Comptes Rendus Mathématique

Safari Mukeru

Some applications of the Menshov–Rademacher theorem

Youpei Zhang

Xianhua Tang

Vicenţiu Rădulescu

Anisotropic Choquard problems with Stein–Weiss potential: nonlinear patterns and stationary waves

Duo Li

Comptes Rendus
Mathématique
Duo Li
Projective bundles and blowing ups
Volume 359, issue 9 (2021), p. 1129-1133
<https://doi.org/10.5802/crmath.249>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 9, p. 1129-1133
https://doi.org/10.5802/crmath.249
Algebraic geometry / Géométrie algébrique
Projective bundles and blowing ups
Duo Lia
a School of Mathematics(Zhuhai), Sun Yat-sen University, Zhuhai, 519082,
Guangdong, China.
E-mail: liduo5@mail.sysu.edu.cn
Abstract. We study the blowing up X̃ of a smooth projective variety X along a smooth center B that is
equipped with a projective bundle structure over a variety Z . If B is a point, then X is a projective space.
If the Picard number ρ(X ) is 1, then dim Z has a lower bound dim X −dimB − 1. Moreover, when dim Z is
dim X −dimB −1, X is a projective space and B is a linear subspace in X . If X is a projective space Pn and B
is a curve, then either n is 3 and B is a twisted cubic curve or n is an arbitrary integer and B is a line in Pn . If
X is a quadric Qn and B is a curve, then n is 3 and B is a line in Q3.
Résumé. Nous étudions l’éclatement X̃ d’une variété projective lisse X le long d’un centre lisse B , munie
d’une structure de fbré projectif. Si B est un point, X est un espace projectif. Si le nombre de Picard ρ(X ) est
1, alors dim Z a une borne inférieure dim X −dimB −1. De plus, lorsque dim Z est dim X −dimB −1, X est un
espace projectif et B est un sous-espace linéaire dans X . Si X est l’espace projectif Pn et B est une courbe, ou
n est égale à 3 et B est une courbe cubique tordue, ou n est un entier arbitraire et B est une ligne droite dans
Pn . Si X est une quadrique et B est une courbe, alors n est égale à 3 et B est une ligne droite dans Q3.
2020 Mathematics Subject Classification. 14J45, 14E30.
Funding. The author is supported by National Natural Science Foundation of China (12001547) and Guang-
dong Basic and Applied Basic Research Foundation(2019A1515110907).
Manuscript received 16th July 2021, accepted 23rd July 2021.
1. Introduction
Let L be a linear subspace of dimension l in a projective space Pn . We consider the rational
map πL : Pn 99K Γ given by the linear projection from L to Γ, where Γ is a linear subspace of
dimension n − l − 1 disjoint from L. Then the blowing up BlL(Pn) of Pn along L is the graph
of πL . More precisely, BlL(Pn) ⊆ Pn ×Γ is the closed subvariety of Pn ×Γ defined as BlL(Pn) =
{(p, q) ∈Pn ×Γ|p ∈ 〈L, q〉}, where 〈L, q〉 is the linear subspace generated by L and q. The projection
p2 : BlL(Pn) → Γ to the second factor is a projective bundle. Actually, for any point q in Γ, the
fiber p−12 (q) is the linear space 〈L, q〉. So any blowing up of a projective space along a linear
subspace is equipped with a projective bundle structure. It is interesting to find more examples
of blowing ups that are equipped with projective bundle structures. In [1, Section 4] and [2], there
are examples of blowing ups of projective spaces along non-linear subvarieties that are equipped
with projective bundle structures.
In this article, we aim to prove some classification results about (X ,B)(where X and B are
projective smooth varieties and B is a subvariety of X ) such that the blowing up BlB (X ) is a
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1130 Duo Li
projective bundle over some variety Z . In Proposition 1, we show that if B is a point, then X
is a projective space. In Theorem 4, we show that if the Picard number ρ(X ) is 1 and dim Z
is dim X − dimB − 1, then X is a projective space and B is a linear subspace in X . Finally, in
Theorem 6, we show that if X is Pn and B is a curve, then either n is 3 and B is a twisted cubic
curve or n is an arbitrary positive integer and B is a line in Pn ; if X is a quadric Qn(n ≥ 3) and B is
a curve, then n is 3 and B is a line in Q3.
Convention. A complex variety is an irreducible integral scheme of finite type defined over C. In
this article, all varieties considered are complex projective varieties.
2. Main results
Setup. We assume that X is a smooth projective variety of dimension n and we denote the
blowing up X̃ of X along a closed smooth subvariety B by ϕ : X̃ → X . Assume that π : X̃ → Z
is a projective bundle over a variety Z . We summarize all the morphisms in a diagram as follows:
X̃
ϕ

