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Equation and Related Equations
Armand Koenig
To cite this version:
Armand Koenig. Lack of Null-Controllability for the Fractional Heat Equation and Related Equations.
SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2020,
58 (6), pp.3130-3160. �10.1137/19M1256610�. �hal-01829289v2�
EQUATION AND RELATED EQUATIONS
2
ARMAND KOENIG† 3
Abstract. We consider the equation (∂t+ρ(√
−∆))f(t, x) =1ωu(t, x),x∈RorT. We prove it is not null- 4
controllable ifρis analytic on a conic neighborhood ofR+ andρ(ξ) =o(|ξ|). The proof relies essentially on geometric 5
optics, i.e. estimates for the evolution of semiclassical coherent states.
6
The method also applies to other equations. The most interesting example might be the Kolmogorov-type equation 7
(∂t−∂2v+v2∂x)f(t, x, v) =1ωu(t, x, v) for (x, v)∈Ωx×Ωvwith Ωx=RorTand Ωv=Ror (−1,1). We prove it is not 8
null-controllable in any time ifωis a vertical bandωx×Ωv. The idea is to remark that, for some families of solutions, 9
the Kolmogorov equation behaves like the rotated fractional heat equation (∂t+√
i(−∆)1/4)g(t, x) =1ωu(t, x), 10
x∈T. 11
Key words. null controllability, observability, fractional heat equation, degenerate parabolic equations 12
AMS subject classifications. 93B05, 93B07, 93C20, 35K65 13
1. Introduction.
14
1.1. Problem of the null-controllability. Consider the following equation, which is called
15
thefractional heat equation, where Ω =RorT,ω is an open subset of Ω,α≥0:
16
(∂t+ (−∆)α/2)f(t, x) =1ωu(t, x) t∈[0, T], x∈Ω
17
Here, we define (−∆)α/2with the functional calculus, that is, (−∆)α/2f =F−1(|ξ|αF(f)) if Ω =R,
18
whereF is the Fourier transform; andcn((−∆)−α/2f) =|n|αcn(f) if Ω =T, wherecn(f) is thenth
19
Fourier coefficient off.
20
It is a control problem with statef ∈L2(Ω) and controlusupported inω. More precisely, we
21
are interested in the exact null-controllability of this equation.
22
Definition 1.1. We say that the fractional heat equation is null-controllable onωin timeT >0
23
if for allf0 inL2(Ω), there existsuin L2([0, T]×ω)such that the solutionf of:
24
(1.1) (∂t+ (−∆)α/2)f(t, x) =1ωu(t, x) t∈[0, T], x∈Ω f(0, x) =f0(x) x∈Ω.
25
satisfies f(T, x, v) = 0for all (x, v)inΩ.
26
The main motivation for this study, apart from studying the fractional heat equation itself, is
27
the null-controllability of a Kolmogorov-type equation. More specifically, we are interested in the
28
following equation, where Ω = Ωx×Ωv with Ωx=R orT, Ωv =Ror (−1,1) andω is an open
29
subset of Ω:
30
(∂t+v2∂x−∂v2)f(t, x, v) =1ωu(t, x, v) t∈[0, T],(x, v)∈Ω.
31
∗This work was partially supported by the ERC advanced grant SCAPDE, seventh framework program, agreement no. 320845. This work is partially supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir”’s program of the Idex PSL reference «ANR-10-IDEX-0001-02 PSL».
†Université Paris-Dauphine, Université PSL, CNRS, CEREMADE, 75016 PARIS, FRANCE. (ar- mand.koenig@dauphine.psl.eu,https://koenig.perso.math.cnrs.fr).
1
For convenience, we will just say in this paper “the Kolmogorov equation”. Note that thanks to
32
Hörmander’s bracket condition [21, Section 22.2], the operatorv2∂x−∂v2 is hypoelliptic. Also, this
33
equation is well-posed. This can be proved by Hille-Yosida’s theorem (see [2, Section 4] in the case
34
Ω =T×(−1,1)). As we will see, this Kolmogorov equation is related to the rotated fractional heat
35
equation.
36
Definition 1.2. We say that the Kolmogorov equation is null-controllable onω in timeT >0
37
if for allf0 inL2(Ω), there exists uin L2([0, T]×ω)such that the solutionf of:
38
(1.2)
(∂t+v2∂x−∂v2)f(t, x, v) =1ωu(t, x, v) t∈[0, T],(x, v)∈Ω f(0, x, v) =f0(x, v) (x, v)∈Ω
f(t, x, v) = 0 t∈[0, T],(x, v)∈∂Ω (if non-empty)
39
satisfies f(T, x, v) = 0for all (x, v)inΩ.
