Metamorphoses of Functional Shapes in Sobolev Spaces

Abstract

In this paper, we describe in detail a model of geometric-functional variability between fshapes. These objects were introduced for the first time by Charlier et al. (J Found Comput Math, 2015. arXiv:1404.6039) and are basically the combination of classical deformable manifolds with additional scalar signal map. Building on the aforementioned work, this paper’s contributions are several. We first extend the original $L^2$ model in order to represent signals of higher regularity on their geometrical support with more regular Hilbert norms (typically Sobolev). We describe the bundle structure of such fshape spaces with their adequate geodesic distances, encompassing in one common framework usual shape comparison and image metamorphoses. We then propose a formulation of matching between any two fshapes from the optimal control perspective, study existence of optimal controls and derive Hamiltonian equations and conservation laws describing the dynamics of geodesics. Secondly, we tackle the discrete counterpart of these problems and equations through appropriate finite elements interpolation schemes on triangular meshes. At last, we show a few results of metamorphosis matchings on several synthetic and real data examples in order to highlight the key specificities of the approach.

Appendices

Proof of Theorem 1

Before the actual proof of Theorem 1, we shall introduce a few definitions and intermediate results. Let $s\ge 0$ and $s'=\max (s,1)$ and we recall that X is a compact submanifold of $\mathbb {R}^n$ of dimension d and class $C^{s}$ and that $V\hookrightarrow C^{s'}_0(\mathbb {R}^n,\mathbb {R}^n)$. For a given coordinate system $(x^i)_{1\le i\le d}$, we will denote by $(\partial _{i})$ and $(dx^i)$ the corresponding frame and coframe, respectively. We introduce the following class of sections over the (a, b) tensor bundle:

Definition 3

We say that $A\in \varGamma _{{\text {pol}}}^{p,s}(T^a_b(U))$ with $a,b,p,s \in \mathbb {N}$, $a,b\le s$ and $p<s'$ if there exists a coordinate system $(x^i)_{1\le i\le d}$ on U such that for any $(\phi ,u)\in {{\mathrm{Diff}}}^s_0(\mathbb {R}^n)\times U$

$$\begin{aligned} A(\phi ,u)=A^\alpha _\beta (\phi ,u)\partial ^a_\alpha \otimes dx^\beta \end{aligned}$$

where for any compact $K\subset U$ there exists two polynomials P and Q such that for any multi-indices $\alpha ,\beta$ and for any $\phi ,\phi '\in {{\mathrm{Diff}}}^{s}_0$ we have

$$\begin{aligned} u\mapsto A^\alpha _\beta (\phi ,u))\in C^p(U,\mathbb {R})\text {, } \sup _{K,k\le p}|\partial ^kA^\alpha _\beta (\phi ,u)|\le P(\rho _{s}(\phi ))\,, \end{aligned}$$

and

$$\begin{aligned} \sup _{K,k\le p}|\partial ^kA^\alpha _\beta (\phi ,u)-\partial ^kA^\alpha _\beta (\phi ',u)|\le \rho _{s}(\phi '\circ \phi ^{-1})Q(\rho _{s}(\phi ),\rho _{s}(\phi ')) \end{aligned}$$

with the notation $\rho _s(\psi )\doteq \sum _{k\le s} \Vert d^k (\psi -{{\mathrm{Id}}})\Vert _\infty +\Vert d^k(\psi ^{-1}-{{\mathrm{Id}}})\Vert _\infty$ for any $\psi \in {{\mathrm{Diff}}}^s_0(\mathbb {R}^n)$.

In the previous, we point out that $\alpha$ and $\beta$ are multi-indices of integers between 1 and d such that $|\alpha |=a, \ |\beta |=b$. When $a=b=0$, the space $\varGamma _{{\text {pol}}}^{p,s}(T^a_b(U))$ will be denoted $C_{{\text {pol}}}^{p,s}(U)$.

Remark 5

A first important remark is that the definition is not dependent on the choice of the coordinate system. Indeed, if $s=0$, we have $a=b=p=0$ and the definition does not depend on any coordinate system. If $s=1$ then $p=0$ and if $(y_1,\ldots ,y^d)$ is another coordinate system, it is sufficient to notice that $\frac{\partial }{\partial x^i}=\frac{\partial y^j}{\partial x^i}\frac{\partial }{\partial y_j}$ and $dx^i=\frac{\partial x^i}{\partial y^j}dy^j$ where the mappings $\frac{\partial y^j}{\partial x^i}$ and $\frac{\partial x^i}{\partial y^j}$ are continuous and bounded on K. Last, if $s\ge 2$, we get $\tilde{A}^{\tilde{\alpha }}_{\tilde{\beta }}(\phi ,u)\frac{\partial ^a}{\partial y^{\tilde{\alpha }}}\otimes dy^{\tilde{\beta }}=A^\alpha _\beta (\phi ,u)\frac{\partial ^a}{\partial x^\alpha }\otimes dx^\beta$ for $\tilde{A}^{\tilde{\alpha }}_{\tilde{\beta }}(\phi ,u)=A^\alpha _\beta (\phi ,u)\frac{\partial y^{\tilde{\alpha }}}{\partial x^\alpha }\frac{\partial x^\beta }{\partial y^{\tilde{\alpha }}}$ with $\frac{\partial y^{\tilde{\alpha }}}{\partial x^\alpha }=\prod _{i=1}^a\frac{\partial y^{{\tilde{\alpha }}_i}}{\partial x^{\alpha _i}}\in C^{s-1}(U,\mathbb {R})$ and $\frac{\partial x^\beta }{\partial y^{\tilde{\beta }}}=\prod _{i=1}^b\frac{\partial x^{\beta _i}}{\partial y^{{\tilde{\beta }}_i}}\in C^{s-1}(U,\mathbb {R})$. Since $p\le s-1$, we deduce that $\tilde{A}^{\tilde{\alpha }}_{\tilde{\beta }}(\phi ,u)\in C^p(U,\mathbb {R})$ for any $({\tilde{\alpha }},{\tilde{\beta }})$ and satisfies the needed polynomial controls in the coordinate system $(y^1,\ldots ,y^d)$ thanks to the Faà di Bruno Formula.