π // Z
X
We introduce some notations here: we let E be the exceptional divisor of ϕ, let F1 be a line
in a fiber π−1(z) (we have π−1(z) ' Pn−dim Z ) and let F2 be a line in a fiber ϕ|−1E (b)(we have
ϕ|−1E (b) 'Pn−dimB−1).
In the remaining part of this section, we keep the assumptions in the Setup and we keep using
the notations there.
We firstly consider the blowing up at a point that is equipped with a projective bundle
structure.
Proposition 1. If B is a point, then X is a projective space Pn .
Proof. We consider the morphism π|E : E → Z . Since E is a projective space Pn−1, the morphism
π|E : E → Z is quasi-finite. Otherwise, π|E will contract E to a single point. So dim Z is n −1 and
hence π|E is surjective. Note that Z is smooth and projective, by the Lazarsfeld’s theorem in [5],
Z is a projective space Pn−1. By [4, Lemma V.3.7.8], X is a projective space. 
In the remaining part of this section, we keep an additional assumption that the Picard
number ρ(X ) of X is 1.
Now let us prove a lemma about the lower bound of dim Z .
Lemma 2. The morphism π|E : E → Z is surjective and the varieties Z , X̃ , X are Fano varieties.
Moreover, dim Z is at least n −dimB −1. If dim Z is n −dimB −1, then Z is a projective space.
Proof. Note that E is a divisor in X̃ . The codimension codim(π(E), Z ) is at most 1. If
codim(π(E), Z ) is 1, the divisor E is the pull-back of π(E). Since the Picard number of Z is 1, E
is a nef divisor, which contradicts to the fact that E is covered by negative curves. Then π|E is
surjective. Now Z is a uniruled variety whose Picard number is 1, so Z is Fano.
To prove X̃ is Fano, we have the canonical bundle formulas:
−K X̃ =ϕ∗(−KX )− (n −dimB −1)E =π∗(−KZ −detE )+Oπ(n −dim Z +1)
where Oπ(1) is the tautological line bundle ofπ. Since the Picard number of X̃ is 2, the cone N E(X̃ )
is generated by two extremal rays as R≥0[F1]+R≥0[F2], where [Fi ](i = 1,2) are the numerically
C. R. Mathématique — 2021, 359, n 9, 1129-1133
Duo Li 1131
equivalent classes of Fi . By calculating the intersection numbers of −K X̃ with F1 and F2, we
deduce that X̃ is Fano. Since X is a uniruled variety whose Picard number is 1, X is Fano.
Let Y 'Pn−dimB−1 be a fiber of ϕ|E : E → B. The morphism π|Y : Y → Z is quasi-finite onto its
image. Otherwise, there would be a curve CY contracted by π. But CY is numerically equivalent
to a positive multiple of a line in Y , then the extremal ray of N E(X̃ ) spanned by [F2] is contracted
by π, which is impossible. So dim Z is at least n −dimB −1. Moreover, if dim Z is n −dimB −1,
π|Y is surjective. By the Lazarsfeld’s theorem in [5], Z is a projective space. 
Now we prove a simple lemma about the intersection numbers, which is very useful in the
proof of our main results.
Lemma 3. Let H1 be the pull-back of the ample generator HZ of Pic(Z ) and let H2 be the pull-
back of the ample generator HX of Pic(X ). We denote by a,b,c,d the intersection numbers H1 ·F2,
Oπ(1) ·F2, H2 ·F1, E ·F1 in the following diagram of intersection numbers:
H1 Oπ(1) H2 E
F1 0 1 c d
F2 a b 0 −1
Then a equals to c and a divides 1+bd .
Proof. The torsion free abelian group Pic(X̃ ) has two bases (H1,Oπ(1)) and (H2,E) which satisfy
the relation
(
H1
Oπ(1)
)
=
( d a
c −a
1+bd
c −b
)
·
(
H2
E
)
.
The matrix A =
( d a
c −a
1+bd
c −b
)
is an element of SL2(Z), so c divides 1+bd and the determinant
det A (= ac ) is 1. 
In the remaining part of this article, we keep using the notations in Lemma 3.
If dim Z is n −dimB −1, we have the following classification.
Theorem 4. Let dimB be m. If dim Z is n −m −1, then X is a projective space Pn , B ' Pm (⊆ Pn)
is a linear subspace in Pn and π◦ϕ−1 :Pn 99KPn−m−1 is the linear projection from B.
Proof. Let Rz be the fiber π−1(z) (' Pm+1) of some general point z ∈ Z . Then the intersection
Yz = E ∩Rz is a hypersurface of degree d in Rz . We claim that d must equal to 1 (the proof of this
claim mainly follows from [7, Lemma 2.1]). Actually, suppose that d is at least 2, then for a general
line l in Rz , the intersection l ∩Yz consists of at least two distinct points y1 and y2. Since ϕ|Rz is
quasi-finite onto its image, ϕ(Yz ) is B. So we can assume that bi = ϕ(yi ) (i = 1,2) are distinct
points in B. By varying Rz and Yz , we can construct a one-dimensional family of lines {lt }t∈C in X̃
such that the intersection of every lt with each ϕ−1(bi ) is not empty. Then the surface S =∪t∈C lt
is a ruled surface. Letϕ−1(bi )∩S be Ci . The curves Ci (i = 1,2) satisfy C1∩C2 =; andϕ(Ci ) = bi . By
the construction of S, S is not contained in E . So ϕ|S is a birational morphism. Hence Ci (i = 1,2)
are exceptional curves, which is impossible.
Now d is 1. Note that E doesn’t contain any fiber of π. Otherwise, there will be a morphism
ϕ|Rz : Rz → B where Rz ' Pm+1 is some fiber of π. Then ϕ contracts Rz to a point, which is
impossible. Hence for any z ∈ Z , the intersection E∩Rz = Yz 'Pm is a linear subspace in Rz . Since
ϕ(Yz ) is B , the variety B is a projective space. Then E has two projective bundle structures over
projective spaces. So by [6, Theorem A], the morphism (ϕ|E ,π|E ) : E → B × Z is an isomorphism.
Hence ϕ|Y : Y → B is an isomorphism. So the intersection number H2 ·F1(= H1 ·F2) equals to 1.
Suppose that π : X̃ → Z is given by |αH2−βE |. We have equalities H1 ·F2 = (αH2−βE) ·F2 = 1 and
H1 ·F1 = (αH2 −βE) ·F1 = 0. So both α and β equal to 1. Note the identities m +2 = −K X̃ ·F2 =
