4.5 Numerical simulation of the optimal control problem
4.5.3 Numerical examples
In all our examples, we choose the numerical domain Ω = [−L, L] withL= 20 and periodic boundary conditions. The number of spatial grid points is N = 256. In addition, we set the final control time to beT = 10, and we useM = 1024 equidistant time steps. In order to avoid the influence of the boundary, we choose a trapping potential U(x) = 30 Lx2
. The initial guess for the control is taken to be just α1 ≡ 0 in the linear case (λ = 0), whereas each algorithm in the nonlinear case (λ6= 0) is started from the control obtained by solving the linear problem. In our tests of the first-order gradient method, we choose TOL = 10−8 in the terminating condition (4.58) for the whole algorithm, µ = 10−3, and a maximum number of 20000 iterations. For the Newton method, we likewise set TOL = 10−8 and we stop the algorithm after at most 45 Newton steps.
Example: shifting a linear wave packet
For validation purposes, we consider the time-evolution of alinearwave packet, i.e.λ= 0, whose center of mass we aim to shift towards a prescribed point y1 ∈ [−L, L]. For this
Algorithm 1: Preconditioned MINRES algorithm
Given δkα0, set r0 :=−Aδkα0−J0(αk),z1 =Rr0, β1 =hr0, z1i12, z1 = zβ1
1,v1 = βr0
1, γ0 =γ1 = 1, σ0 =σ1 = 0, η=β1 foreach `= 1,2, . . . do
if η <TOLM IN RES then
Stop,δαk :=δα`k is the solution to (4.60).
else
Set µ` =hz`,Rz`i,
v`+1 =Az`−β`v`−µ`v`, z`+1 =Rv`+1,
β`+1 =hz`+1,Rz`+1i12, z`+1 = βz`+1
`+1, v`+1 = vβ`+1
`+1,
ρ0 =γ`µ`−σ`γ`−1β`, ρ1 =q
ρ20+β`+12 , ρ2 =σ`µ`+γ`−1γ`β`, ρ3 =σ`−1β`,
γ`+1 = ρρ0
1, σ`+1 = β`+1ρ
1 , w`+1 = ρ1
1 z`+1−ρ3w`−1 −ρ2w` , x`+1 =x`+γ`+1ηw`+1,
η=−σ`+1η end
end
purpose consider a control potential V(x) = 3
10+ 3x
200 ≥0, ∀x∈[−L, L], and the observable
A(x) = 1−e−(κ(x−y1))2/L2.
In this case, we find that the algorithm converges well even if we only invoke the first order gradient method. Indeed, as we decrease the regularization parameters γ1, γ2 1, we approach an optimal solution which, as it seems, cannot be improved upon. This optimal solution, or, more precisely, its spatial density ρ = |ψ|2, is depicted in Figure 4.1 (right plot), where we denote by “target” the function proportional to 1−A(x) with κ= 0.07 and y1 = −2L/8, such that it has the same L2–norm as ψ0. The left plot shows the associated control.
Since this solution seems optimal, the choice ofγ1, γ2 becomes negligible below a certain threshold. Thus, it suffices to considerγ1 = 0 and only include the cost term proportional toγ2. Similar results hold for any other given pointy1 ∈Ω, providedy1 stays sufficiently
Figure 4.1: Shifting a linear wave paket far away from the boundary.
Example: splitting a linear wave paket
We still consider the linear case, i.e., λ = 0, and aim to split a given initial wave packet into two separate packets centered around y1 and y2, respectively. The control potential is chosen as
V(x) =e−8x2/L2 ≥0, and the observable
A(x) = 1−
e−(κ(x−y1))2/L2 +e−(κ(x−y2))2/L2 .
In the following we fix κ = 0.07, y1 =−2L/8, and y2 = 2L/8. In this case we find that the residual of the first order gradient method does not drop below the tolerance given in (4.58) before the maximum number of iterations is reached. With the Newton method, however, we find a (local) minimum of the objective functional J(ψ, α) in less than 20 Newton iterations. Of course there is no guarantee that this is a global minimum.
In order to illustrate our results, we consider the case where γ1 = 0, γ2 = 1.5×10−6. At the final control time T = 10 we then obtain:
hAψ(T), ψ(T)i2L2
x ≈2.261×10−3.
The spatial density ρ = |ψ|2 of the corresponding solution is shown in the right plot of Figure 4.2. The associated control is depicted in the left plot. If, instead, we choose γ1 = 4×10−5, γ2 = 1×10−9, we find
hAψ(T), ψ(T)i2L2
x ≈2.269×10−3,
Figure 4.2: Splitting a linear wave paket with γ1 = 0
and the corresponding solution is given in Figure 4.3. Here the intermediate state is a plot of ρ(t) at t= 4 = 0.4×T.
Figure 4.3: Splitting a linear wave paket with γ1 >0
A direct comparison of the (spatial densities of the) resulting wave functions and the respective controls is given in Figure 4.4. We see that the spatial densities are nearly identical, but the variability of the respective control parameters is not the same. This is, of course, related to time–evolution of the weight factor ω(t), defined in (4.9), which is shown in Figure 4.5 for the case of γ1 = 4×10−5 and γ2 = 1×10−9.