40
1.2. Statement of the results. We will prove that the rotated fractional heat equation is
41
never null controllable if Ω\ω has nonempty interior, and that the Kolmogorov equation is never
42
null-controllable ifω=ωx×Ωv where Ωx\ωx has nonempty interior.
43
Theorem 1.3. Let0≤α <1and Ω =Ror Ω =T. Letω be a strict open subset ofΩ. The
44
fractional heat equation (1.1)is not null controllable in any time onω.
45
This Theorem still holds in higher dimension, with Ω = Rd ×Td
0, but our method seems
46
ineffective to treat the case where Ω is, say, an open subset ofR. This may be because we are using
47
the spectral definition of the fractional Laplacian, and our method might be adapted if instead we
48
used a singular kernel definition of the fractional Laplacian.
49
Actually, we prove the non-null controllability of a class of equations of the form (∂t +
50
ρ(√
−∆))f(t, x) =1ω(t, x).
51
Theorem 1.4. LetK >0, C={ξ∈C,<(ξ)> K,|=(ξ)|< K−1<(ξ)} andρ: C ∪R+→Csuch
52
that
53
1. ρis holomorphic on C,
54
2. ρ(ξ) =o(|ξ|)in the limit |ξ| →+∞,ξ∈ C,
55
3. ρis measurable on R+ andinfξ∈R+<(ρ(ξ))>−∞.
56
Let Ω =RorT,ω be a strict open subset of ΩandT >0. Then the equation
57
(1.3) (∂t+ρ(
√
−∆))f(t, x) =1ωu(t, x), t∈[0, T], x∈Ω
58
is not null-controllable onω in time T.
59
For lack of a better name, we will call the equation (1.3) thegeneralized fractional heat equation.
60
This Theorem can be generalized to the case Ω =Rd×Td
0. The hypothesis infR+<(ρ)>−∞is
61
only used to ensure that the equation is well-posed.
62
The fractional heat equation is the case ρ(ξ) = ξα. Note that if α= 0, then the fractional
63
heat equation is just a family of decoupled ordinary differential equations, and the conclusion of
64
Theorem 1.3is unimpressive. At the other end, the method used in this article does not work as-is
65
ifα= 1, but we still expect non-null-controllability, even if this remains a conjecture if Ω is not the
66
one-dimensional torus.
67
Some equations behave like the fractional heat equation, at least in some regimes. This is the
68
case of the Kolmogorov equation, and if the control acts on a vertical band, we will prove it is not
69
null-controllable with the same method.
70
Theorem 1.5. Let Ωx = R or T, let Ωv =R or (−1,1), and let Ω = Ωx×Ωv. Let T > 0
71
and ωx be a strict open subset of Ωx. The Kolmogorov equation (1.2) is not null-controllable on
72
ω=ωx×Ωv in time T.
73
This Theorem can be extended to higher dimension in xandv if Ωv =Rd. If we want, say
74
Ωv= (−1,1)d, we lack information on the eigenvalues and eigenfunctions of−∂v2+inv2on (−1,1)d,
75
but this is the only obstacle to the generalization of the Theorem to this case. We also give a
76
non-null-controllability result in small time for more general control region.
77
Theorem 1.6. Let Ωx = R or Ωx = T and Ωv = R or Ωv = (−1,1). Let Ω = Ωx×Ωv.
78
Let ω⊂Ω. Assume that there exist x0∈Ωx and a >0such that the symmetric vertical interval
79
{(x0, v),−a < v < a}is disjoint fromω. Then, the Kolmogorov equation(1.2)is not null-controllable
80
onω in timeT < a2/2.
81
Whether this conditionT < a2/2 is optimal or not is an open question, but we conjecture that
82
it is optimal, at least for some geometries. Ifω=T×(a, b) with 0< a < b,Theorem 1.6proves that
83
null-controllability does not hold in timeT < a2/2. This special case was already known [6, Theorem
84
1.3], it is also proved in the same reference that null-controllability holds for some T > 0. Our
85
Theorem 1.6sharpens the lower-bound on the minimal time of null-controllability if the geometry of
86
ω is different than a cartesian product.
87
While the fractional heat equation and the Kolmogorov equation are the main focus of this
88
article, the method can be used to treat other equations: those that behave like the fractional heat
89
equation for α <1. In Appendix A, we briefly discuss the fractional Schrödinger equation, and
90
sketch the proof for the Kolmogorov-type equation (∂t−∂v2−v∂x)f(t, x, v) =1ωu(t, x, v) (notice
91
the vinstead of the v2), and theimproved Boussinesq equation (∂t2−∂x2−∂x2∂t2)f(t, x) =1ωu(t, x).