iX − (n − m − 1) (where iX is the index of X ). We deduce that iX is n + 1. By [3, Corollary of
C. R. Mathématique — 2021, 359, n 9, 1129-1133
1132 Duo Li
Theorem 1.1], X is a projective space. Since dim H 0(X̃ ,OX̃ (H2 −E)) = dim H 0(X , IB (1)) is at least
n −m, the subvariety B is a linear subspace in X 'Pn . 
When B is a smooth curve, we have the following criterion.
Proposition 5. If B is a smooth curve and n is at least 3, the following conditions are equivalent:
(1) π maps F2 birationally to a line in Z ,
(2) ϕ maps F1 birationally to a line in X ,
(3) (X ,B) is one of the following cases:
(a) (Pn , line) (π◦ϕ−1 :Pn 99KPn−2 is the linear projection from B).
(b) (P3, twisted cubic curve) (π◦ϕ−1 : P3 99K P2 is given by sections of |OP3 (2)| vanishing
along B).
(c) (Q3, line) (π◦ϕ−1 : Q3 99KP2 is the linear projection from B).
Proof. By Lemma 3, conditions (1) and (2) are equivalent. It is obvious that condition (3) implies
conditions (1) and (2).
Now suppose condition (1) or (2) holds. By Lemma 2, dim Z is at least n −2. If dim Z is n −2,
then by Theorem 4, (X ,B) is case (a).
If dim Z is n−1, then π is a P1-bundle. So we have identities −K X̃ ·F1 = iX −d(n−2) = 2. Then
iX is 2+d(n −2), which is at most n +1. Hence there are only two possibilities: when d is 2, n is 3
and iX is 4; when d is 1, iX is n.
If d is 2, then it is easy to see that X is P3 and Z is P2. Note that π is given by the linear system
|2H2 −E |. Then dim H 0(P3, IB (2)) is at least 3. We claim that B is not a plane curve. Otherwise,
there exists a plane L in P3 containing B and the strict transform L̃ of L is numerically equivalent
to H2 − kE for some positive integer k. So we have equalities 1 = H2 · F1 = L̃ · F1 + kE · F1 =
L̃ ·F1 + 2k, which is impossible. If degB is 4, then B is a complete intersection of two quadrics
and dim H 0(P3, IB (2)) is 2. So degB is 3.
If d is 1, then iX is n, hence by [3, Corollary of Theorem 2.1], X is a quadric Qn . We consider
the morphism π|E : E → Z . Since d is 1, π|E is a birational morphism. Note that π is given by
the linear system |H2 −E |. So dim H 0(Qn , IB (1)) is at least n, which implies that B is a line. The
exceptional divisor E of ϕ is isomorphic to PP1 (O ⊕O (1)⊕(n−2)). So the birational morphism π|E
contracts the minimal section of ϕ|E corresponding to the surjection O ⊕O (1)⊕(n−2) → O → 0.
Since Z is smooth, the exceptional locus of π|E should be a divisor, which implies the equality
n −2 = 1. So X is Q3 and B is a line in Q3. 
Now let us prove another main result of this section.
Theorem 6. Assume that X is Pn and B is a curve, then either n is 3 and B is a twisted cubic curve
or n is an arbitrary integer and B is a line in Pn . Assume that X is Qn and B is a curve, then n is 3
and B is a line in Q3.
Proof. If X is Pn , there are equalities: 1 = H n2 = ϕ∗(H nX ) = (−bH1 + aOπ(1))n = (−b)n H n1 +
a
(∑n−1
k=0 a
n−k−1C kn (−bH1)k ·Oπ(1)n−k
)
. Since H n1 vanishes, the integer a divides 1, hence a is 1.
Then by Proposition 5, either n is 3 and B is a twisted cubic curve or n is an arbitrary integer and
B is a line in Pn .
If X is Qn , then there are equalities: 2 = H n2 = ϕ∗(H nX ) = (−bH1 + aOπ(1))n =
a
(∑n−1
k=0 a
n−k−1C kn (−bH1)k ·Oπ(1)n−k
)
. Suppose that a is 2, then
∑n−1
k=0 a
n−k−1C kn (−bH1)k ·Oπ(1)k
is 1. Assume that there is a vector bundle E such that π : X̃ → Z is the projectization of E . If rkE
is 2, then by the canonical bundle formulas:
−K X̃ =ϕ∗(−KX )− (n −2)E =π∗(−KZ −detE )+ rk(E ) ·Oπ(1),
we have (iX H2 − (n − 2)E) · F2 = ((iZ −degE )H1 + 2Oπ(1)) · F2. By Lemma 3, n equals to 2(iZ −
degE +b+1), hence n is an even number. Then ∑n−1k=0 an−k−1C kn (−bH1)k ·Oπ(1)n−k is n(−bH1)n−1 ·
C. R. Mathématique — 2021, 359, n 9, 1129-1133
Duo Li 1133
Oπ(1)+a ∑n−2k=0 an−k−2C kn (−bH1)k ·Oπ(1)n−k , which is an even number. So rkE is 3. By Theorem 4,
X should be a projective space, which is impossible. So a is 1. By Proposition 5, X is Q3 and B is a
line in Q3. 
Acknowledgments
The author would like to thank Baohua Fu, Qifeng Li, Yi Gu and Yong Hu for many inspiring
discussions and suggestions. The author would like to thank Yewon Jeong for her valuable
contributions to the first version of this article and for her generosity allowing me to be the only
author of this article. Especially, the referee gave the statements and proof of Theorem 4 and
Proposition 5 which hugely strengthen the main results in the first version of this article.
References
[1] L. Ein, N. Shepherd-Barron, “Some special Cremona transformations”, Am. J. Math. 111 (1989), p. 783-800.
[2] S. Galin, D. S. Nagaraj, “Projective Bundles and blow-ups of projective spaces”, https://arxiv.org/abs/2006.12112,
2020.
[3] S. Kobayashi, T. Ochiai, “Characterizations of complex projective spaces and hyperquadrics”, J. Math. Kyoto Univ. 13
(1973), p. 31-47.
[4] J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 32,
Springer, 1995.
[5] R. Lazarsfeld, “Some applications of the theory of positive vector bundles”, in Complete intersections (Acireale, 1983),
Lecture Notes in Mathematics, Springer, 1984, p. 29-61.
[6] E. Sato, “Varieties which have two projective space bundle structures”, J. Math. Kyoto Univ. 25 (1985), p. 445-457.
[7] T. Tsukioka, “Del Pezzo surface fibrations obtained by blow-up of a smooth curve in a projective manifold”, C. R. Math.
Acad. Sci. Paris 340 (2005), p. 581-586.
C. R. Mathématique — 2021, 359, n 9, 1129-1133


