This approach is based on a Duhamel-type formula in the space of observables, here taken to be the space of continuous functions in the state space. In the remainder of this introduction, I will give a brief mathematical account of the results obtained in this thesis.
Quantitative uniform hydrodynamic limits
The convergence of local particle densities can be quantified in terms of weak convergence of the empirical measure. Finally, we will present a new argument which makes it possible to prove convergence of the microscopic entropy to a macroscopic entropy.
The nonlinear Schr¨ odinger equation
Existence theory with an angular momentum rotation term
To obtain global existence in the defocus case λ > 0, we change to a rotating coordinate system. It is an NLS with time-dependent potential and can be treated as such as in [18], where it was shown that global existence in the presence of time-dependent potential holds if λ >0.
Optimal control of nonlinear Schr¨ odinger equations
Throughout this chapter, H will be a subspace of the space M+(Td) of positive Radon measures on the torus. In Section 2.3, we present our method using the particularly easy case of independent random walks.
Previous results
The convergence of the relative entropy can thus be thought of as a stronger notion of the hydrodynamic limit. Therefore even if one manages to prove exponential decay in time of the relative entropy HN µNt |νfN.
A toy model: independent random walks
Thus, it appears that a quantitative estimate of the rate of convergence is available in the stronger form of the hydrodynamic limit given by the convergence of the entropy with respect to the local Gibbs state. Before proceeding with the proof of Theorem 2.3.1, let us collect some semigroups related to the evolution of the particle system and the boundary equation.
Hydrodynamic limit for zero range processes
The hydrodynamic limit
As before, Theorem 2.4.6 gives convergence in the hydrodynamic limit, conditional on the convergence of the initial data. The semigroup relations can be summarized as StN :P(XN)→P(XN) with double TtN :Cb(XN)→Cb(XN),.
Regularity of the limit equation
Moreover L∞(Td) is only the critical case, in which simple interpolation arguments do not give a bound on Dkft of the form of Lemma 2.4.13. The regularity of the filtering equation at d = 1,2 was known even before the famous results of de Giorgi and Nash on the older H¨ continuum of solutions of parabolic equations.
Consistency and stability
This is standard except for the time dependence through ft of the coefficients of the linearized equation (2.40). It remains to find a limit on R1 and R2 to complete the proof of the consistency result. Closer inspection of the proof shows that to prove the prior relationship between G∞ andTt∞, we simply need to use θ >0 in the stability result, Lemma 2.4.14.
Proof of the hydrodynamic limit
As in the proof of the hydrodynamic limit for independent random walks, Theorem 2.3.1, the term T1 is a measure of the difference of the particle semigroup and the limit semigroup at the observer level and is bounded (at fixed Ψ) by a consistency result on both generators. Furthermore, we have seen that this compliance estimate is carried along the boundary equation flow with the stability result. The last term T3 did not appear in the proof of Theorem 2.3.1 and measures the error due to the relaxation of the empirical measure.
Convergence of the entropy
Then, both the microscopic entropy and the time-averaged Fisher information converge to the corresponding macroscopic quantities. Given the hydrodynamic limit and Yau's results using the relative entropy method, we expect µNt to be close to the local Gibbs state νfN. Note that so far we have not proved that the rate of convergence of the microscopic entropy is uniform in time.
The replacement lemma
Proof of the replacement lemma from the block estimates
The two-block estimate estimates the error introduced by averaging over a smaller field of size l instead of a field of size N (with a weighted average given by χ). Recall that, similar to the earlier replacement of the integral over u ∈ Td by a discrete sum, the identity R. However, we are still missing one important ingredient to be able to present a proof of block estimates.
Equivalence of ensembles
We carefully monitor the dependence on the integer l and now prove the equivalence of the ensembles. This completes the proof of equivalence of ensembles in the case of limited densities. The proof is similar to the proof of equivalence of ensembles in the case of bounded densities, but relies on Assumption 1 (iii) to obtain estimates of the growth of moments η(x) under νρL.