A second useful remark is that $C_{{\text {pol}}}^{p,s}(U)$ is an algebra over the field $\mathbb {R}$.

Lemma 4

Assume here that $s\ge 2$. For any coordinate system $(x^i)_{1\le i\le d}$ on an open set $U\subset X$, we have for any $1\le i\le d$ that

$$\begin{aligned} \nabla \partial _i\in \varGamma _{{\text {pol}}}^{s-2,s}(T^1_1(U))\text { and }\nabla dx^i\in \varGamma _{{\text {pol}}}^{s-2,s}(T^0_2(U)) \end{aligned}$$

where for $\phi \in {{\mathrm{Diff}}}^s_0(\mathbb {R}^n)$, $\nabla =\nabla ^\phi$ is the Levi–Civita covariant derivative associated with the pullback metric $g=g^\phi$ on X of the induced metric $g^{\phi (X)}$ on $Y=\phi (X)$ by the Euclidean metric on $\mathbb {R}^n$.

Proof

First we have $\nabla \partial _j=\varGamma ^l_{ij}\partial _l\otimes dx^i$ where the $\varGamma ^l_{ij}$ are the Christoffel symbols of second kind so that it is sufficient to prove that $\varGamma ^k_{ij}\in C_{{\text {pol}}}^{s-2,s}(U)$. For given $\phi \in G_V$, as a function of $u\in U$ we have $g_{ij}=\langle d \phi .\partial _i,d\phi .\partial _j\rangle \in C^{s-1}(U,\mathbb {R})$. Using the chain rule, we get easily for $u\in K$ that, for any $k\le s-1$, $|\partial ^kg_{ij}|\le P_k(\Vert \phi \Vert _{k+1,\infty })$ where P is a polynomial. Moreover, introducing $\psi =\phi '\circ \phi ^{-1}-{{\mathrm{Id}}}$,

$$\begin{aligned} g_{ij}(\phi ')-g_{ij}(\phi )&=\langle d(\psi +{{\mathrm{Id}}})\circ \phi \cdot d\phi \cdot \partial _i,d(\psi +{{\mathrm{Id}}})\circ \phi \cdot d\phi \cdot \partial _j\rangle \\&\quad -\langle d \phi \cdot \partial _i,d\phi \cdot \partial _j\rangle \\&=\langle d\psi \circ \phi \cdot d\phi \cdot \partial _i,d\psi \circ \phi \cdot d\phi \cdot \partial _j\rangle +\langle d\phi \cdot \partial _i,d\psi \circ \phi \cdot d\phi \cdot \partial _j\rangle \\&\quad +\langle d\psi \circ \phi \cdot d\phi \cdot \partial _i,d\phi \cdot \partial _j\rangle \end{aligned}$$

we get that $|\partial ^kg_{ij}(\phi ')-\partial ^kg_{ij}(\phi )|\le \Vert \psi \Vert _{k+1,\infty }Q_k(\Vert \psi \Vert _{k+1,\infty },\Vert \phi \Vert _{k+1,\infty })$ and we deduce immediately that $g_{ij}\in C_{{\text {pol}}}^{s-1,s}(U)$.

We need now a similar control for the cometric $g^{ij}$. Denoting $\varvec{g} =(g_{ij})_{1\le i,j\le d}$, we have $\varvec{g} ^{-1}=(g^{ij})_{1\le i,j\le d}$ and $\varvec{g} ^{-1}={{\mathrm{com}}}(\varvec{g} )^T/\det (\varvec{g} )$ where ${{\mathrm{com}}}(\varvec{g} )$ is the comatrix of the matrix $\varvec{g}$. Since ${{\mathrm{com}}}(\varvec{g} )^T$ is a polynomial expression in the coefficients $g_{ij}$ we get, using the algebra structure property of Remark 5, that all the coefficients of ${{\mathrm{com}}}(\varvec{g} )$ are in $C_{{\text {pol}}}^{s-1,s}(U)$. Similarly, $\det (\varvec{g} )\in C_{{\text {pol}}}^{s-1,s}(U)$ so that, in order to get $\det (\varvec{g} )^{-1}\in C_{{\text {pol}}}^{s-1,s}(U)$, it is sufficient to prove that for any compact $K\subset U$, there exists a polynomial P such that

$$\begin{aligned} \det (\varvec{g} )^{-1}\le P(\rho _{s}(\phi ))\,. \end{aligned}$$