92
1.3. Bibliographical comments.
93
1.3.1. Control of partial differential equations. Let Abe on operator on a Hilbert space
94
H such that the equation∂tf +Af = 0 is well-posed (i.e. −A generates a strongly continuous
95
semigroup of bounded linear operators on H, see for instance [34, Ch. 2] or [15, Sec. 2.3 and
96
Appendix A] for the definition).
97
Let U be a Hilbert space and B:U →H a bounded linear operator. With the right choice
98
of H, A, U andB, the problems we are interested in can be stated the following way: for every
99
f0∈H, does there existu∈L2(0, T;U) such that the solution of∂tf+Af=Bu,f(0) =f0satisfies
100
f(T) = 0?
101
For the fractional heat equation (1.1) on Rn, we choose H = L2(Rn), A = (−∆)α/2 with
102
domainHα(Rn),U =L2(ω) andB:u7→u1ω. For the Kolmogorov equation (1.2) onR2, we choose
103
H =L2(R2), A =−∂v2+v2∂x (the domain of A is a bit complicated to define, see [2, Sec. 4]),
104
U =L2(ω) andB:u7→u1ω.
105
Whether there exists a u∈L2(0, T;U) such that the solution of ∂tf +Af =Bu, f(0) =f0 106
satisfiesf(T) = 0 depends of course ofA, B and on the spacesH andU. Let us discuss existing
107
results whenAis a parabolic operator related to the fractional heat equation or the Kolmogorov
108
equation.
109
1.3.2. Null-controllability of the (fractional) heat equation in dimension one: the
110
moment method. Let us first look at the heat equation in dimension one with Dirichlet boundary
111
conditions, i.e. H =L2(0, π), D(A) =H2(0, π)∩H01(0, π) and A:f ∈D(A)7→ −∂x2f ∈L2(0, π).
112
Let us also denoteλn the eigenvalues ofA, and assume that λn is increasing, so thatλn=n2.
113
A possible strategy to control the heat equation in dimension one is to look for a control of the
114
form ˜u(t, x) =u(t)v(x). In the framework ofsubsection 1.3.1, this is the choiceU =RandButhe
115
function x∈(0, π)7→uv(x). We will call this kind of controlsshaped controls. This is the strategy
116
pioneered by Fattorini and Russel [18]. Let us describe it briefly.
117
Let v: (0, π) → R. Let f0 ∈ L2(0, π), let u ∈ L2(0, T) and let f be the solution of (∂t−
118
∂x2)f(t, x) =u(t)v(x),f(t,0) =f(t, π) = 0 with initial conditionf(0, x) =f0(x). Finally, for every
119
g∈L2(0, π), letcn(g) :=Rπ
0 g(x) sin(nx) dxbe then-th Fourier coefficient of g. Then, the relation
120
f(T,·) = 0 is equivalent to themoment problem
121
∀n∈N\ {0}, Z T
0
e(t−T)λnu(t) dt=−e−T λncn(f0) cn(v) .
122
Fattorini and Russel prove such auexists by constructing a biorthogonal family to (e−tλn)n∈N\{0},
123
i.e. a family of functions (gn)n∈N\{0}such thatRT
0 gn(t)e−λmtdt= 1 if n=mand 0 ifn6=m (see
124
also [33] for a more streamlined proof that this family exists). Then the functionudefined by
125
u(t) =−X
n>0
e−T λncn(f0)
cn(v) gn(T−t)
126
formally solves the moment problem. Moreover, we can prove some estimates on the functions
127
(gn)n>0, and if cn(v) does not decay too fast when n→ +∞, the series that defines uactually
128
converges.
129
This strategy can be adapted for the fractional heat equation (1.1) whenα >1, as Micu and
130
Zuazua [26] already remarked. Indeed, the construction and estimate on the biorthogonal family
131
relies on the hypotheses P
n>0|λn|−1<+∞andλn+1−λn≥c >0. These hypotheses still hold if
132
we replace the operatorA=−∂x2on (0, π) by (−∂x2)α/2as long asα >1. Indeed, the eigenvalues
133
are nowλn=nα.
134
On the other hand, ifα≤1, this proof does not work anymore. In fact, Micu and Zuazua [26,
135
Sec. 5] proved that ifα≤1, the fractional heat equation (1.1) is not null-controllable with controls
136
of the formu(t)v(x). Miller [27, Sec. 3.3] (see also [17, Appendix]) also gets similar results, with
137
similar methods.