Restriction to bounded particle configurations
The second term on the right-hand side of (2.81) can be evaluated from the entropy inequality (2.32), which gives The presence of the exponential function means that the integral of the exponential on the right-hand side can be evaluated with
Proof of the one block estimate
Due to the convexity and translation invariance of the Fisher information, it also holds that (2.85) DN fN|νρN. The two functional inequalities, the logarithmic Sobolev inequality and the Csisz'ar-Kullback-Pinsker inequality, were thus instrumental in the local substitution in the infinitesimal volume element Λl of the particle distribution from its local thermodynamic equilibrium with a clear error estimate. This error term together with Lemma 2.6.10 and the error term appearing in (2.96) give the estimate of a block.
Proof of the two blocks estimate
By construction, the Fisher information (2.99) is equivalent to the ZRP Fisher information on a box of side 4l + 2. The measure ν2,l,K is invariant with respect to this zero-range process and we again define the canonical Fisher Information. Furthermore, we have constructed our canonical Fisher information such that it is the canonical Fisher information of the null-range process on the square lattice Λy,l with both faces glued together.
First make the following conjecture about the propagation of higher regularity in Hk(Td) for general dimensions. Rate of convergence on the hydrodynamic boundary: Theorem 2.4.6 remains true if we change rHL to be . The difficulty lies in obtaining explicit limits for the propagation of the Sobolev norms inHk(Td).
Mathematical setting and main result
At this point it is not clear whether these additional constraints are only due to our test strategy, or whether they indicate an actual change in the behavior of the solutions of (3.4). In terms of physics, the latter would correspond to the case where the spin is stronger than the trap and thus a similar behavior (at least qualitatively) to the "free" case would be expected, i.e. This chapter is now organized as follows. : Section 3.3 is devoted to the proof of Assertion (1) of Theorem 3.2.3.
Local and global existence
Finally, to prove the augmentation alternative, we first note that the local existence argument above can be repeated as long as askψ(t)kΣremains bounded. For quadratic potentials of the form (3.2), the Strichart estimates can be obtained explicitly by referring to the generalization of Mehler's formula for the kernel S(t), cf. In this case, the linear part of the energy appears to be the sum of non-negative terms, and the global existence can be inferred as in the case of NLS with a quadratic constraint [17].
Finite time blow-up
Note that the conditions for the initial data energy ψ0 are not identical in both cases. To use this piece of information, we first add and subtract from (3.29) a multiple of the angular momentum LΩ(t), i.e. Under the conditions of the theorem, we obtain I(T)≤0 with another integration with respect to time, thus introducing a contradiction.
Numerical simulations of a rotating Bose-Einstein condensate
Physics background
The aim of the current paper is to consider quantum control systems within the framework of optimal control, cf. The goal of the control process is thereby quantified by means of an objective functional J = J (ψ, α), which is minimized provided that the time evolution of the quantum state is governed by the Gross-Pitaevskii equation (GPE). Furthermore, cost terms based on, for example, the L2 norm of α tend to yield highly oscillatory optimal controls due to the oscillatory nature of the underlying (nonlinear) Schrödinger equation.
Mathematical setting
In the present work, we will present a new choice for the cost term, which is based on the corresponding physical work performed by the control process. It therefore seems natural to include such a term in the cost functional of our problem to quantify the control action. A typical choice for A would be A=A0−awherea∈Rise some prescribed expectation value for the observableA0 in the state ψ(T, x).
Relation to other works and organization of the chapter
Throughout this chapter we will denote strong convergence of a sequence (xn)n∈N with xn → x and weak convergence with xn * x. For simplicity, we will often write ψ(t) ≡ ψ(t,·) and also use the shorthand LptLqx instead of Lp(0, T;Lq(Rd)).