Projective bundles and blowing ups

Vivette Girault

L. Ridgway Scott

Tanner Duality Between the Oldroyd–Maxwell and Grade-two Fluid Models

Ludovic Godard-Cadillac

Smooth traveling-wave solutions to the inviscid surface quasi-geostrophic equations

Yunhyung Cho

Comptes Rendus
Mathématique
Yunhyung Cho
Remark on the Betti numbers for Hamiltonian circle actions
Volume 359, issue 2 (2021), p. 113-117.
<https://doi.org/10.5802/crmath.127>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 2, p. 113-117
https://doi.org/10.5802/crmath.127
Symplectic geometry / Géométrie symplectique
Remark on the Betti numbers for Hamiltonian
circle actions
Yunhyung Choa
a Department of Mathematics Education, Sungkyunkwan University, Seoul, Republic
of Korea.
E-mail: yunhyung@skku.edu
Abstract. In this paper, we establish a certain inequality in terms of Betti numbers of a closed Hamiltonian
S1-manifold with isolated fixed points.
Résumé. Dans cet article, nous établissons une certaine inégalité en termes de nombres de Betti d’une S1-
variété hamiltonienne avec des points fixes isolés.
2020 Mathematics Subject Classification. 53D20, 53D05.
Funding. This work is supported by the National Research Foundation of Korea(NRF) grant funded by the
Korea government(MSIP; Ministry of Science, ICT & Future Planning) (NRF-2020R1C1C1A01010972).
Manuscript received 12th March 2020, revised 2nd September 2020 and 5th October 2020, accepted 2nd Octo-
ber 2020.
1. Introduction
Let (M ,ω) be a 2n-dimensional closed symplectic manifold admitting a Hamiltonian torus
action with only isolated fixed points. It has been a long-standing open problem whether M
admits a Kähler metric or not. Historically, Delzant [9] proved that if M admits a Hamiltonian
T n-action, where the fixed point set is automatically discrete, then M admits a T n-invariant
Kähler metric. Restricting to an S1-action case, several results on the existence of a Kähler
metric were provided in some special cases. For instance, Karshon [13] proved that every closed
symplectic four manifold admitting a Hamiltonian circle action admits a Kähler metric. (In
fact, the S1-action is induced from a toric action when the fixed points are isolated.) Also if
dim M = 6 with b2(M) = 1, then it turned out that M admits a Kähler metric, which was proved by
Tolman [18] and McDuff [15]. Recently, the author has shown that any 6-dimensional monotone
closed semifree Hamiltonian S1-manifold admits a Kähler metric, see [5–7].
As a counterpart, there were “candidates” of closed Hamiltonian T -manifolds (with isolated
fixed points) which possibly fail to admit Kähler metrics. Tolman [17] and Woodward [19] con-
structed a six-dimensional closed Hamiltonian T 2-manifold with only isolated fixed points and
with no T 2-invariant Kähler metric. Surprisingly Goertsches–Kostantis–Zoller [10] have recently
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
114 Yunhyung Cho
shown that the examples of Tolman and Woodward indeed admit Kähler metrics that are not
T 2-invariant. Thus their result provides a positive evidence for the conjecture of the existence of
Kähler metrics.
On the other hand, it seems reasonable to ask whether (M ,ω) enjoys Kählerian properties,
such as the hard Lefschetz property of the symplectic form ω or the unimodality of even Betti
numbers. Recall that every closed Kähler manifold (M ,ω, J ) satisfies the hard Lefschetz property,
that is,
[ω]n−k : H k (M ;R) → H 2n−k (M ;R)
α 7→α∪ [ω]n−k
is an isomorphism for every k = 0,1, · · · , n. This implies that
[ω] : H k (M ;R) → H k+2(M ;R)
is injective for every k with 0 ≤ k < n, and therefore the sequence of even (as well as odd) Betti
numbers of M is unimodal. In other words,
bk ≤ bk+2, k = 0,1, · · · , n −1
where bi denotes the i th Betti number of M . In this paper we deal with the following conjecture.
Conjecture 1 ([12]). Let (M ,ω) be a 2n-dimensional closed symplectic manifold equipped with a
Hamiltonian S1-action with only isolated fixed points. Then the sequence of even Betti numbers is
unimodal, i.e.,
b2i ≤ b2i+2 for every 0 ≤ i <
⌊n
2
⌋
.
It is worth mentioning that every odd Betti number of M vanishes by Frankel’s theorem which
states that a moment map is a Morse function whose critical points are of even indices. (See [2,
Theorem IV.2.3].) Therefore we only need to care about even Betti numbers of M .
In [8], the author and Kim proved Conjecture 1 when dim M = 8. The main goal of this article
is to improve the result of [8] and prove the following inequality, which is automatically satisfied
when Conjecture 1 is true.
Theorem 2. Let (M ,ω) be a closed symplectic manifold admitting a Hamiltonian circle action
with only isolated fixed points where dim M = 8n or 8n +4. Then
b2 + ·· · +b2+4(n−1) ≤ b4 + ·· · +b4+4(n−1).
In particular when dim M = 8 or 12, we have
b2 ≤ b4.
2. Proof of the main Theorem 2
The main technique for proving Theorem 2 is the ABBV-localization due to Atiyah–Bott and
Berline–Vergne. Recall that for an S1-manifold M , the equivariant cohomology is defined by
H∗
S1
(M) := H∗(M ×S1 ES1) where ES1 is a contractible space on which S1 acts freely. Then H∗S1 (M)
inherits an H∗(BS1)-module structure induced from the projection
π : M ×S1 ES1 → BS1 := ES1/S1.
Note that H∗(BS1;R) ∼= H∗(CP∞;R) = R[u]. Moreover, for the inclusion map i : M S1 ,→ M , we
have an induced ring homomorphism
i∗ : H∗S1 (M ;R) → H∗S1
(
M S
1
;R
)∼= H∗ (BS1;R)⊗H∗ (M S1 ;R) .
C. R. Mathématique, 2021, 359, n 2, 113-117
Yunhyung Cho 115
When M S
1 = {p1, · · · , pm} is discrete, we may express as
H∗
(
BS1;R
)⊗H∗ (M S1 ;R)∼= m⊕
i=1
H∗
(
BS1;R
)
and so
i∗(α) = ( f1, · · · , fm) , fi ∈R[u]
for α ∈ H∗
S1
(M ;R). We denote by α|pi := fi and call it the restriction of α to pi . By the Kirwan’s
injectivity theorem [14], the map i∗ is injective and hence H∗
S1
(M ;R) is a free H∗(BS1;R)-module.
Theorem 3 (ABBV Localization Theorem [1,3]). Let M be a closed S1-manifold with only isolated
fixed points and α ∈ H∗
S1
(M ;R). Then we have∫
M
α= ∑
p ∈M S1
α|p(
Πni=1wi (p)
)
un
.
