Let us now list a few questions which, to the best of our knowledge, are still open and seem worth investigating.
• The qualitative properties of time optimal controls:
As suggested in [33] one might try to optimize the control with respect to the controllability time. Indeed, its is known that, under constraints on the control, parabolic equations have a minimal controllability time, see for instance [40,32].
For constrained controllability it is known that there exists a minimal controllability time to control, for instance, from 0 to θ (see [33]). We may try to optimize the control strategies so as to minimize the controllability time. In our case, that is, the spatially heterogeneous case, are these controls of bang-bang type? Another qualitative question that is relevant in this context is that of symmetry: in the one dimensional case, when working on an interval [−L, L], are time-optimal controls symmetric? In the multi-dimensional case, when the domain Ω is a ball, is it possible to prove radial symettry of time optimal controls?
• The influence of spatial heterogeneity on controllability time:
Adding a drift (which corresponds to the spatially heterogeneous model) modifies the con-trollability time. As we have seen, such heterogeneities might lead to a lack of concon-trollability.
However, it is also suggested in the numerical experiments shown below that adding a drift might be beneficial for the controllability time. It might be interesting to consider the fol-lowing question: given L∞and L1 bounds on the spatial heterogeneity N , which is the drift yielding the minimal controllability time? In other terms: how can we design the domain so as to minimize the controllability time? In the simulation below, we thus considered the following optimization problem: letting, for any drift N , T (N ) be the minimal controllability time from 0 to θ of the spatially heterogeneous equation (2) (with T (N ) ∈ (0; +∞]), solve
inf
−M 6M61 ,RL
−LN =0
T (N ).
We obtain the following graph with M = 250 and L = 2.5:
-2.5 2.5
Space -250
-200 -150 -100 -50 0 50 100 150 200 250
Optimal drift
Figure 11: Time optimal spatial heterogeneity.
A Proof of Lemma 1
Proof of Lemma1. Let us first remark that (13) has a variational structure. Indeed, u is a solution of
−∆u + εh∇n , ∇ui = f (u) , u ∈ W01,2(Ω) if and only if
− ∇ · (eεn∇u) = f (u)eεn, u ∈ W01,2(Ω). (39)
Following the arguments of [6, Remark II.2], we introduce the energy functional associated with
From standard arguments in the theory of sub and super solutions [6], if there exists v ∈ W01,2(Ω) such that
E1(v) < 0 (40)
then there exists a non-trivial solution to (13). We now prove that there exists v ∈ W01,2(Ω) such that (40) holds, by adapting the construction of [35]: let B(x; ρΩ) be one of the ball of maximum radius inscribed in Ω. Up to a translation, we assume that x = 0.
Let δ > 0. We define vδ as follows for some constant C > 0, and
Z
Ω
F (v) = F (1)ρdΩ+ O
δ→0(ρd−1Ω ).
Hence, since n is bounded, the conclusion: as ρΩ→ ∞ ad δ → 0 the energy of v1is negative.
Acknowledgment. The authors wish to thank B. Geshkovhki for his numerous comments and P. Lissy for fruitful conversations.
This project has received funding from the European Union?s Horizon 2020 research and inno-vation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex. The work of the authors has been funded by the Alexander von Humboldt-Professorship program, the European Research Council (ERC) under the European Union’s Horizon 2020 research and in-novation programme (grant agreement No. 694126-DyCon), grant MTM2017-92996 of MINECO (Spain), ELKARTEK project KK-2018/00083 ROAD2DC of the Basque Government, ICON of the French ANR and ”Nonlocal PDEs: Analysis, Control and Beyond”, AFOSR Grant FA9550-18-1-0242.
I. Mazari was partially supported by the ANR Project ANR-18-CE40-0013 - SHAPO on Shape Optimization.
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