Periodic trajectory tracking for control-affine driftless systems on compact Lie groups

TL;DR

This paper addresses the problem of asymptotic periodic trajectory tracking for control-affine, driftless systems on compact Lie groups, providing local solutions under specific controllability and regularity conditions.

Contribution

It introduces a local solution to the periodic trajectory tracking problem on compact Lie groups for semisimple cases with controllability, and establishes conditions for trajectory existence.

Findings

Solution exists locally for semisimple Lie groups and controllable systems.

Provides sufficient conditions for the existence of periodic trajectories.

Addresses tracking for a class of regular periodic trajectories.

Abstract

We treat the periodic trajectory tracking problem: given a periodic trajectory of a control-affine, left-invariant driftless system in a compact and connected Lie group $G$ and an initial condition in $G$, find another trajectory of the system satisfying the initial condition given and that asymptotically tracks the periodic trajectory. We solve this problem locally (for initial conditions in a neighborhood of some point of the periodic trajectory) when $G$ is semisimple and the system is Lie-determined (i.e. controllable), and only for a class of periodic trajectories (which we call regular). Finally we present a set of sufficient conditions to ensure the existence of such trajectories.

Equations

$${\mathbf{x}}^{\prime }$$

$$X(t,x)$$

$$\underset{t\to \infty }{\lim }\mathbf{x}(t)\cdot {\mathbf{x}}_r(t{)}^{-1}$$

$$\underset{t\to \infty }{\lim }\mathbf{x}(t)\cdot \mathbf{y}(t{)}^{-1}=e$$

$$\underset{k\to \infty }{\lim }\mathbf{x}(kT)$$

$$x_{\infty }$$

$${\mathbf{z}}^{\prime }$$

$$\mathbf{x}$$

$$u_k$$

$$({\mathbf{x}}_1\cdot {\mathbf{x}}_2{)}^{\prime }$$

$${\mathbf{x}}^{\prime }=(\mathbf{z}\cdot {\mathbf{x}}_r{)}^{\prime }=\mathrm{d}{R}_{{\mathbf{x}}_r}{\mathbf{z}}^{\prime }+\mathrm{d}{L}_{\mathbf{z}}{\mathbf{x}}_r^{\prime }=\mathrm{d}{R}_{{\mathbf{x}}_r}\sum _{k=1}^m{v}_k\mathrm{Ad}({\mathbf{x}}_r)X_k(\mathbf{z})+\mathrm{d}{L}_{\mathbf{z}}\sum _{k=1}^m{u}_k^r{X}_k({\mathbf{x}}_r)$$

$$\mathrm{d}{R}_{{\mathbf{x}}_r}\sum _{k=1}^m{v}_k\mathrm{Ad}({\mathbf{x}}_r)X_k(\mathbf{z})$$

$$=\mathrm{d}{R}_{{\mathbf{x}}_r}\sum _{k=1}^m{v}_k\mathrm{d}{R}_{{\mathbf{x}}_r^{-1}}{X}_k(R_{{\mathbf{x}}_r}\mathbf{z})$$

$$=\mathrm{d}{R}_{{\mathbf{x}}_r}\mathrm{d}{R}_{{\mathbf{x}}_r^{-1}}\sum _{k=1}^m{v}_k{X}_k({\mathbf{x}}_r\cdot \mathbf{z})$$

$$=\sum _{k=1}^m{v}_k{X}_k(\mathbf{x})$$

$$\mathrm{d}{L}_{\mathbf{z}}\sum _{k=1}^m{u}_k^r{X}_k({\mathbf{x}}_r)=\sum _{k=1}^m{u}_k^r\mathrm{d}{L}_{\mathbf{z}}{X}_k({\mathbf{x}}_r)=\sum _{k=1}^m{u}_k^r{X}_k(L_{\mathbf{z}}{\mathbf{x}}_r)=\sum _{k=1}^m{u}_k^r{X}_k(\mathbf{x})$$

$$\mathbf{x}(0)=\mathbf{z}(0)\cdot {\mathbf{x}}_r(0)=x_0\cdot x_{\infty }^{-1}\cdot x_{\infty }=x_0$$

$$\underset{t\to \infty }{\lim }\mathbf{x}(t)\cdot {\mathbf{x}}_r(t{)}^{-1}=\underset{t\to \infty }{\lim }\mathbf{z}(t)=e.$$

$$V(x)$$

$$W(t,w)$$

$$a_k(t,w)$$

$$v_k(t)$$

$${\mathcal{C}}_W$$

$$\mathbf{x}(t)$$

$$u_k(t)$$

$$V(\mathbf{x}(t_n^j))$$

$$V(\mathbf{x}(t_k^1))\le V(\mathbf{x}(t_{n_k}^2))\le V(x_2),\forall k\in \mathbb{N}.$$