By construction, the time–integral of ω(t) can be interpreted as the physical work performed during the control process. We find that compared to the case γ1 > 0, the term kE(·)k2L2
t is around 30% larger (64.5 versus 49.1) and kE(·)k˙ 2L2
t is around twice as large (95.0 versus 43.4) in the case where γ1 = 0, yielding a significant advantage of our control cost over terms considering theH1-norm only; see [40] for the latter.
Finally, Figure 4.6 shows an example of the evolution of the objective functional J(ψ, α)
Figure 4.4: Direct comparison between results
Figure 4.5: The weight factor ω=R
V|ψ|2dx over time
over the number of iterations of the Newton method, here for the case where γ1 = 0.
Example: splitting a Bose–Einstein condensate
We consider the same situation as in the previous example, but with an additional (cubic) nonlinearity. More precisely, we choose λ= 8 >0. It turns out that the conclusions are similar to the ones found in the linear case (λ = 0). Qualitatively, the main difference is that during the time–evolution, the wave function spreads out more because of the additionally repulsive (defocusing) nonlinearity. In the linear case, the widest extension of the wave packet is always comparable to its final value. Choosing as beforeγ1 = 4×10−5 and γ2 = 1×10−9, we obtain the solution depicted in the right plot of Figure 4.7, where we show the spatial density at the times t= 0, t =T = 10 and at the intermediate time t= 4. The control is shown in the left plot. In comparison to the linear case (λ= 0), the
Figure 4.6: Value of J(ψ, α) over number of iterations
observable term in the objective functionalJ(ψ, α) is found to be slightly larger. Indeed, we obtain
hAψ(T), ψ(T)i2L2
x ≈3.720×10−3.
This seems to indicate that nonlinear effects counteract the influence of the control po- tential.
Figure 4.7: Splitting a condensate with γ1 >0
We again compare the present case with the one where γ1 = 0 (i.e. no cost term propor- tional to the physical work) andγ2 = 1.5×10−6. First, we find that
hAψ(T), ψ(T)i2L2
x ≈3.382×10−3. Moreover,kEk˙ 2L2
t is about 150% larger (172.1 versus 68.5) than in the case whereγ1 6= 0.
Similarly, the total energy kEk2L2
t is around 15% larger (91.8 versus 79.5).
Example: splitting an attractive Bose–Einstein condensate
Our numerical method allows us to go beyond the rigorous mathematical theory developed in the early chapters. In particular we may try to control the behavior of attractive condensates, which are modeled by (4.1) with λ < 0, i.e. a focusing nonlinearity. Here we choose λ=−1, whereas the parameters γ1 = 4×10−5,γ2 = 1×10−9 are the same as before. The results are shown in Figure 4.8 (control in the left plot and the state at times t= 0,10,4 in the right plot). The observable part of the objective functional satisfies
hAψ(T), ψ(T)i2L2
x ≈2.143×10−3.
Figure 4.8: Splitting a focusing condensate with γ1 >0
In comparison to the case of a repulsive (defocusing) nonlinearity the final value for the observable term hAψ(T), ψ(T)i2L2
x is much smaller, confirming the basic intuition that an attractive condensate does not tend to spread out as much as in the repulsive case.
Chapter 5
Concluding remarks and future work
5.1 On the Cauchy-problem of nonlinear Schr¨ odinger equations with angular momentum rotation term
In Chapter 3, we have investigated local and global existence for the nonlinear Schr¨odinger equation with angular momentum rotation term, generalizing earlier results in the liter- ature [37, 38]. As we have seen there, equation (3.4) can be considered (upon a change of coordinates) as a special case of NLS with time-dependent potentials (sub-quadratic in x). This class of models has recently been studied in [18]. Following the arguments given therein, one could infer global in-time existence of (3.4) forsufficiently small initial dataψ0 ∈Σ, regardless of the sign of the nonlinearity. Moreover, growth rates for higher order (weighted) Sobolev norms can also be obtained as in [18]. In addition, we note that for a repulsive, isotropic quadratic potential V(x) = −γ22|x|2, the time-dependent change of coordinates is trivial and we could henceforth conclude global in-time existence for sufficiently largeγ >0 by following the arguments given in [16].
We also want to point out that for the usual NLS withσ = 2/dthere is an extra symmetry which has been successfully deployed in the study of blow-up (yielding explicit blow-up solutions and blow-up rates), see e.g. [76]. Using the so-called Lens transform [45] one can transfer (most of) these results to the case of NLS with isotropic time dependent quadratic potential W(t, x) = γ(t)|x|2, see [18]. However, it is argued in [18] that such an approach is only feasible in the case of isotropic potentials and thus, we cannot expect from it any further insight on the possibility of blow-up in our case, when (L·Ω)V(x)6= 0 and |Ω|> γ.
Finally, it is worth noting that the effect of the angular momentum rotation term in our model is very different from other situations. For example, it has been shown for the Euler equations with Coriolis force that blow-up can be delayed through a sufficiently
strong rotation term [58] (see also as [4] for a related result). Clearly, the situation in our model is much more involved, and we can not expect an analogous result to be true (the counterexample being the case where V(x) is axially symmetric).
In future work, we would like to numerically test the blow-up conditions of Theorem 3.2.3.
In particular, we are intested in establishing whether the conditions are due to technical difficulties or present real obstacles to blow-up.