where w1(p), · · · , wn(p) denote the weights of the tangential S1-representation at p.
To obtain Theorem 2, we will apply Theorem 3 to canonical classes which form a basis of
H∗
S1
(M ;R) as an H∗(BS1;R)-module.
Theorem 4 ([16, Lemma 1.13]1). Let (M ,ω) be a 2n-dimensional closed Hamiltonian S1-manifold
with only isolated fixed points. For each fixed point p ∈ M S1 of index 2k, there exists a unique class
αp ∈ H 2kS1 (M ;Z) such that
• αp |q = 0 for every q( 6= p) ∈ M S1 with either H(q) ≤ H(p) or ind(q) ≤ 2k,
• αp |p =∏ki=1λi u, where λ1, · · · , λk are negative weights of the S1-action at p.
Moreover, the set {αp | p ∈ M S1 } is a basis of H∗S1 (M ;R) as an H∗(BS1;R)-module.
Now we are ready to prove Theorem 2.
Proof of Theorem 2. We first consider the case dim M = 8n. Suppose that
b2 +·· ·+b2+4(n−1) > b4 + ·· · +b4+4(n−1). (1)
Since H∗
S1
(M) is a free module over H∗(BS1), we have
H 4n−2S1 (M)
∼= u0 ⊗H 4n−2(M)⊕u1 ⊗H 4n−4(M)⊕·· ·⊕u(2n−1) ⊗H 0(M)
which implies that
• dimR H 4n−2S1 (M ;R) ∼= b0 +b2 +·· ·+b4n−2, and
• {αp · u2n−1− 12 ind(p) | p ∈ M S1 , ind(p) ≤ 4n − 2} is a basis of H 4n−2S1 (M ;R) (as an R-vector
space) by Theorem 4.
Now, consider the following map
Φ : H 4n−2S1 (M ;R) →
(
Rb0 ⊕Rb4 ⊕ ·· · ⊕Rb4(n−1)
)
⊕
(
Rb4n ⊕ ·· · ⊕Rb8n−4
)
α 7→ (α0, · · · , α4n−4,α4n , · · · , α8n−4)
with the identification
Rb4i = ⊕
ind(p)=4i
R · u2n−1 and α4i :=
(
α|p
)
ind(p)=4i ∈
⊕
ind(p)=4i
R · u2n−1 (2)
for each i = 1, · · · , n. Since the dimension of the range of the map Φ satisfies
dimImΦ≤ b0 + ·· · +b4n−4 + (b4n +b4n+4 + ·· · +b8n−4) < b0 + ·· · +b4n−4 + (b4n−2 + ·· · +b2)
1See also [18, Proposition 2.2] and [11, Lemma 2.10]
C. R. Mathématique, 2021, 359, n 2, 113-117
116 Yunhyung Cho
by (1) and Poincaré duality, the map Φ has a non-trivial kernel. In other words, there exists a
nonzero element α ∈ H 4n−2
S1
(M ;R) such that
α|p = 0
for every fixed point p ∈ M S1 of index 0,4, · · · , 8n −4.
Now fix a moment map H for the S1-action on (M ,ω) such that H attains the maximum value
0. Denote by pmax the maximal fixed point and so ind(pmax) = 8n. The equivariant extension
[ωH ] ∈ H 2S1 (M ;R) of ω with respect to the moment map H satisfies
[ωH ] |p =−H(p)u ∈R[u]
for every p ∈ M S1 , see [4, Proposition 2.6]. Since H(p) < 0 for every p 6= pmax by the choice of H ,
we obtain
0 =
∫
M
α2 · [ωH ] =
∑
p ∈M S1
−α2|p · H(p)u(
4n∏
i=1
wi (p)
)
u4n
= ∑
ind(p)≡2 (mod4)
−α2|p · H(p)u(
4n∏
i=1
wi (p)
)
u4n
(3)
by Theorem 3 and the fact [ωH ]|pmax = −H(pmax)u = 0. Moreover, there exists at least one fixed
point p ∈ M S1 such that
α|p 6= 0 and ind(p) < 8n
because
• α|p 6= 0 for some p ∈ M S1 by the Kirwan’s Injectivity Theorem [14], and
• if α|p = 0 for every p ∈ M S1 with p 6= pmax, then α|pmax 6= 0 and it violates Theorem 3
0 =
∫
M
α= α|pmax(
4n∏
i=1
wi (p)
)
u4n
6= 0.
Consequently, each summand of the rightmost equation of (3) has non-positive coefficient (of 1u )
and at least one of those should be negative. Therefore it leads to a contradiction.
Now it remains to consider the case of dim M = 8n + 4. Under the same assumption (1), we
similarly define
Φ : H 4nS1 (M ;R) →
(
Rb0 ⊕Rb4 ⊕ ·· · ⊕Rb4n
)
⊕
(
Rb4n+4 ⊕ ·· · ⊕Rb8n
)
α 7→ (α0, · · · , α4n ,α4n+4, · · · , α8n)
with the same identification as in (2). Note that dimR H 4nS1 (M ;R) = b0 +b2 + ·· · +b4n−2 +b4n and
dimImΦ≤ b0 + ·· · +b4n + (b4n+4 +·· ·+b8n) = b0 + ·· · +b4n + (b4n + ·· · +b4)
< b0 + ·· · +b4n + (b4n−2 + ·· · +b2) = dimR H 4nS1 (M ;R)
by (1) and Poincaré duality. Thus Φ has a non-trivial kernel α ∈ H 4n
S1
(M ;R) and so there exists a
nonzero α ∈ H 4n
S1
(M ;R) such that α|p = 0 for every fixed point p of index 0,4, . . . , 4n. Therefore,
we obtain
0 =
∫
M
α2 · [ωH ] =
∑
ind(p)≡2 (mod4)
−α2|p · H(p)u(
4n+2∏
i=1
wi (p)
)
u4n+2
6= 0
which leads to a contradiction. This completes the proof of Theorem 2. 
C. R. Mathématique, 2021, 359, n 2, 113-117
Yunhyung Cho 117
References
[1] M. F. Atiyah, R. Bott, “The moment map and equivariant cohomology”, Topology 23 (1984), no. 1, p. 1-28.
[2] M. Audin, Topology of Torus actions on symplectic manifolds Second revised edition, Progress in Mathematics, vol. 93,
Birkhäuser, 2004.
[3] N. Berline, M. Vergne, “Classes charactéristiques équivariantes. Formule de localisation en cohomologie équivari-
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[4] Y. Cho, “Unimodality of Betti numbers for Hamiltonian circle actions with index-increasing moment maps”, Int. J.
Math. 27 (2016), no. 5, article no. 1650043 (14 pages).
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https://arxiv.org/abs/1904.10962, to appear in International Journal of Mathematics, 2021.
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https://arxiv.org/abs/1905.07292, to appear in International Journal of Mathematics, 2021.
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no. 3, p. 315-339.
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368 (2020), article no. 107141.
[11] R. F. Goldin, S. Tolman, “Towards Generalizing Schubert Calculus in the Symplectic Category”, J. Symplectic Geom. 7
(2009), no. 4, p. 449-473.
[12] L. Jeffrey, T. Holm, Y. Karshon, E. M. Lerman, E. Meinrenken, “Moment maps in various geometries”, 2005, available
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[13] Y. Karshon, Periodic Hamiltonian flows on four-dimensional manifolds, Memoirs of the American Mathematical
Society, vol. 672, American Mathematical Society, 1999.
[14] F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton
University Press, 1984.
[15] D. McDuff, “Some 6-dimensional Hamiltonian S1-manifolds”, J. Topol. 2 (2009), no. 3, p. 589-623.
[16] D. McDuff, S. Tolman, “Topological properties of Hamiltonian circle actions”, Int. Math. Res. Pap. 2006 (2006), no. 4,
article no. 72826.
[17] S. Tolman, “Examples of non-Kähler Hamiltonian torus actions”, Invent. Math. 131 (1998), no. 2, p. 299-310.
[18] ——— , “On a symplectic generalization of Petrie’s conjecture”, Trans. Am. Math. Soc. 362 (2010), no. 8, p. 3963-3996.
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C. R. Mathématique, 2021, 359, n 2, 113-117