$$\underset{t\to \infty }{\lim }V(\mathbf{x}(t))$$

$$\underset{t\to \infty }{\lim }\Vert {\mathbf{x}}^{\prime }(t)\Vert$$

$$\underset{n\to \infty }{\lim }\mathbf{x}(t_n+ϵ)$$

$$\underset{n\to \infty }{\lim }d(\mathbf{x}(t_n+ϵ),x)$$

$$d(\mathbf{x}(t_n+ϵ),\mathbf{x}(t_n))\le \left|{\int }_{I(t_n,t_n+ϵ)}\Vert {\mathbf{x}}^{\prime }(t)\Vert \mathrm{d}t\right|\le \left(\underset{t\in I(t_n,t_n+ϵ)}{\sup }\Vert {\mathbf{x}}^{\prime }(t)\Vert \right)|ϵ|$$

$$n\ge n_0⟹\max \{t_n,t_n+ϵ\}>R⟹\underset{t\in I(t_n,t_n+ϵ)}{\sup }\Vert {\mathbf{x}}^{\prime }(t)\Vert <\delta$$

Full text

This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, grants 131876/2010-4 and 140838/2012-0) and the São Paulo Research Foundation (FAPESP, grant 2018/12273-5).

1. Introduction

The present work addresses the problem of periodic trajectory tracking for control-affine driftless systems, specifically in the case when the ambient manifold is a Lie group $G$ (which we will further assume to be compact and connected) and the system is left-invariant (see below). It is heavily inspired by [7] (see also its first author’s PhD thesis [6]), in which the problem is studied in $\mathrm{SU}(n)$ aiming applications to Quantum Computing, and can indeed be considered as a (tentative) extension of their methods to abstract Lie groups. We do not, however, rely on any of their results or even notations directly, but rather on their ideas; nor we aim at any applications whatsoever.

In Section 2 we describe the periodic trajectory tracking problem (PTTP) for our system (2.1) and reduce it to the problem of stabilization of an auxiliary system (2.2). The main conclusion here is that if the identity element of $e$ of $G$ is a critical point, and moreover a local attractor of this new system, then one can solve the PTTP locally i.e. for initial conditions close to the reference periodic trajectory. This leads us to investigate some aspects of the stability of time-dependent vector fields on compact Riemannian manifolds, which we do in Section 3, and then apply our conclusions to characterize the $\omega$-limit points of an auxiliary vector field $W$ associated to (2.2): they are precisely the critical points of $W$. We also conclude that every central point of $G$ is critical to $W$, so a necessary condition for our approach to work is that $G$ is semisimple e.g. $\mathrm{SU}(n)$.

In Section 4 we restrict our attention to a class of periodic trajectories, which we call regular, for which an even simpler characterization of the critical points of the associated $W$ is achieved: they are the critical points of a Lyapunov-like function $V$; and moreover central points of $G$ are non-degenerate critical points of $V$ provided $G$ is semisimple. A little more effort then allows us to conclude that, in that case, the latter points are also local attractors of $W$, and since $e$ is obviously central we solve the PTTP locally. We close this work (Section 5) discussing a condition that ensures the existence of periodic trajectories, including a more or less concrete construction of them.

We refer the reader to [5] and [1] for the basics of Control Theory on Lie groups. For more sophisticate aspects of Lie group theory – notably some results regarding the adjoint representation of $G$, to which we are naturally led by the change of variables that produces the auxiliary system (2.2) and that stalks us until the end, revealing how semisimplicity is an essential feature to the problem – the reader is referred to a less introductory text on the subject e.g. [3]; more paramount results and definitions, as well as possibly non-standard notation, are also briefly explained in the footnotes.

Acknowledgments

I wish to thank H. B. Silveira and P. A. Tonelli for reading the original manuscript and making invaluable suggestions, and the latter also for many long discussions and for proposing the problem.

2. The periodic trajectory tracking problem

Let $G$ be a compact, connected Lie group, whose Lie algebra of left-invariant vector fields we denote by $\mathfrak{g}$. Given $X_{1},\ldots,X_{m}\in\mathfrak{g}$ we consider the left-invariant driftless system

[TABLE]

where $u_{1},\ldots,u_{m}\in\mathbb{R}$ are controls. We shall work exclusively with smooth trajectories: $(m+1)$-uples $(\mathbf{x},u_{1},\ldots,u_{m})$ where $u_{1},\ldots,u_{m}:\mathbb{R}\rightarrow\mathbb{R}$ are smooth (i.e. $C^{\infty}$) functions – the controls – and $\mathbf{x}:\mathbb{R}\rightarrow G$ is an integral curve of the time-dependent vector field

[TABLE]

The trajectory is said to be $T$-periodic ($T>0$) provided $\mathbf{x},u_{1},\ldots,u_{m}$ are $T$-periodic functions.

For simplicity, we shall assume that $\Gamma\mathrel{\dot{=}}\operatorname{\mathrm{span}}\{X_{1},\ldots,X_{m}\}$ is bracket-generating i.e. the Lie algebra generated by $\Gamma$ is $\mathfrak{g}$, and hence $\Gamma$ has a single orbit thanks to Sussmann’s Theorem.