(40)

However, since $T_{\phi (u)}Y=\text {Span}\{d\phi (u)\cdot \partial _i,\ 1\le i\le d\}$ where $Y=\phi (X)$, then for $(e_1,\ldots ,e_d)$ an orthonormal basis of $T_{\phi (u)}Y$, we have $\det (\varvec{g} )^{-1}=\det ((\langle d\phi ^{-1}(\phi (u)).e_i,d\phi ^{-1}(\phi (u)).e_j\rangle )_{ij})\le \Vert d\phi ^{-1}\Vert ^{2d}_\infty$. Using the fact that $\varGamma ^k_{ij}=\frac{1}{2}(\partial _ig_{mj}+\partial _j g_{mi}-\partial _mg_{ij})g^{mk}$ we get immediately that $\varGamma ^{k}_{ij}\in C_{{\text {pol}}}^{s-2,s}(U)$ and $\nabla \partial _i\in \varGamma _{{\text {pol}}}^{s-2,s}(T^1_1(U))$. Now since $0=\nabla (dx^i(\partial _j))=\nabla dx^i(\partial _j)+dx^i(\nabla \partial _j)$ we get $\nabla dx^i(\partial _j)=-\varGamma ^i_{lj}dx^l$ and $\nabla dx^i=-\varGamma ^i_{lj}dx^j\otimes dx^l$. Since we have just proved that $\varGamma ^i_{lj}\in C_{{\text {pol}}}^{s-2,s}(U)$, we get the result. $\square$

Lemma 5

Let $\mathcal {I}=\{\ (\alpha ,\beta )\ |\ \alpha \in \llbracket 1,d\rrbracket ^a,\ \beta \in \llbracket 1,d\rrbracket ^b,\ 1\le a<b\le s\ \}$ and $(x^i)_{1\le i\le d}$ be a coordinate system defined on an open set $U\subset X$.

There exists a family of functions $(c^\alpha _\beta )_{(\alpha ,\beta )\in \mathcal {I}}$ such that

• 1. for any $(\alpha ,\beta )\in \mathcal {I}$, we have $c^\alpha _\beta \in C_{{\text {pol}}}^{s-(1+|\beta |-|\alpha |),s}(U)\subset C_{{\text {pol}}}^{0,s}(U)$

• 2. for any $0\le k\le s$ any $f\in H^k_{loc}(U)$ and any $\beta \in \llbracket 1,d\rrbracket ^k$, we have (a.e.) on U

  $$\begin{aligned} \partial ^k_\beta f=\nabla ^k_\beta f+\sum _{l=1}^{k-1}\sum _{\alpha ,|\alpha |=l}c^\alpha _\beta \nabla ^l_\alpha f \end{aligned}$$

  (41)

where for $s\ge 1$ and $\phi \in {{\mathrm{Diff}}}^s_0(\mathbb {R}^n)$, $\nabla =\nabla ^\phi$ is Levi–Civita covariant derivative associated with the pullback metric $g=g^\phi$ on X on the Euclidean metric on $\phi (X)$ and where $\partial ^k_\beta f=\partial ^k\!f(\partial ^k_\beta )$ and $\nabla ^l_\alpha f=\nabla ^l\! f(\partial ^l_\alpha )$.

Proof

For $k=0$ or $k=1$ the result is trivial. Let consider a proof by induction for $k\ge 1$. We have for $\beta \in \llbracket 1,d\rrbracket ^k$ and ${\tilde{\beta }}=(i,\beta )$ that

$$\begin{aligned} \partial ^{k+1}_{\tilde{\beta }}f=\partial _i(\partial ^k_\beta f)=\partial _i\left[ \nabla ^k_\beta f+\sum _{l=1}^{k-1}\sum _{\alpha ,|\alpha |=l}c^\alpha _\beta \nabla ^l_\alpha f\right] \,. \end{aligned}$$

However, $\partial _i(\nabla ^k_\beta f)=\nabla ^{k+1}_{\tilde{\beta }}f+\nabla ^k f(\nabla _{\partial _i}\partial ^k_\beta )$. Moreover, since we have

$$\begin{aligned} \nabla _{\partial _i}\partial ^k_\beta =\sum _{l=1}^k\otimes _{j=1}^{l-1}\partial _{\beta _l}\otimes \nabla _{\partial _i} \partial _{\beta _l}\otimes _{j=l+1}^{k}\partial _{\beta _j}=\sum _{l=1}^k\varGamma _{i\beta _l}^m\otimes _{j=1}^{l-1}\partial _{\beta _j}\otimes \partial _{m}\otimes _{j=l+1}^{k}\partial _{\beta _j} \end{aligned}$$