Existence of minimizers
Convergence of infimizing sequences
By reflexivity of L2(0, T; Σ), we consequently deduce the existence of a sequence (denoted by the same symbol) such that. On the other hand, the first term on the right is bounded by C/R2, since (ωn)n∈N is bounded in Σ by assumption. Given any > 0, we therefore first choose R > 0 large enough so that the first term is bounded by and then n large enough so that the second term is bounded by to show that ωn → ω in L2(Rd) as n .
Minimizers as mild solutions
The limit in L∞(0, t;L2σ+2(Rd)) then follows from the uniform-in-time limit inH1(Rd) and the Gagliardo-Nirenberg inequality (4.17). We have thus shown that the second term on the right-hand side of (4.19) vanishes in the limit n→. For the remaining term, we use the fact that V ∈L∞(Rd) and H¨older's inequality to obtain that.
Lower semicontinuity of objective functional
The first term on the right-hand side of (4.24) is convex in αn and thus satisfies. Finally, the cost term involving γ2 is lower semicontinuous due to the convexity and weak convergence of αn in H1 (0, T). A possible way around this problem would be to assume that A is positive definite, which, however, is not true for general observers of the form A =A0 -a, with a ∈ R.
Derivation and analysis of the adjoint equation
- Identification of the derivative of J(ψ, α)
- Local and global existence theory for solutions of higher regularity . 167
- Lipschitz continuity with respect to the control
- Proof of differentiability and characterization of critical points
Now we are ready to prove the Lipschitz continuity of the solution ψ(α) with respect to the control parameter α∈H1(0, T) over the entire control interval [0, T]. This shows that the first term in the second line has the form given in (4.4.2). Here the second term on the right is linear of (˜α−α) and therefore of the desired form.
Numerical simulation of the optimal control problem
Gradient-related descent method
Once an appropriate solver for the state and the adjoint equations is at hand, our gradient-related descent scheme works as follows. With this choice of a descent direction, we then perform a line search to determine the length of the step taken along δak. Of course, more elaborate strategies based on interpolation or alternative line search criteria are possible; see for example [69] for more details.
Newton method
Consequently, to calculate the Hessian function, it is necessary to solve several linearized Schr¨odinger-type equations with different source terms and boundary data. The choice of ∂α2J(ψ(αk), αk) is easy to implement and can be expected to contain some features of the full Hessian Dα2J, making the inversion problem for the MINRES algorithm better conditioned. 2) Here we chose the MINRES algorithm over alternatives such as the conjugate gradient (CG) method because the Hessian D2αJ is symmetric but not necessarily positive definite. The match between the scalar product in dual space and the preconditioner of the MINRES algorithm was investigated in [35].
Numerical examples
The spatial density ρ = |ψ|2 of the corresponding solution is shown in the right plot in Figure 4.2. A direct comparison of the (spatial densities of the) resulting wave functions and the respective controls is given in Figure 4.4. In the linear case, the widest extension of the wave packet is always comparable to its final value.
Optimal bilinear control of Gross-Pitaevskii equations
In future work, we would like to numerically test the inflation conditions of Theorem 3.2.3. Finally, we would like to mention that it is possible to extend our results (with some technical effort) to the case of focusing of non-linearities,λ <0, provided σ < 2/d. The latter prohibits the appearance of finite-time blow-up in the dynamics of the Gross-Pitaevskii equation.
Uniform quantitative hydrodynamic limits
Gauthier, Optimal control of the Schr¨odinger equation with two or three levels, in Nonlinear and Adaptive Control (Sheffield, 2001), vol. Boscain, Controllability of the Schr¨odinger equation with a discrete spectrum driven by an external field, Ann. Glassey, On the augmentation of solutions of the Cauchy problem for nonlinear Schr¨odinger equations, J .
Embedding T d N into T d yields lattice sites of distance 1/N . In the limit, we
Contour plots of the density function |ψ(t, x)| 2 for dynamics of a vortex
Contour plots of the density function |ψ(t, x)| 2 for dynamics of a vortex
Shifting a linear wave paket
Direct comparison between results
The weight factor ω = R
Value of J(ψ, α) over number of iterations
