⟨Q(Ψ(t)−Ψd(t)),z(t)⟩dt+R⟨QT(Ψ(T)−ΨdT),z(T)⟩.

We deduce then (47) from lemma 3.4. ∎

Let for and and be the associated contact sets defined, up to a zero-measure set, as

 (49) {Im(u):={t∈(0,T):u(t)=um},IM(u):={t∈(0,T):u(t)=uM}.

The first order optimality necessary condition is given as follows.

###### Proposition 4.4.

Let be a local solution of problem (P). Then, up to a set of measure zero there holds

 (50) {t;Λ(t)>0}⊂Im(^u),{t;Λ(t)<0}⊂IM(^u).
###### Proof.

Same proof as in [3, Proposition 2.17]. ∎

## 5. Second order optimality conditions

### 5.1. Technical results

Set Since , we have, in the semigroup sense:

 (51) ddtδΨ(t)+AδΨ(t)=^u(s)B2δΨ(s)+v(t)(B1+B2^Ψ(t)+B2δΨ(s)).

Thus, is solution of

 (52) ˙η(t)+Aη(t)=^uB2η(t)+v(s)B2δΨ(s).

We get the following estimates.

###### Lemma 5.1.

The linearized state solution of (25), the solution of (51), and solution of (52) satisfy, whenever remains in a bounded set of :

 (53) ∥z∥L∞(0,T;¯¯¯¯H) = O(∥v∥1), (54) ∥δΨ∥L∞(0,T;¯¯¯¯H) = O(∥v∥1), (55) ∥η∥L∞(0,T;¯¯¯¯H) = O(∥δΨv∥L1(0,T;¯¯¯¯H))=O(∥v∥21).
###### Proof.

Similar to the proof of lemma 2.18 in [3]. ∎

For solution of (17), the corresponding solution of (41), , and , let us set

 (56) Q(z,v):=∫T0(q(z(t))+α2v(t)2+2v(t)R⟨^p(t),B2z(t)⟩)dt+qT(z(T)).
###### Proposition 5.2.

Let belong to . Set , , . Then

 (57) F(u)=F(^u)+DF(^u)v+12Q(δΨ,v).
###### Proof.

We can expand the cost function as follows:

 (58) F(u)=F(^u)+12∫T0α2v(t)2+q(δΨ(t)))dt+12qT(δΨ(T))+∫T0(α1+α2^u(t))v(t)dt+R(∫T0⟨Q(^Ψ(t)−Ψd(t)),δΨ)⟩dt+⟨QT(^Ψ(T)−Ψd(T)),δΨ(T)⟩).

Applying lemma 3.4 to the pair , where is solution of the linearized equation (25), and using the expression of in (46), we obtain the result. ∎

###### Corollary 5.3.

We have that

 (59) F(u)=F(^u)+DF(^u)v+12Q(z,v)+O(∥v∥31),

where

###### Proof.

We have that

 (60) Q(δΨ,v)−Q(z,v)=R(∫T0⟨Q(δΨ(t)+z(t)),η(t)⟩+2v(t)⟨p(t),B2η(t)⟩dt)+R(⟨QT(δΨ(T)+z(T)),η(T)⟩).

By (53)-(55) this is of order of . The conclusion follows. ∎

### 5.2. Second order necessary optimality conditions

Given a feasible control , the critical cone is defined as

 (61) C(u):={v∈L1(0,T)| Λ(t)v(t)=0 a.e. on [0,T],v(t)≥0a.e. on Im(u),v(t)≤0 a.e. % on IM(u)}.
###### Theorem 5.4.

Let be a local solution of (P) and be the corresponding costate. Then there holds,

 (62) Q(z[v],v)≥0for all v∈C(^u).
###### Proof.

The proof is similar to the one of theorem 3.3 in [3]. ∎

### 5.3. Second order sufficient optimality conditions

In this subsection we assume that , and obtain second order sufficient optimality conditions. Consider the following condition: there exists such that

 (63) Q(z,v)≥α0∫T0v(t)2dt,for all % v∈C(^u).
###### Theorem 5.5.

Let satisfy the first order optimality conditions of (P), being the corresponding costate, as well as (63) Then is a local solution of problem , that satisfies the quadratic growth condition.

###### Proof.

It suffices to adapt the arguments in say [15, Thm. 4.3] or Casas and Tröltzsch [24]. ∎

Using the technique of Bonnans and Osmolovskiĭ [16] we can actually deduce from theorem 5.4 that is a strong solution in the following sense (natural extension of the notion of strong solution in the sense of the calculus of variations).

###### Definition 5.6.

We say that a control is a strong solution if there exists such that, if and , then .

In the context of optimal control of PDEs, sufficient conditions for strong optimality were recently obtained for elliptic state equations in Bayen et al. [9], and for parabolic equations by Bayen and Silva [10], and by Casas and Tröltzsch [24].

We consider the part of the Hamiltonian depending on the control:

 (64) H(t,u):=α1u+12α2u2+uR⟨^p(t),B(t)⟩,

where