we get that $\nabla _{\partial _i}\partial ^k_\beta \in \varGamma _{{\text {pol}}}^{s-2,s}(T^k_0(U))$ and $\nabla ^k f(\nabla _{\partial _i}\partial ^k_\beta )$ can be written as $\sum _{\alpha ,|\alpha |=k}c^\alpha _{\tilde{\beta }}\nabla ^k_\alpha f$ for functions $c^\alpha _{\tilde{\beta }}\in C_{{\text {pol}}}^{s-2,s}(U)$. Similarly, we have for $1\le l\le k$ and $\alpha \in \llbracket 1,d\rrbracket ^l$ that

$$\begin{aligned} \partial _i(c^\alpha _\beta \nabla ^l_\alpha f)=\partial _i(c^\alpha _\beta )\nabla ^l_\alpha f+ c^\alpha _\beta \nabla ^{l+1}_{(i,\alpha )}f+c^\alpha _\beta \nabla ^lf(\nabla _{\partial _i}\partial ^l_\alpha ). \end{aligned}$$

Denoting $c^\alpha _{{\tilde{\beta }},1}=\partial _i(c^\alpha _\beta )\in C_{{\text {pol}}}^{s-(1+|{\tilde{\beta }}|-|\alpha |),s}(U)$, $c^{(i,\alpha )}_{{\tilde{\beta }},2}=c^\alpha _\beta \in C_{{\text {pol}}}^{s-(1+|\beta |-|\alpha |),s}(U)=C_{{\text {pol}}}^{s-(1+|{\tilde{\beta }}|-|(i,\alpha )|),s}(U)$ and since $\nabla _{\partial _i}\partial ^l_\alpha \in \varGamma _{{\text {pol}}}^{s-2,s}(T^{l+1}_0(U))$, $c^\gamma _{{\tilde{\beta }},3}=c^\alpha _\beta dx^\gamma (\nabla _{\partial _i}\partial _\alpha ^l)\in C_{{\text {pol}}}^{s-(1+|\beta |-l),s}(U)\subset C_{{\text {pol}}}^{s-(1+|{\tilde{\beta }}|-|\gamma |),s}(U)$ we get that $\partial _i(c^\alpha _\beta \nabla ^l_\alpha f)$ can be written as $\sum _{m=l}^{l+1}\sum _{\gamma ,|\gamma |=m}c^\gamma _{{\tilde{\beta }}}\nabla ^m_\gamma f$ for some appropriate functions $c^\gamma _{\tilde{\beta }}\in C_{{\text {pol}}}^{s-(1+|{\tilde{\beta }}|-|\gamma |),s}(U)$ and decomposition (41) holds for the rank $k+1$. $\square$

We finally get to the main result itself.

Proof of Theorem 1

The starting point is to recast the Sobolev norm on $Y=\phi (X)$ as an integral on X through the pullback metric and pullback covariant derivative. Up to the introduction of a finite partition of unity $(\chi _l)$ subordinated to finite covering of X with charts $(U_l,\psi _l)$, we can restrict to one open set $U=U_l$ and show that for $\chi =\chi _l$ and $K=\text {supp}(\rho )$, there exists a polynomial P such that

$$\begin{aligned} \sum _{k=0}^s\int _K \chi .g^0_k(\nabla ^k f,\nabla ^k f){{\mathrm{vol}}}(g)\le P(\rho _s(\phi ))\sum _{k=0}^s\int _K \chi .\overline{g}^0_k(\overline{\nabla }^k f,\overline{\nabla }^k f){{\mathrm{vol}}}(\overline{g}) \end{aligned}$$

(42)

where $\overline{g}=g^{\text {Id}}$ and $\overline{\nabla }=\nabla ^{\text {Id}}$. For $s=0$ the results comes from the inequalities (40). Let assume that $s\ge 1$ (and thus $s'=s$). From Lemma 5, there exists universal functions $c^\alpha _\beta \in C_{{\text {pol}}}^{0,s}(U)$ for any pair $(\alpha ,\beta )\in \mathcal {I}$ such that $\partial ^k_\beta f=\nabla ^k_\beta f+\sum _{l=1}^{k-1}\sum _{\alpha ,|\alpha |=l}c^\alpha _\beta \nabla ^l_\alpha f$. In particular, if we denote $\mathcal {J}\doteq \{(k,\gamma )\ |\ 0\le k\le d,\ \gamma =\llbracket 1,d\rrbracket ^k\ \}$, $\varvec{f} =(\partial ^k_\gamma f)_{(k,\gamma )\in \mathcal {J}}$ and $\varvec{\tilde{f}} =(\nabla ^k_\gamma f)_{(k,\gamma )\in \mathcal {J}}$, then there exists $\varvec{M} \in C_{{\text {pol}}}^{0,s}(U,L(\mathbb {R}^\mathcal {J},\mathbb {R}^\mathcal {J}))$ (invertible since triangular with ones on the diagonal) with coefficients in $C_{{\text {pol}}}^{0,s}(U)$ such that $\varvec{f} =\varvec{M} \varvec{\tilde{f}}$. Moreover, since $\sum _{k=0}^sg^0_k(\nabla ^k f,\nabla ^k f){{\mathrm{vol}}}(g)$ can be rewritten as $\varvec{q}(\varvec{\tilde{f}} )$ where $\varvec{q}$ is a non-degenerate positive quadratic form continuously depending on the location $u\in U$ and coefficients in $C_{{\text {pol}}}^{s-1,s}(U)\subset C_{{\text {pol}}}^{0,s}(U)$, we get that there exists a polynomial $\tilde{P}$ such that $\varvec{q}(\varvec{\tilde{f}} )=\varvec{q}(\varvec{M} ^{-1}\varvec{f} )\le P(\rho _s(\phi ))|\varvec{f} |^2$ so that