Remark on the Betti numbers for Hamiltonian circle actions

Xu Pan

Liangchen Wang

Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production

Pierre-Cyril Aubin-Frankowski

Comptes Rendus
Mathématique
Pierre-Cyril Aubin-Frankowski
Interpreting the dual Riccati equation through the LQ reproducing
kernel
Volume 359, issue 2 (2021), p. 199-204.
<https://doi.org/10.5802/crmath.174>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 2, p. 199-204
https://doi.org/10.5802/crmath.174
Control theory / Théorie du contrôle
Interpreting the dual Riccati equation through
the LQ reproducing kernel
Pierre-Cyril Aubin-Frankowskia
a École des Ponts ParisTech and CAS, MINES ParisTech, PSL Research University,
France
E-mail: pierre-cyril.aubin@mines-paristech.fr
Abstract. In this study, we provide an interpretation of the dual differential Riccati equation of Linear-
Quadratic (LQ) optimal control problems. Adopting a novel viewpoint, we show that LQ optimal control
can be seen as a regression problem over the space of controlled trajectories, and that the latter has a very
natural structure as a reproducing kernel Hilbert space (RKHS). The dual Riccati equation then describes
the evolution of the values of the LQ reproducing kernel when the initial time changes. This unveils new
connections between control theory and kernel methods, a field widely used in machine learning.
Manuscript received 19th November 2020, accepted 17th December 2020.
1. Introduction
We consider the problem of finite-dimensional time-varying linear quadratic (LQ) optimal con-
trol with finite horizon and quadratic terminal cost as in
V (t0,x0) := min
u( · )
x(T )>JT x(T )+
∫ T
t0
[
x(t )>Q(t )x(t )+u(t )>R(t )u(t )]dt (1)
s.t. x′(t ) = A(t )x(t )+B(t )u(t ), a.e. in[t0,T ] (1a)
x(t0) = x0, (1b)
where the state x(t ) ∈RN and the control u(t ) ∈RM . We shall henceforth assume that JT Â 0,1 and
for all t ∈ [t0,T ], R(t )< r IdM with r > 0, as well as A( · ) ∈ L1([t0,T ],RN ,N ), B( · ) ∈ L2([t0,T ],RN ,M ),
Q( · ) ∈ L1([t0,T ],RN ,N ), and R( · ) ∈ L2([t0,T ],RN ,N ). To have a finite objective, we restrict our
attention to measurable controls satisfying R( · )1/2u( · ) ∈ L2([t0,T ],RN ). Problem (1) is intimately
related to the differential Riccati equation,2 expressed as
−∂1J(t ,T ) = A(t )>J(t ,T )+ J(t ,T )A(t )− J(t ,T )B(t )R(t )−1B(t )>J(t ,T )+Q(t ) ; J(T,T ) = JT , (2)
1Here< (resp. Â) denotes the (strict) partial order over positive semi-definite matrices.
2The index T in J( · ,T ) is kept as a reminder that (2) is defined w.r.t. a given terminal time T . We denote by ∂1 the
derivative w.r.t. the first variable.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
200 Pierre-Cyril Aubin-Frankowski
which solution J( · ,T ) satisfies V (t0,x0) = x>0 J(t0,T )x0. It is well-known (e.g. [3, p. 31, 408]) that
under the above positivity assumptions, J(t ,T ) is a symmetric positive definite matrix, which
inverse M(t ,T ) := J(t ,T )−1 satisfies a dual Riccati equation
∂1M(t ,T ) = A(t )M(t ,T )+M(t ,T )A(t )>−B(t )R(t )−1B(t )>+M(t ,T )Q(t )M(t ,T ) ; M(T,T ) = J−1T . (3)
This inverse matrix has been used as a tool to obtain a representation formula in infinite-
dimensional LQ control [2] but it has not received the deserved interest yet. Whereas the solution
of (2) is equal to the Hessian of the value function V (t0, ·), i.e. J(t0,T ) = ∂x,xV (t0, ·), we show
(Theorem 4 below) that the solution of (3) is equal to the diagonal element of a matrix-valued
reproducing kernel K ( · , · ), naturally associated with (1). Owing to this interpretation, the dual
Riccati equation (3) is thus no less fundamental and effectively allows to reverse the perspective
between the adjoint vector and the optimal trajectory.
We first need to bring trajectories to the fore in (1). In his seminal book, Luenberger [4,
p. 255] already discussed that an optimal control problem such as (1) can be seen as either
optimizing over the set of controls u( · ), or jointly over the set of trajectories x( · ) and controls
u( · ), connected through the dynamic constraint (1a). Luenberger also alluded without details to
a third possibility, that of optimizing directly over the controlled trajectories. We follow this last
viewpoint and consequently introduce the vector space S[t0,T ] of controlled trajectories of the
linear system:
S[t0,T ] :=
{
x : [t0,T ] →RN
∣∣∣∣∃ u( · ) s.t. x′(t ) = A(t )x(t )+B(t )u(t ) a.e.and R( · )1/2u( · ) ∈ L2([t0,T ],RN )
}
. (4)
There is not necessarily a unique choice of u( · ) for a given x( · ) ∈ S[t0,T ].3 Therefore, with each
x( · ) ∈ S[t0,T ], we associate the control u( · ) having minimal norm based on the pseudoinverse
B(t )ª of B(t ) for the RM -norm ‖ ·‖R(t ) := ‖R(t )1/2 · ‖:
u(t ) = B(t )ª[x′(t )−A(t )x(t )] a.e. in[t0,T ]. (5)
Problem (1) then induces a natural inner product over S[t0,T ]. As a matter of fact, the expression
〈x1( · ),x2( · )〉K := x1(T )>JT x2(T )+
∫ T
t0
[
x1(t )
>Q(t )x2(t )+u1(t )>R(t )u2(t )
]
dt (6)
is bilinear and symmetric over S[t0,T ] ×S[t0,T ]. It is positive definite over S[t0,T ] as ‖x( · )‖2K = 0
implies that u( · ) a.e.≡ 0 and, as JT Â 0, x(T ) = 0, hence x( · ) ≡ 0. Therefore
V (t0,x0) = min
x( · )∈S[t0,T ]
‖x( · )‖2K s.t. x(t0) = x0 (7)
In other words the value function V (t0,x0) of (1) coincides with the optimal value of a constrained
norm minimization over S[t0,T ]. The solution of (7) can be made explicit as (S[t0,T ],〈 · , · 〉K ) is not
an arbitrary Hilbert space, but a vector-valued reproducing kernel Hilbert space (vRKHS).
2. Vector spaces of linear controlled trajectories as vRKHSs
Definition 1. Let T be a non-empty set. A Hilbert space (HK (T),〈 · , · 〉K ) of RN -vector-valued
functions defined on T is called a vRKHS if there exists a matrix-valued kernel KT : T×T → RN ,N
such that the reproducing property holds: for all t ∈ T, p ∈ RN , KT( · , t )p ∈ HK (T) and for all
f ∈HK (T), p>f(t ) = 〈f,KT( · , t )p〉K .
3This is the case for instance if B(t ) is not injective for a set of times t with positive measure.
C. R. Mathématique, 2021, 359, n 2, 199-204
Pierre-Cyril Aubin-Frankowski 201
Remark. It is well-known that by Riesz’s theorem, an equivalent definition of a vRKHS is that,
for every t ∈T and p ∈RN , the evaluation functional f ∈HK (T) 7→ p>f(t ) ∈R is continuous. There
is also a one-to-one correspondence between the kernel KT and the vRKHS (HK (T),〈 · , · 〉K ) (see
e.g. [5, Theorem 2.6]). Moreover, by symmetry of the scalar product, the matrix-valued kernel has
a Hermitian symmetry, i.e. KT(s, t ) = KT(t , s)> for any s, t ∈T.
Lemma 2. (S[t0,T ],〈 · , · 〉K ) is a vRKHS over [t0,T ] with a reproducing kernel K[t0,T ] which we call
the LQ kernel.
Proof of Lemma 2. The proof is identical to the one of Lemma 1 in [1] where S[t0,T ] was equipped
with the following inner product
〈x1( · ),x2( · )〉K ,init := x1(t0)>x2(t0)+
∫ T
t0
[
x1(t )
>Q(t )x2(t )+u1(t )>R(t )u2(t )
]
dt . (8)