138
But these negative results, based on Müntz Theorem, only concernscalar controls, i.e. the case
139
where the control space isU =R(orC). If the control space is larger, sayU =L2(ω), we cannot
140
rule out the existence of a control with Müntz Theorem. Indeed, there are many equations which
141
are not null-controllable with scalar controls, but that are null-controllable with a larger control
142
space. One of them is the heat equation in dimension larger than one. Let us discuss it now.
143
1.3.3. Null-controllability of the (fractional) heat equation and the spectral inequal-
144
ity. Let Ω be a bounded open subset ofRd. Let (λn)n≥0 be the sequence of the eigenvalues of
145
−∆ on Ω with Dirichlet boundary conditions.1 According to Weyl’s law, if the dimension d is
146
greater than 1, then P
n≥0λ−1n = +∞, so we cannot prove null-controllability with the moment
147
method. To build a control for the heat equation we have to choose a control space that is infinite
148
dimensional [15, Th. 2.79]. We chooseU =L2(ω) andB:u∈L2(ω)7→u1ω∈L2(Ω). We call this
149
kind of controlsinternal controls.
150
To prove the null-controllability of the heat equation in any dimension, Fursikov and Im-
151
mananuvilov [20] use parabolic Carleman inequalities, which are weighted energy estimates, to prove
152
1I.e.D(A) ={f∈H01(Ω), ∆f∈L2(Ω)}where ∆f is in the sense of distributions.
(more or less) directly the observability inequality that is equivalent to the null-controllability (see for
153
instance [15, Th. 2.44 or Th. 2.66] for the equivalence between null-controllability and observability).
154
Independently, Lebeau and Robbiano [25,23] developped another strategy to prove the null-
155
controllability of the heat equation. This strategy yields more insight for our purpose, so let us give
156
more details.
157
By means of elliptic Carleman inequalities, Lebeau and Robbiano proved a spectral inequality,
158
which is the following: letM be a connected compact riemannian manifold with boundary, letω
159
be an open subset of M, and let (φi)i∈N be an orthonormal basis of eigenfunctions of−∆ with
160
associated eigenvalues (λi)i∈N, then there existsC >0 andK >0 such that for every sequence of
161
complex numbers (ai)i∈Nand everyµ >0
162
(1.4)
X
λi<µ
aiφi
L2(M)≤CeK√µ
X
λi<µ
aiφi
L2(ω).
163
The key point to deduce the null-controllability of the heat equation from this spectral inequality
164
is that if one takes an initial condition of the form f0 = P
λi≥µaiφi with no component along
165
frequencies less thanµ, the solution of the heat equation decays likee−T µ|f0|L2(M), and the exponent
166
in µin this decay (i.e. 1) is larger than the one appearing in the spectral inequality (i.e. 1/2).2
167
For the fractional heat equation (1.1), the dissipation stays stronger that the spectral inequality
168
as long asα >1. Thus, forα >1, we can prove the null-controllability with Lebeau and Robbiano’s
169
method, as already mentioned by Micu and Zuazua [26] and Miller [27] (see also [28,17]).
170
Our Theorem 1.3 proves that the thresholdα > 1 is optimal: if α <1, then the fractional
171
heat equation is not null-controllable (at least for Ω =Tn). Note that the caseα= 1 and Ω =T
172
has already been proved to lack null-controllability with internal controls [22, Th. 4]. Let us also
173
mention an article where the null-controllability of an equation closely related to our fractional heat
174
equation have been investigated [11].
175
So it seems that the Lebeau-Robbiano method is in some sense optimal: if the dissipation is
176
not stronger than the spectral inequality, then we do not have null-controllability. Let us finish
177
with another class of parabolic equations for which Lebeau and Robbiano’s method does not work:
178
degenerate parabolic equations.
179
1.3.4. Null-controllability of degenerate parabolic partial differential equations. De-
180
generate parabolic equations are equations of the form∂tf(t, x) +Af(t, x) =1ωu(t, x), 0≤t≤T,
181
x∈Ω, whereAis a second-order differential operator which is degenerate elliptic, i.e. its principal
182
symbolP(x, ξ) satisfiesP(x, ξ)≥0 but is zero for somex∈Ω andξ6= 0.
183
The interest in the null-controllability of degenerate parabolic equations is more recent. We now
184
understand the null-controllability of parabolic equations degenerating at the boundary in dimension
185
one [12] and two [13] (see also references therein), where the authors found that these equations
186
where null-controllable if the degeneracy is not too strong, but might not be if the degeneracy is too
187
strong. To the best of our knwoledge, the only other general family of degenerate parabolic equations
188
whose null-controllability has been investigated is hypoelliptic quadratic differential equations [8,7].