$$\begin{aligned} \sum _{k=0}^s\int _K \chi .g^0_k(\nabla ^k f,\nabla ^k f){{\mathrm{vol}}}(g)\le \tilde{P}(\rho _s(\phi ))\sum _{k=0}^s\int _K \chi .|\partial ^k f|^2\mathrm{d}x. \end{aligned}$$

Furthermore, considering $\varvec{M}$ for $\phi =\text {Id}$ there exists a constant $R\ge 0$ such that we have $\sum _{k=0}^s\int _K \chi .|\partial ^k f|^2\mathrm{d}x\le R\sum _{k=0}^s\int _K \chi .\overline{g}^0_k(\overline{\nabla }^k f,\overline{\nabla }^k f){{\mathrm{vol}}}(\overline{g})$ so that (42) holds with $P=R\tilde{P}$ and we have obtained Theorem 1. $\square$

We conclude this appendix by adding an extra property of continuity with respect to $\phi$ of the pullback $H^s$ metrics, which is used in the proof of Theorem 2. From the previous developments, we get that for any chart $(U,\varphi )$ on X associated with a coordinate system $(x^1,\ldots ,x^d)$ on U there exists a family of functions $c^\alpha _{\beta }$ such that for any $f\in H^s_{loc}(U)$

$$\begin{aligned} \partial ^k_\beta f=\nabla ^k_\beta f+\sum _{l=1}^{k-1}\sum _{\alpha ,|\alpha |=l}c^\alpha _\beta \nabla ^l_\alpha f\,. \end{aligned}$$

(43)

Let us denote $E\doteq \bigoplus _{k=0}^s ({\mathop {\otimes }\limits ^{k}}T^*X)$, E is a $C^{s-1}$ vector bundle over X. For any local chart $(U,\varphi )$ with coordinate functions $(x^1,\ldots ,x^d)$, $(q_k(dx^\beta ))_{\beta \in \llbracket 1,d\rrbracket ^k,1\le k\le d}$ is a local frame of E over U where $q_k:{\mathop {\otimes }\limits ^{k}}T^*X\rightarrow E$ denotes the canonical embedding. We will also consider ${{\mathrm{End}}}(E)\rightarrow X$ the endomorphism vector bundle where ${{\mathrm{End}}}(E)_x\doteq {{\mathrm{End}}}(E_x)$.

Definition 4

We say that $M\in \varGamma ^{0,s}({{\mathrm{End}}}(E))$ where $M:{{\mathrm{Diff}}}^s_0(\mathbb {R}^n)\rightarrow \varGamma ^0({{\mathrm{End}}}(E)))$ if for any coordinate system $(x^1,\ldots ,x^d)$ defined on a open set $U\subset M$, all the coefficients of M in the local frame $(dx^\beta )_\beta$ are in $C_{{\text {pol}}}^{0,s}(U)$.

Definition 5

We say that $G\in \varGamma ^{0,s}(E^*\otimes E^*)$ where $G:{{\mathrm{Diff}}}^s_0(\mathbb {R}^n)\rightarrow \varGamma ^0(E^*\otimes E^*)$ if for any coordinate system $(x^1,\ldots ,x^d)$ defined on a open set $U\subset M$, all the coefficients of G in the local frame $(q_k(\partial ^k_\alpha )\otimes q_{k'}(\partial ^{k'}_{\alpha '}))$ for $0\le k,k'\le s$ and $(\alpha ,\alpha ')\in \llbracket 1,d\rrbracket ^k\times \llbracket 1,d\rrbracket ^{k'}$ are in $C_{{\text {pol}}}^{0,s}(U)$, where $q_k:{\mathop {\otimes }\limits ^{k}}TM\rightarrow E^*$ is the canonical embedding.

Now, writing

$$\begin{aligned} \Vert f\Vert _{H^{s,\phi }(X)} \doteq \Vert f \circ \phi ^{-1} \Vert _{H^s(\phi (X))} \end{aligned}$$

as the pullback $H^s$ metric on X induced by $\phi \in {{\mathrm{Diff}}}^s_0(\mathbb {R}^n)$, we have the following property:

Lemma 6

For any $f\in H^{s}(X)$, the application ${{\mathrm{Diff}}}^s_0(\mathbb {R}^n) \rightarrow \mathbb {R}_{+}$, $\phi \mapsto \Vert f\Vert _{H^{s,\phi }(X)}$ is continuous.