Owing to Lemma 2, we can look for a “representer theorem”, i.e. a necessary condition to
ensure that the solutions of an optimization problem like (7) live in a finite dimensional subspace
of S[t0,T ] and consequently enjoy a finite representation.
Theorem 3 (cf. [1]). Let (HK (T),〈 · , · 〉K ) be a vRKHS defined on a set T. For a given I ∈ N, let
{ti }i∈[[1,I ]] ⊂T. Consider the following optimization problem with “loss” function L :RI →R∪{+∞},
strictly increasing “regularizer” functionΩ :R+ →R, and vectors {ci }i∈[[1,I ]] ⊂RN
min
f∈HK (T)
L
(
c>1 f(t1), . . . ,c
>
I f(tI )
)+Ω (‖f‖K ) .
Then, for any minimizer f, there exists {pi }i∈[[1,I ]] ⊂RN such that f =∑Ii=1 KT( · , ti )pi with pi =αi ci
for some αi ∈R.
Taking L(e>1 x(t0), . . . ,e
>
N x(t0)) := χx0 (x(t0)) and Ω(y) = y2, with ei the i -th basis vector of RN ,
χx0 the indicator function of x0, we apply Theorem 3 to (7). Since ‖·‖2K is strongly convex and
there exists x( · ) ∈ S[t0,T ] satisfying x(t0) = x0, the solution of (7) is unique and can be written as
x(t ) = K[t0,T ](t , t0)p0, with p0 = K[t0,T ](t0, t0)ªx0 ∈RN , where K[t0,T ](t0, t0)ª is the pseudoinverse of
K[t0,T ](t0, t0) for the R
N -seminorm ‖K[t0,T ](t0, t0)1/2 · ‖. Thus, owing to the reproducing property,
V (t0,x0) = ‖x( · )‖2K = 〈K[t0,T ]( · , t0)p0,K[t0,T ]( · , t0)p0〉K = p>0 K[t0,T ](t0, t0)p0 (9)
= x>0 K[t0,T ](t0, t0)ªx0 = p>0 x0.
So we conjecture that K[t0,T ](t0, t0)
ª = J(t0,T ). We actually have a stronger result:
Theorem 4. Let Kd : t0 ∈ ]−∞,T ] 7→ K[t0,T ](t0, t0). Then Kd (t0) = J(t0,T )−1.
The proof of Theorem 4 (in Section 3 below) boils down to identifying the reproducing kernel
of (S[t0,T ],〈 · , · 〉K ). Informally, the inverse relation comes from inverting the graph of the (x,p)-
relation. As a matter of fact, consider the solution p(t ) of the adjoint equation
p′(t ) =−A(t )>p(t )+Q(t )x(t ) p(T ) =−JT x(T ). (10)
Then we have p(t ) = −J(t ,T )x(t ). In other words, the solution J( · ,T ) of the differential Ric-
cati equation maps the optimal trajectory x( · ) to its adjoint vector p( · ). On the contrary, since
x(t ) = K[t0,T ](t , t0)p0, the kernel K[t0,T ]( · , t0) maps an initial covector p0 ∈ Rn to the optimal tra-
jectory x( · ). This effectively inverts the graph of the relation between x( · ) and p( · ). The inver-
sion performed is related to yet another change of perspective, from an online and differential
approach to an offline and integral one.
Through Pontryagine’s Maximum Principle (PMP), it is well known that the optimal control
u( · ) satisfies u(t ) = R(t )−1B(t )>p(t ) =−R(t )−1B(t )>J(t ,T )x(t ) =: G(t )x(t ). Hence, based on J(t ,T ),
one has a closed feedback loop, with gain matrix G(t ), and knows the control to apply based only
on the present time and state. However the optimal trajectory x( · ) is not encoded as simply as in
C. R. Mathématique, 2021, 359, n 2, 199-204
202 Pierre-Cyril Aubin-Frankowski
the kernel formula x(t ) = K[t0,T ](t , t0)p0. It has to be derived through numerical approximations
of the dynamics (1a). Conversely, the kernel K[t0,T ] performs the integration of the Hamiltonian
system (1a)-(10) and sparsely encodes x( · ) over [t0,T ] by p0. This sparsity partly stems from the
smaller number of constraints in (7) w.r.t. (1) since the dynamics (1a) were incorporated in the
definition of S[t0,T ]. Unlike in the PMP, the adjoint vector p(t ) disappears in the kernel perspective
and only the initial condition (or some intermediate rendezvous points) induce a covector pi .
More generally, for a given interval [t0,T ], Theorem 3 states that to encode the optimal tra-
jectories one needs at most as many covectors pi as there are points ti where the trajectory is
evaluated in the optimization problem. It is a classical property of “kernel machines”, frequently
leveraged in classification tasks (e.g. SVMs in [7]). This result was exploited in [1] to tackle affine
state constraints. From the PMP perspective, it resulted in focusing only on the measures sup-
ported on the constraint boundary. Unlike the adjoint vector p(t ) associated with the equal-
ity constraint (1a), which never vanishes except for abnormal trajectories, the covectors corre-
sponding to inequality constraints are null whenever the constraint is not active. This led to ex-
tremely sparse encoding of the optimal trajectory by specifying only the active covectors on the
[t0,T ] time interval.4 Offline computation of the kernel is indeed well suited for path-planning
problems. The kernel formalism however conflicts with the online perspective since varying t0
changes the domain of K[t0,T ]. As the correspondence between the kernel KT and the vRKHS
(HK (T),〈 · , · 〉K ) is one-to-one (e.g. [5, Theorem 2.6]), varying T = [t0,T ] or modifying the inner
product changes the kernel. In general, restricting the domain leads to complicated relations be-
tween a vRKHS and its kernel [6, p. 78–80]. In our case, the dual Riccati equation (3) precisely
describes how the values of the LQ kernel change when varying t0.
3. Proof of Theorem 4
The proof corresponds to the identification of the reproducing kernel of (S[t0,T ],〈 · , · 〉K ). Since
we shall proceed with fixed initial time t0, we drop the corresponding index and set K ( · , · ) =
K[t0,T ]( · , · ). By existence and unicity of the reproducing kernel, we just have to exhibit a function
K ( · , · ) which satisfies the requirements of Definition 1.
Let us denote by ΦA(t , s) ∈ RN ,N the state-transition matrix of z′(τ) = A(τ)z(τ), defined from