189
Other equations have been studied on a case-by-case basis. For instance, the Kolmogorov
190
equation has been investigated since 2009 [9, 2, 6]. In these papers, the authors found that if
191
Ω =T×(−1,1) andω=T×(a, b) with 0< a < b <1, the Kolmogorov equation is null-controllable
192
in large times, but not in time smaller than a2/2, and that if −1 < a < 0 < b < 1, it is null-
193
controllable in arbitrarily small time [2]. If in the Kolmogorov equation (1.2) we replace v2 by
194
2The exponent 12 ofµin the spectral inequality is optimal ifωis a strict open subset ofM[23, Proposition 5.5].
v, the null-controllability holds in arbitrarily small time if ω = T×(a, b) [9, 2]. On the other
195
hand, if we replacev2 by vγ whereγ is an integer greater than 2 and ω =T×(a, b), it is never
196
null-controllable [6]. In this last article, the null-controllability of a model of the equation we are
197
interested in, namely the equation (∂t+iv2(−∆x)1/2−∂v2)g= 0, is also investigated.
198
Another degenerate parabolic equation is the Grushin equation (∂t−∂x2−x2∂y2)f(t, x, y) =
199
1ωu(t, x, y) on Ω = (−1,1)×(0, π) with Dirichlet boundary conditions. If the control domain
200
is a vertical band ω = (a, b)×(0, π) with 0 < a < b, there exists a minimum time for the null-
201
controllability to hold [4]. This minimum time has since been computed [5]. On the other end, if
202
the domain control is an horizontal bandω= (0, π)×(a, b) with (a, b)((0, π), then the Grushin
203
equation is not null controllable [22].
204
Let us finally just mention an article on the heat equation on the Heisenberg group since 2017 [3],
205
and that some parabolic equations on the real half-line, some of them related to the present work,
206
have been shown to strongly lack controllability [16].
207
1.4. Outline of the proof, structure of the article. As usual in controllability problems, we
208
focus onobservability inequalitieson theadjoint systems, that are equivalent to the null-controllability
209
(see [15, Theorem 2.44]).
210
Specifically, the null-controllability of the fractional heat equation (1.1) is equivalent to the
211
existence ofC >0 such that for every solutiong of
212
(1.5) (∂t+ (−∆)α/2)g(t, x) = 0 t∈(0, T), x∈Ω
213
we have
214
(1.6) |g(T,·)|L2(Ω)≤C|g|L2((0,T)×ω).
215
So, to disprove the null controllability, we only have to find solutions of (1.5) that are concentrated
216
outsideω. To construct such solutions, we consider initial states that are (essentially)semiclassical
217
coherent states, i.e. initial states of the formg0,h:x7→h−1/4e−(x−x0)2/2h+ixξ0/h. We will prove that
218
solutions of Eq. (1.5) with these initial conditions stay concentrated aroundx0. More precisely, we
219
get asymptotic expansion of these solutions thanks to the saddle point method. We do this informally
220
at first, insection 2, then rigorously in subsection 4.1in the case Ω =Rand insubsection 4.2in the
221
case Ω =T. This proof relies on some technical computations that are done insection 3. These
222
computations are carried over in a slightly general framework, that allows to directly treat the other
223
equations, namely the Kolmogorov-type equations and the improved Boussinesq equation. We also
224
sketch the proof of the generalization of Theorem 1.4in higher dimension insubsection 4.3.
225
Let us finish this introduction by explaining how the Kolmogorov equation for Ωx= Ωv=Rand
226
the fractional heat equation are related. The first eigenfunction of−∂2v+iξv2onR, is3e−
√
iξ v2/2(up to
227
a normalization constant), with eigenvalue √
iξ. So, Φξ: (x, v)∈R27→eiξx−
√
iξ v2/2 is a generalized
228
eigenfunction of the Kolmogorov operatorv2∂x−∂v2, with eigenvalue√
iξ. So, the solution of the
229
Kolomogorov equation (∂t+v2∂x−∂2v)f = 0 with initial conditionf(0, x, v) =R
Ra(ξ)Φξ(x, v) dξis
230
f(t, x, v) =R
Ra(ξ)Φξ(x, v)e−
√iξ tdξ. This suggests that, dropping thevvariable for the moment, the
231
Kolmogorov equation behaves like an equation where the eigenfunctions are theeiξx with eigenvalue
232 √
iξ, i.e. the equation (∂t+√
i(−∆x)1/4)f(t, x) = 0 withx∈R.
233
3Here and in all this paper, we choose the branch of the square root with positive real part.
Based on this observation and the non-null-controllability result of the rotated fractional heat
234
equation on the whole real line, we prove insubsection 5.2that the Kolmogorov equation is not null-
235
controllable in the case Ω =R×R. For Kolmogorov’s equation on Ω = Ωx×(−1,1), we need some
236
information on the eigenvalues and the eigenfunctions, which are not explicit anymore. We already
237
proved most of what we need in another article [22, Section 4]. We prove the non-null-controllability
238
of Kolmogorov equation with Ωv= (−1,1) insubsection 5.4.