Proof

Let us introduce $p_E:{{\mathrm{Diff}}}^s_0(\mathbb {R}^n)\times H^s(X)\rightarrow L^2(X,E)$ such that

$$\begin{aligned} p_E(\phi ,f)\doteq \bigoplus _{k=0}^s q_k(\nabla ^k f) \end{aligned}$$

where $\nabla =\nabla ^\phi$ for $\phi \in {{\mathrm{Diff}}}^s_0(\mathbb {R}^n)$ as well as the pullback of the metric $g_E(\phi )\doteq \oplus _{k=0}^s g^0_k$ with once again $g=g^\phi$. Then, we have by definition

$$\begin{aligned} \Vert f\Vert _{H^{s,\phi }(X)}^2=\int _X g_E(\phi )\left( p_E(\phi ,f),p_E(\phi ,f)\right) {{\mathrm{vol}}}(g^\phi ) \ . \end{aligned}$$

Now, with (43), we see that there exists $M\in \varGamma ^{0,s}({{\mathrm{End}}}(E))$ such that $p_E(\phi ,f)=M(\phi )\cdot p_E({{\mathrm{Id}}},f)$. Similarly, thanks to the previously derived expressions of the metric $g^\phi$, we have $g_E\in \varGamma ^{0,s}(E^*\otimes E^*)$ and thus there exists $S\in \varGamma ^{0,s}({{\mathrm{End}}}(E))$ such that

$$\begin{aligned}&\int _X g_E(\phi )\left( p_E(\phi ,f),p_E(\phi ,f)\right) {{\mathrm{vol}}}(g^\phi )=\int _X \overline{g}_E\left( S(\phi )\cdot p_E(\phi ,f),p_E(\phi ,f)\right) {{\mathrm{vol}}}(\overline{g})\\&\quad =\int _X \overline{g}_E\left( S(\phi )\cdot M(\phi )\cdot \overline{p}_E(f),M(\phi )\cdot \overline{p}_E(f)\right) {{\mathrm{vol}}}(\overline{g})\\&\quad =\int _X \overline{g}_E\left( \varLambda (\phi )\cdot \overline{p}_E(f),\overline{p}_E(f)\right) {{\mathrm{vol}}}(\overline{g}) \end{aligned}$$

with $\varLambda \in \varGamma ^{0,s}({{\mathrm{End}}}(E))$ and $\overline{g}_E=g_E({{\mathrm{Id}}})$, $\overline{p}_E(f) = p_E({{\mathrm{Id}}},f)$. Since the coefficients of $\varLambda$ in a local frame belong to $C_{{\text {pol}}}^{0,s}$, they are in particular continuous with respect to $\phi$ for the norm of uniform convergence of $\phi$ and its derivatives up to order s on the compact X. As a consequence, if $\phi ^n \rightarrow \phi$, then $\varLambda (\phi ^n) \rightarrow \varLambda (\phi )$ in $\varGamma ^0({{\mathrm{End}}}(E))$ and:

$$\begin{aligned} \Vert f\Vert _{H^{s,\phi ^n}(X)}^2 \xrightarrow [n\rightarrow \infty ]{} \int _X \overline{g}_E\left( \varLambda (\phi )\cdot \overline{p}_E(f),\overline{p}_E(f)\right) {{\mathrm{vol}}}(\overline{g}) = \Vert f\Vert _{H^{s,\phi }(X)}^2 \end{aligned}$$

which completes the proof. $\square$

Proof of Theorem 6

The proof follows similar steps as the pure diffeomorphic case derived in [5]. Let us introduce the total cost functional:

$$\begin{aligned} J(q,\check{f},v,\check{h})&= \int _0^1 \left[ \tfrac{\gamma _V}{2} \Vert v_t\Vert _{V}^2 +\tfrac{\gamma _f}{2} \Vert \check{h_t}\Vert _{H^s_q}^2 \right] \mathrm{d}t +g(q_1,\check{f_1}) \\&=\int _0^1 L(q_t,v_t,\check{h}_t)\mathrm{d}t +g(q_1,\check{f_1}) \end{aligned}$$

where L is by definition the Lagrangian function. It is differentiable with respect to $v \in V$, $\check{h} \in H^s(M)$ as well as $q \in C^{s'}(M,\mathbb {R}^n)$ since $s'\ge s$. The variation of J writes:

$$\begin{aligned}&\Big (\delta J(q,\check{f},v,\check{h})\Big |(\delta q, \delta \check{f},\delta v, \delta \check{h})\Big ) \\&=\int _0^1 \left[ \Big (\partial _q L(q_t,v_t,\check{h}_t)\Big |\delta q_t\Big ) + \Big (\partial _{v} L(q_t,v_t,\check{h}_t)\Big |\delta v_t\Big ) + \Big (\partial _{\check{h}} L(q_t,v_t,\check{h}_t)\Big |\delta \check{h}_t\Big )\right] \mathrm{d}t \\&\qquad + \Big (\partial _{q}g(q_1,\check{f_1})\Big |\delta q_1\Big ) + \Big (\partial _{\check{f}}g(q_1,\check{f_1})\Big |\delta \check{f}_1\Big ) \end{aligned}$$