s to t , i.e. z(t ) = ΦA(t , s)z(s). The key property used throughout this section is the variation of
constants, a.k.a. Duhamel’s principle, stating that for any absolutely continuous x( · ) such that
x′(t ) = A(t )x(t )+B(t )u(t ) a.e., we have for any σ, t ∈ [t0,T ]
x(t ) =ΦA(t ,σ)x(σ)+
∫ t
σ
ΦA(t ,τ)B(τ)u(τ)dτ. (11)
Setting ∂1K (s, t ) : p 7→ dds (K (s, t )p), let us define formally U(s, t ) := B(s)ª[∂1K (s, t )−A(s)K (s, t )].
The reproducing property for K then writes as follows, for all t ∈ [t0,T ], p ∈RN , x( · ) ∈S[t0,T ],
p>x(t ) = (K (T, t )p)>JT x(T )+
∫ T
t0
(K (s, t )p)>Q(s)x(s)ds +
∫ T
t0
(U(s, t )p))>R(s)u(s)ds. (12)
4In [1], we considered the inner product (8) rather than (6) which assumes a terminal quadratic cost. The choice of (8)
was more appropriate for fixed initial time and general lower semicontinuous convex terminal costs, alongside affine
state constraints.
C. R. Mathématique, 2021, 359, n 2, 199-204
Pierre-Cyril Aubin-Frankowski 203
By the Hermitian symmetry of K and the variation of constants (11) written for σ = T , we can
rewrite (12) as, for all t ∈ [t0,T ], x( · ) ∈S[t0,T ],
x(t ) = K (t ,T )JT x(T )+
∫ T
t0
K (t , s)Q(s)
(
ΦA(s,T )x(T )+
∫ s
T
ΦA(s,τ)B(τ)u(τ)dτ
)
ds
+
∫ T
t0
U(s, t )>R(s)u(s)ds.
Regrouping terms,
x(t ) =
(
K (t ,T )JT +
∫ T
t0
K (t , s)Q(s)ΦA(s,T )ds
)
x(T )+
∫ T
t0
K (t , s)Q(s)
∫ s
T
ΦA(s,τ)B(τ)u(τ)dτds
+
∫ T
t0
U(s, t )>R(s)u(s)ds
Setting K̃ (t , s) := ∫ st0 K (t ,τ)Q(τ)ΦA(τ, s)dτ, and applying Fubini’s theorem, we get
x(t ) = (K (t ,T )JT + K̃ (t ,T ))x(T )+∫ T
t0
[
U(s, t )>R(s)−
∫ s
t0
K (t ,τ)Q(τ)ΦA(τ, s)B(s)dτ
]
u(s)ds.
Identifying with (11) for σ= T , we derive that
K (t ,T )JT = ΦA(t ,T )− K̃ (t ,T ),
U(s, t )>R(s) =
{
(−ΦA(t , s)+ K̃ (t , s))B(s) ∀ s ≥ t ,
K̃ (t , s)B(s) ∀ s < t . (13)
Let us introduce formally an adjoint equation defined for a variable Π(s, t ) ∈ RN ,N . For any given
t ∈ [t0,T ],
∂1Π(s, t ) =−A(s)>Π(s, t )+Q(s)K (s, t ) Π(t0, t ) =− IdN .
This is the matrix version of (10) but with an initial rather than terminal condition. Again,
applying the variation of constants (11) with σ = t0 to Π(s, t ), taking the transpose and owing
to the symmetries of K ( · , · ) andΦ( · , · ), we derive that
Π(s, t ) =Φ(−A>)(s, t0)Π(t0, t )+
∫ s
t0
Φ(−A>)(s,τ)Q(τ)K (τ, t )dτ
Π(s, t )> =−ΦA(t0, s)+
∫ s
t0
K (t ,τ)Q(τ)ΦA(τ, s)dτ=−ΦA(t0, s)+ K̃ (t , s).
Since ∂1K (s, t ) = A(s)K (s, t ) + B(s)U(s, t ), by (13), for any given time t ∈ [t0,T ], we have two
coupled differential equations for K ( · , t ) andΠ( · , t )
∂1K (s, t ) = A(s)K (s, t )+B(s)R(s)−1B(s)>
{
Π(s, t )+ΦA(t0, s)>−ΦA(t , s)> ∀ s ≥ t ,
Π(s, t )+ΦA(t0, s)> ∀ s < t .
∂1Π(s, t ) =−A(s)>Π(s, t )+Q(s)K (s, t ),
Π(t0, t ) =− IdN K (t ,T )JT =−Π(T, t )>+ΦA(t ,T )−ΦA(t0,T ).
(14)
Equations (14) seem quite intricate but they become simpler for t = t0, and as seen in (9), t0 is
actually the only time that interests us to solve the LQ optimal control problem (7). For t = t0, the
equations boil down to
∂1K (s, t0) = A(s)K (s, t0)+B(s)R(s)−1B(s)>Π(s, t0)
∂1Π(s, t0) =−A(s)>Π(s, t0)+Q(s)K (s, t0)
Π(t0, t0) =− IdN ; Π(T, t0) =−JT K (T, t0).
(15)
C. R. Mathématique, 2021, 359, n 2, 199-204
204 Pierre-Cyril Aubin-Frankowski
Let us solve (15) by variation of the constant JT . We thus look for a function J( · ,T ), which we will
prove solves (2), such that J(T,T ) = JT and Π(s, t0) =−J(s,T )K (s, t0). We take the derivative in s of
the latter expression to obtain
−A(s)>Π(s, t0)+Q(s)K (s, t0) =−J(s,T )
(
A(s)K (s, t0)+B(s)R(s)−1B(s)>Π(s, t0)
)− (∂1J(s,T ))K (s, t0)
Therefore, applying to the right of the equations any pseudo-inverse of K (s, t0),
0 = [A(s)>J(s,T )>+ J(s,T )A(s)− J(s,T )B(s)R(s)−1B(s)>J(s,T )+Q(s)+∂1J(s,T )]projImK (s,t0) .
So it suffices that J( · ,T ) solves the differential Riccati equation (2) and that, by symmetry of K ,
Π(t0, t0) =− IdN =−J(t0,T )K (t0, t0) =−K (t0, t0)J(t0,T ).
Consequently K (t0, t0) = J(t0,T )−1 which formalizes our intuition of the inverse relation between
J and K . Now let us vary the initial time and consider the function Kd : t0 7→ K[t0,T ](t0, t0). Taking
the derivative w.r.t. t0 ofΠ(t0, t0) =− IdN =−J(t0,T )K[t0,T ](t0, t0), we get
0 = (∂1J(t0,T )Kd (t0)+ J(t0,T )(∂1Kd (t0)),
applying Kd (t0) to the left of the equation and using that Kd (t0) = J(t0,T )−1, we obtain
0 =−Kd (t0)A(t0)>−A(t0)Kd (t0)+B(t0)R(t0)−1B(t0)>−Kd (t0)Q(t0)Kd (t0)+∂1Kd (t0).
This concludes our proof as Kd ( · ) solves the dual matrix Riccati equation (3) which has a unique
solution.
References
[1] P.-C. Aubin-Frankowski, “Linearly-constrained Linear Quadratic Regulator from the viewpoint of kernel methods”,
https://arxiv.org/abs/2011.02196, 2020.
[2] V. Barbu, G. D. Prato, “A representation formula for the solutions to operator Riccati equation”, Differ. Integral Equ. 5
(1992), no. 4, p. 821-829.
[3] A. Bensoussan, G. D. Prato, M. C. Delfour, S. K. Mitter, Representation and Control of Infinite Dimensional Systems,
Birkhäuser, 2007.
[4] D. Luenberger, Optimization by vector space methods, John Wiley & Sons, 1968.
[5] M. Micheli, J. A. Glaunès, “Matrix-valued kernels for shape deformation analysis”, Geometry, Imaging and Computing
1 (2014), no. 1, p. 57-139.
[6] S. Saitoh, Y. Sawano, Theory of Reproducing Kernels and Applications, Developments in Mathematics, vol. 44, Springer,
2016.
[7] B. Schölkopf, J. Smola, Alexander, Learning with kernels : support vector machines, regularization, optimization, and
beyond, MIT Press, 2002.
C. R. Mathématique, 2021, 359, n 2, 199-204


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