239
Finally, we sketch the proof for the Kolmogorov equation with v instead of v2 and for the
240
improved Boussinesq equation inAppendix A.
241
2. Informal presentation of the proof. As we explained insubsection 1.4, we will try to
242
disprove the observability inequality (1.6). We only discuss here the case Ω =R.
243
Since the fractional heat equation is invariant by translation, we may assume thatω⊂ {|x|> δ}
244
for some δ >0. Then, forh >0, we consider the solutiongh of the fractional heat equation (1.5)
245
with initial conditiong0,h(x) =e−x2/2h+iξ0/h with someξ0>0. The solution of the fractional heat
246
equation is then
247
gh(t, x) = 1 2π
Z
R
e−t|ξ|α+ixξF(g0,h)(ξ) dξ,
248
whereF is the Fourier transform defined byFf(ξ) =R
Rf(x)e−ixξdx. But the Fourier transform of
249
g0,h has a closed-form expression. Indeed,F(e−x2/2)(ξ) =√
2πe−ξ2/2, and using the properties of
250
the Fourier transform (scaling and translation), we findF(g0,h)(ξ) =√
2πhe−(hξ−ξ0)2/2h. Thus
251
gh(t, x) = r h
2π Z
R
e−(hξ−ξ0)2/2h+ixξ−t|ξ|αdξ.
252
If we make the change of variablesξ0 =hξ, we find
253
(2.1) gh(t, x) = 1
√ 2πh
Z
R
e−(ξ−ξ0)2/2h+ixξ/h−t|ξ|α/hαdξ.
254
Notice that if his small, the term e−(ξ−ξ0)2/2h is concentrated around ξ =ξ0, and so is the
255
integrand. Thus, the major part of this integral comes from a neighborhood ofξ0. In this situation,
256
we can compute asymptotic expansion with the saddle point method.
257
More precisely, the saddle point method (see for instance [32, Ch. 2]) is a way to compute
258
asymptotic expansion of integrals of the formI(h) =R
eφ(x)/hu(x) dxin the limith→0+, whereφ
259
anduare entire functions.
260
If the main contribution in the integralI(h) comes from a nondegenerate critical point ofφat
261
x= 0, the “standard” saddle point method gives
262
(2.2) I(h)∼eφ(0)/hX
k
√
2πu˜2khk+1/2
263
where the ˜u2k are of the form A2ku(0), and A2k are differential operators of order 2k, with in
264
particular ˜u0=u(0)|φ00(0)|−1/2.
265
The term e−|ξ|α/hα prevents us from applying the saddle point method to Eq. (2.1) as-is, but
266
let us pretend we can do it anyway (the rigorous computation will be carried insection 3). Since
267
the critical point ofξ7→ −(ξ−ξ0)2/2 +ixξ isξc=ξ0+ix, we get from the saddle point method
268
gh(t, x)≈e−(ξc−ξ0)2/2h+ixξc/h−tξcα/hα=e−x2/2h+ixξ0/h−t(ξ0+ix)α/hα.
269
By integrating this asymptotic expansion, we get the following lower bound on the left-hand side of
270
the observability inequality (1.6):
271
(2.3) |gh(T,·)|2L2(R)≥ |gh(T,·)|L2(|x|<δ)≥ch−1 Z
|x|<δ
e−x2/h−Ch−αdx≥c0e−C0h−α.
272
Again by integrating the asymptotic expansion ongh, we get the following upper on the right-hand
273
side of the observability inequality
274
(2.4) |gh|L2([0,T]×ω)≤ |gh|2L2([0,T]×{|x|>δ})≤Ch−1 Z
[0,T]×{|x|>δ}
e−x2/h−cth−αdtdx≤C0e−δ2/h.
275
Comparing this upper bound (2.4) and the lower bound (2.3), and taking the limit h→0+, we see
276
that the observability inequality (1.6) cannot be true if α <1. Thus the fractional heat equation is
277
not null-controllable.
278
3. Some technical computations. Before we make rigorously the proof outlined insection 2,
279
we carry here some computations as a technical preparation.
280
3.1. Perturbation of the saddle point method. The “standard” saddle point method can
281
be stated in the following way [32, Th. 2.1].