Note that the previous expression involves different duality brackets, in $(C^{s'}(M,\mathbb {R}^n)^*,C^{s'}(M,\mathbb {R}^n))$ for variation with respect to $\delta q$, in $(V^*,V)$ for the variation with respect to $\delta v$ and in $(H^{s}(M)^*,H^s(M))$ for the variation with respect to $\delta \check{h}$ and $\delta \check{f}$. Formally, the optimality of solutions $(q_t,\check{f}_t,v_t,\check{h}_t)$ means that $\delta J$ should vanish under variations satisfying the control evolutions $\dot{q_t} = \xi _{q_t}v_t$ and $\dot{\check{f_{t}}} = \check{h}_t$.

Let $H_{(q_0,\check{f}_0)}^{1}([0,1],C^{s'}(M,\mathbb {R}^n)\times H^s(M))$ be the space of time-dependent states with $H^1$ regularity in time and initial conditions $(q_0,\check{f}_0)$. We define the constraint application

$$\begin{aligned}&\varUpsilon : \ H_{(q_0,\check{f}_0)}^{1}([0,1],C^{s'}(M,\mathbb {R}^n)\times H^s(M)) \times L^2([0,1],V\times H^s(M)) \\&\quad \rightarrow L^2([0,1],C^{s'}(M,\mathbb {R}^n)) \times L^2([0,1],H^s(M)) \end{aligned}$$

by $\varUpsilon (q,\check{f},v,\check{h}) \doteq (\dot{q}-\xi _{q}v,\dot{\check{f}}- \check{h})$. It is clearly differentiable with respect to $\check{f},v$ and $\check{h}$. Now, since it is assumed that $V\hookrightarrow \varGamma ^{s+1}$, the application $q \mapsto \xi _q v = v \circ q$ is differentiable with respect to $q \in C^{s'}(M,\mathbb {R}^n)$ and equal to $(\partial _q \xi _q v | \delta q) = d_{q} v(\delta q)$. It results that $\varUpsilon$ is differentiable with respect to q as well.

With these notations, we are considering minimizers of J in the constraint set $\varUpsilon ^{-1}(\{0\})$. In order to invoke Lagrange multipliers theorem in this infinite-dimensional setting (Theorem 4.1 in [31]), it needs to be checked that $d_{(q,\check{f},v,\check{h})}\varUpsilon$ is surjective for all $(q,\check{f},v,\check{h})$. Writing $\varUpsilon _{1}(q,v) = \dot{q}-\xi _{q}v$ and $\varUpsilon _{2}(\check{f},\check{h}) = \dot{\check{f}}- \check{h}$, we have from Lemma 3 of [5] that $d_{(q,v)}\varUpsilon _1$ is surjective and it is straightforward to verify that so is $d_{(\check{f},\check{h})}\varUpsilon _2$. We deduce the existence of Lagrange multipliers $p \in L^2([0,1],C^{s'}(M,\mathbb {R}^n))^*$ and $p^f \in L^2([0,1],H^s(M))^*$ such that:

$$\begin{aligned} 0&= \Big (d_{(q,\check{f},v,\check{h})}J + (d_{(q,\check{f},v,\check{h})}\varUpsilon )^*(p,p^f) \Big | (\delta q,\delta \check{f},\delta v, \delta \check{h}) \Big ) \nonumber \\&= (p|\dot{\delta q}) - (p|\partial _q \xi _q v.\delta q) - (p|\xi _q \delta v) + (p^f|\dot{\delta \check{f}}) - (p^f|\delta \check{h}) \nonumber \\&\quad +\int _0^1 \left[ (\partial _q L(q_t,v_t,\check{h}_t)|\delta q_t) + (\partial _{v} L(q_t,v_t,\check{h}_t)|\delta v_t) + (\partial _{\check{h}} L(q_t,v_t,\check{h}_t)|\delta \check{h}_t)\right] \mathrm{d}t \nonumber \\&\quad + (\partial _{q}g(q_1,\check{f_1})|\delta q_1) + (\partial _{\check{f}}g(q_1,\check{f_1})|\delta \check{f}_1) \end{aligned}$$

(44)

Moreover, as $H^s(M)$ is reflexive, it satisfies the Radon–Nikodym property and we have $L^2([0,1],H^s(M))^* = L^2([0,1],H^s(M)^*)$ which allows to identify $p^f$ as a square-integrable function in $H^s(M)^*$. The case of the geometric momentum p is, however, slightly more involved but was addressed separately in Lemma 4 of [5], leading to an equivalent identification $p \in L^2([0,1],C^{s'}(M,\mathbb {R}^n)^*)$. It is then straightforward from the expression of the Hamiltonian in (20) that $\dot{q_t} = \xi _{q_t} v_t = \partial _p H(q_t,\check{f}_t,p_t,p_t^{f},v_t,\check{h}_t)$ and $\dot{\check{f}} = \check{h}_t = \partial _{p^f} H(q_t,\check{f}_t,p_t,p_t^{f},v_t,\check{h}_t)$.