282
Theorem 3.1. Let U be an open bounded neighborhood of 0 in Rd. For every N ∈N, there
283
existsCN >0such that for everyh >0and every holomorphic functionuon a complex neighborhood
284
V ofU,
285
Z
U
e−x2/2hu(x) dx=
N−1
X
j=0
hd/2+j
(2π)d/2j!2j(∆)ju(0) +RN(h),
286
where
287
|RN(h)| ≤CNλhd/2+N sup
z∈V
|u(z)|.
288
By using the Morse lemma (see for instance [21, Lemma C.6.1]), one can often transform integral of
289
the formR
eφ(x)/hu(x) dxinto integrals of the form ofTheorem 3.1, plus some exponentially small
290
error. Notice that in this theorem, the phase−x2/2 does not depend onh. However, to rigorously
291
treat the integral of Eq. (2.1), we need to allow the phase to depend onh.
292
Proposition 3.2 (Perturbation of the saddle point method). Let h0>0 and: (0, h0]→R+ 293
such that (h)→0ash→0. LetU ⊂R be an open interval around0. LetV be a complex open
294
bounded neighborhood ofU inC.
295
For every 0< h≤h0, letrh:V →C such that
296
1. ∀0< h≤h0,rh is holomorphic onV,
297
2. there existsC >0 such that for every0< h≤h0 andξ∈V,|rh(ξ)| ≤C(h).
298
For suchrh, we define|r|:= inf{C >0,∀0< h≤h0,∀ξ∈V,|rh(ξ)| ≤C(h)}.
299
For every 0< h≤h0, letuh be a holomorphic bounded function on V. We consider
300
Ih,r(u) :=
Z
U
e−ξ2/2h+rh(ξ)/huh(ξ) dξ.
301
Let M >0. We have uniformly in|r|< M, and uniformly in uh holomorphic bounded onV
302
Ih,r(u) =
√
2πherh(0)/h+O((h)2/h)
uh(0) +O (h+(h))|uh|L∞(V)
.
303
Proof of Proposition 3.2. The strategy is to seeϕh,r(ξ) =−ξ2/2 +rh(ξ) as the phase, and to
304
change the integration variable and the integration path to rewriteIh,r(u) in the formIh,r(u) =
305
R e−ξ2/2hvh(ξ) dξ, even if this change of variables depends onh, and then applyTheorem 3.1.
306
In this proof, M >0 is fixed. We also chooseV0⊂Cconvex such that4 U bV0 bV. Also, we
307
use the convention thatC denotes a constant, that depends only on,M,V andV0, but not onh
308
small enough,|r|≤M orξ∈V0, and that may be different each line.
309
Step 1: finding the critical point. We claim that forhsmall enough, for every|rh|≤M, there exists
310
a unique critical pointξc,h,r ofϕh,r inV0, and that this critical point is non-degenerate.
311
Indeed, ξ ∈ V0 is a critical point of ϕh,r if and only if ∂ξrh(ξ) = ξ. But for every5 ξ ∈V0,
312
|∂ξrh(ξ)| ≤ C|rh|L∞(V) ≤C|r|(h). Moreover, sinceV0 is convex, according to the mean value
313
inequality, forξ, ξ0 ∈V0, 0< h≤h0and|r|≤M,
314
|∂ξrh(ξ)−∂ξrh(ξ0)| ≤sup
V0
|∂ξ2rh||ξ−ξ0| ≤Csup
V
|rh||ξ−ξ0| ≤C|r|(h)|ξ−ξ0| ≤CM (h)|ξ−ξ0|.
315
Thus, if h is small enough such that CM (h) < 1, ξ 7→ ∂ξrh(ξ) takes its value in V0 and is a
316
contraction. Then, according to the contraction mapping theorem, there exists a unique ξc,h,r∈V0
317
such that∂ξrh(ξc,h,r) =ξc,h,r. This is the unique critical point ofϕh,r inV0.
318
Note that we have|ξc,h,r| ≤C|r|(h), whereC depends only on, M, V andV0. Also, the
319
critical value ch,r:=ϕh,r(ξc,h,r) satisfies
320
|ch,r−rh(0)|=
−ξ2c,h,r
2 +rh(ξc,h,r)−rh(0)
≤ |ξc,h,r|2
4 +|ξc,h,r|sup
V0
|∂ξrh| ≤C(h)2|r|.
321
Moreover, we have|∂ξ2ϕh,r(ξc,h,r) +I|=|∂ξ2rh(ξc,h,r)| ≤C|r|(h). So, ifhis small enough, the
322
critical pointξc,h,r is nondegenerate.
323
Step 2: change of variables. Now that we know where the critical point is, and what the critical
324
value of the phase is, we want to change the intergation variables to rewriteIh,r(u) in the form
325
Ih,r(u) =R
e−ξ2/2hu˜h(ξ) dξ.