Considering the variation on $\delta q$ only (i.e., with $\delta v = 0$, $\delta \check{f} = 0$ and $\delta \check{h} =0$) in (44), we obtain for all $\delta q \in C^s(M,\mathbb {R}^n)$:

$$\begin{aligned} (p|\dot{\delta q})&= (p|\partial _q \xi _q v.\delta q)-\int _0^1 (\partial _q L(q_t,v_t,\check{h}_t)|\delta q_t) \mathrm{d}t - (\partial _{q}g(q_1,\check{f_1})|\delta q_1) \nonumber \\&=\int _0^1 (p_t|(\partial _{q} \xi _{q_t} v_t)(\delta q_t)) \mathrm{d}t - \int _0^1 (\partial _q L(q_t,v_t,\check{h}_t)|\delta q_t) \mathrm{d}t - (\partial _{q}g(q_1,\check{f_1})|\delta q_1) \nonumber \\&=\int _0^1 (\underbrace{(\partial _{q} \xi _{q_t} v_t)^*p_t-\partial _q L(q_t,v_t,\check{h}_t)}_{\doteq \alpha _t}|\delta q_t) \mathrm{d}t - (\partial _{q}g(q_1,\check{f_1})|\delta q_1) \end{aligned}$$

(45)

Let’s denote $r_t = \dot{\delta q}_t$ so that $\delta q_t = \int _{0}^{t} r_s \mathrm{d}s$ and:

$$\begin{aligned} \int _0^1 (\alpha _t|\delta q_t) \mathrm{d}t&= \int _0^1 \int _0^t (\alpha _t|r_s) \mathrm{d}t \\&= \int _{0}^1 \left( \int _{s}^1 \alpha _t \mathrm{d}t \Big | r_s \right) \mathrm{d}s \end{aligned}$$

This together with (45) shows that $p_t = \int _{t}^1 \alpha _s \mathrm{d}s - \partial _{q}g(q_1,\check{f_1})$ for almost all $t\in [0,1]$. Now since $\alpha \in L^2([0,1],C^{s'}(M,\mathbb {R}^n)^*) \subset L^1([0,1],C^{s'}(M,\mathbb {R}^n)^*)$, it results that $p \in H^1([0,1],C^{s'}(M,\mathbb {R}^n)^*)$ and:

$$\begin{aligned} \dot{p}_t = -\alpha _{t} = \partial _q L(q_t,v_t,\check{h}_t) - (\partial _{q} \xi _{q_t} v_t)^*p_t = -\partial _q H(q_t,\check{f}_t,p_t,p_t^{f},v_t,\check{h}_t) \end{aligned}$$

with the endpoint condition $p_1 = -\partial _{q}g(q_1,\check{f_1})$.

Similarly, the variation with respect to $\delta \check{f}$ in (44) leads to:

$$\begin{aligned} (p^f|\dot{\delta \check{f}}) + (\partial _{\check{f}}g(q_1,\check{f_1})|\delta \check{f}_1) =0 \end{aligned}$$

If we write $\rho _t = \dot{\delta \check{f_t}}$, we obtain:

$$\begin{aligned} \int _0^1 \left( p^f_t + \partial _{\check{f}}g(q_1,\check{f_1}) \big | \rho _t \right) \mathrm{d}t =0 \end{aligned}$$

which thus holds for all $\rho \in L^2([0,1],H^s(M))$. It results that for almost all $t \in [0,1]$, $p^f_t = -\partial _{\check{f}}g(q_1,\check{f_1})$ or in other words $p^f \in H^1([0,1],H^s(M)^*)$ and:

$$\begin{aligned} \dot{p}_t^f = 0 = -\partial _{\check{f}} H(q_t,\check{f}_t,p_t,p_t^{f},v_t,\check{h}_t) \end{aligned}$$

Finally, the variations with respect to v and $\check{h}$ give:

$$\begin{aligned}&\int _0^1 \left( \xi _{q_t}^* p_t - (\partial _{v} L(q_t,v_t,\check{h}_t)|\delta v_t) \right) \mathrm{d}t =0 \\&\int _0^1 \left( p^f_t - \partial _{\check{h}} L(q_t,v_t,\check{h}_t)|\delta \check{h}_t \right) \mathrm{d}t =0 \end{aligned}$$

for all $\delta v \in L^2([0,1],V), \delta \check{h} \in L^2([0,1],H^s(M))$. Therefore

$$\begin{aligned}&\xi _{q_t}^* p_t - (\partial _{v} L(q_t,v_t,\check{h}_t) = \partial _{v} H(q_t,\check{f}_t,p_t,p_t^{f},v_t,\check{h}_t) = 0\\&p^f_t - \partial _{\check{h}} L(q_t,v_t,\check{h}_t) = \partial _{\check{h}} H(q_t,\check{f}_t,p_t,p_t^{f},v_t,\check{h}_t) = 0 \end{aligned}$$

and the proof of Theorem 6 is complete.