326
We defineψh,r(ξ) onV0, forhsmall enough by
327
ψh,r(ξ+ξc,h,r) =ξ
2 Z 1
0
∂x2ϕh,r(sξ) ds 1/2
.
328
According to Taylor’s formula, for everyξ∈V0, we haveϕh,r(ξ) =ch,r+ψh,r(ξ)2/2.
329
So, by the change of variables/integration pathη=ψh,r(ξ), we have:
330
(3.1) Ih,r(u) =ech,r/h Z
ψh,r(U)
e−η2/2huh(ξ(η))dξ
dη dη=ech,r/h Z
ψh,r(U)
e−η2/2hu˜h(ξ(η)) dη,
331
where ˜uh(η) :=uh(ξ(η)) dξ/dη.
332
Note that according to the definition of ψh,r, forξ ∈ V0, we have |ψh,r(ξ)−ξ| ≤C(h)|r|.
333
Thus, we have for everyη ∈ψh,r(V0),|ψh,r−1(η)−η| ≤C(h)|r|. Thus, dη/dξ= 1 +O((h)|r|).
334
4We denoteAbBfor “Acompact andA⊂B”.
5We will frequently use the following standard consequence of the integral Cauchy formula: iffis holomorphic on D(z0, r), then|f(n)(z0)| ≤Cn,r|f|L∞(D(z0,r)).
Step 3: conclusion. So, by the standard saddle point method (Theorem 3.1):
335
(3.2) Ih,r(u) =ech,r/h√
2πh u˜h(0) +O h|˜uh|L∞(V)
=ech,r/h√
2πh uh(0) +O (h+(h))|uh|L∞(V) 336
and sincech,r/h=rh(0)/h+O((h)2/h) the Proposition is proved.
337
3.2. The framework: truncated coherent states. As we said in the introduction, our aim
338
is to prove the lack of null-controllability of several equations, that behave in some sense like the
339
fractional heat equation. However, treating these equations requires more precise estimates than the
340
fractional heat equation does.
341
To avoid making similar computations several times, we do them in a somewhat general
342
framework. This way, we will be able to treat the other equations (Kolmogorov, etc.) directly.
343
Hypothesis 3.3. We consider the following domain, constants and functions:
344
1. letK >0 andC={ξ∈C,<(ξ)> K,|=(ξ)|< K−1<(ξ)},
345
2. letξ0>0 large enough (for instanceξ0= 4(K+ 1)),
346
3. letδ >0 small enough such that for everyξ∈Randx∈R,|ξ−ξ0|< δ and|x|< δ implies
347
ξ+ξ0+ix∈ C,
348
4. letχ∈Cc∞(−δ, δ) such that 0≤χ≤1 andχ≡1 on a neighborhood of 0, say (−δ2, δ2),
349
5. let X be a topological space, and 0< h0 <1, and for everyγ ∈X and 0< h≤h0, let
350
ργ,h:C →Cbe a holomorphic function,
351
6. we assume that uniformly inγ∈X and 0< h≤h0,ργ,h(ξ) =o(|ξ|) in the limit|ξ| →+∞,
352
ξ∈ C,
353
7. finally, for every 0< h≤h0, we define
354
(h) := sup
|ξ|<δ,|x|<δ,γ∈X
h
ργ,h
ξ+ξ0+ix h
.
355
The goal of the next subsections is to get upper and lower bounds on the following integral,
356
where (vh) is a family of bounded holomorphic functions onC:
357
(3.3) Iγ,h,v(x) =
Z
R
χ(ξ−ξ0)e−(ξ−ξ0)2/2h+ixξ/h+ργ,h(ξ/h)vh(ξ) dξ.
358
Remark 3.4. 1. The hypothesis 6 implies that(h)→0 ash→0.
359
2. For instance, ifρis independent of γandh, withρ(ξ) =|ξ|αand 0≤α <1, then we have
360
C−1h1−α< (h)< Ch1−α for someC >0.
361
3. In the applications, we will typically choose X = [0, T] and fort ∈ X, ρt,h(ξ) = −tρ(ξ)
362
with some ρ:C → C such that ρ(ξ) = o(|ξ|). We will also usually choose vh = 1. In
363
that case, gh(t, x) := It,h,1(x) is solution of (∂t+ρ(√
−∆))gh = 1 with initial condition
364
g0,h(x) = √
2πhχ(−ih∂x−ξ0)e−x2/2h+ixξ0/h, which belongs in L2(R). However, some
365
applications will require a larger parameter spaceX andvh6= 1.
366
3.3. Asymptotics for the evolution of coherent states.
367
Proposition 3.5. Assuming Hypothesis 3.3, we have uniformly in γ∈X,|x| small enough,
368