Definition 2.1**.**

The periodic trajectory tracking problem (PTTP) for system (2.1) is stated as follows: given a $T$-periodic reference trajectory $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ and an initial state $x_{0}\in G$, find another (non-periodic) trajectory $(\mathbf{x},u_{1},\ldots,u_{m})$ of (2.1) such that $\mathbf{x}(0)=x_{0}$ and111$e$: the identity element of $G$.

[TABLE]

We call $\mathbf{x}\cdot\mathbf{x}_{r}^{-1}$ the tracking error between the two trajectories.

Remark 2.2*.*

The motivation for our definition of tracking error comes from concrete examples. If $G$ is a subgroup of $\operatorname{\mathrm{GL}}(n,\mathbb{C})$ and $\mathbf{x},\mathbf{y}:\mathbb{R}\rightarrow G$ are curves then clearly

[TABLE]

where $\|\cdot\|$ is any matrix norm.

Moreover, consider the following asymptotic controllability problem (also sometimes called the $T$-sampling stabilization problem, see for instance [8]) for system (2.1):

Given an initial state $x_{0}\in G$ and a target state $x_{\infty}\in G$, find a trajectory $(\mathbf{x},u_{1},\ldots,u_{m})$ of (2.1) such that, for some $T>0$, we have

[TABLE]

It is clear that this problem can be solved if we are able to find

1. (1)

$(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ a periodic reference trajectory for (2.1) with $\mathbf{x}_{r}(0)=x_{\infty}$ and 2. (2)

$(\mathbf{x},u_{1},\ldots,u_{m})$ a trajectory of (2.1) that tracks $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ i.e. solving the PTTP.

While the second question above is the main subject of the present paper, we shall discuss the first one – the existence of periodic reference trajectories passing through arbitrary points of $G$ – in Section 5.

The very definition of the tracking error suggests that we can reduce the PTTP associated to a given reference trajectory $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ to a stabilization problem, via a change of coordinates that we describe below. From now on we denote

[TABLE]

Proposition 2.3**.**

Assume that $(\mathbf{z},v_{1},\ldots,v_{m})$ is a trajectory of the system222The adjoint map $\operatorname{\mathrm{Ad}}:G\rightarrow\operatorname{\mathrm{GL}}(\mathfrak{g})$ is the group homomorphism that associates to each $x\in G$ an invertible linear map $\operatorname{\mathrm{Ad}}(x):\mathfrak{g}\rightarrow\mathfrak{g}$ as follows: if $I_{x}$ stands for the map $y\in G\mapsto x\cdot y\cdot x^{-1}\in G$ then $\operatorname{\mathrm{Ad}}(x)$ corresponds to $\mathrm{d}(I_{x})_{e}:T_{e}G\rightarrow T_{e}G$ via the canonical isomorphism $\mathfrak{g}\cong T_{e}G$.

[TABLE]

such that $\mathbf{z}(0)=x_{0}\cdot x_{\infty}^{-1}$ and $\lim_{t\to\infty}\mathbf{z}(t)=e$. If we define

[TABLE]

then $(\mathbf{x},u_{1},\ldots,u_{m})$ is a trajectory of (2.1) solving the PTTP i.e. $\mathbf{x}(0)=x_{0}$ and $\lim_{t\to\infty}\mathbf{x}(t)\cdot\mathbf{x}_{r}(t)^{-1}=e$.

Proof.

It is essentially based on the following simple fact – a kind of Leibniz rule for curves on $G$ – which the reader can easily verify: given $\mathbf{x}_{1},\mathbf{x}_{2}:\mathbb{R}\rightarrow G$ two smooth curves we have333For $x\in G$ we denote by $L_{x}$ (resp. $R_{x}$) the left (resp. right) translation map $y\in G\mapsto x\cdot y\in G$ (resp. $y\in G\mapsto y\cdot x\in G$).

[TABLE]

Let $(\mathbf{x},u_{1},\ldots,u_{m})$ be defined by (2.3)-(2.4). Then

[TABLE]

where the first sum can be rewritten as

[TABLE]

while the second is

[TABLE]

Summing it up and using (2.4) we conclude that $\mathbf{x}$ solves (2.1). Moreover

[TABLE]

and

[TABLE]

∎

Thanks to Proposition 2.3, in order to solve the PTTP our main concern shall be, from now on, to find a trajectory $(\mathbf{z},v_{1},\ldots,v_{m})$ of system (2.2) satisfying $\mathbf{z}(0)=x_{0}\cdot x_{\infty}^{-1}$ and $\lim_{t\to\infty}\mathbf{z}(t)=e$: the solution $(\mathbf{x},u_{1},\ldots,u_{m})$ of the PTTP for (2.1) can thus be recovered from our knowledge of $(\mathbf{z},v_{1},\ldots,v_{m})$ and $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$.

We define a Lyapunov-like function $V:G\rightarrow\mathbb{R}$ by

[TABLE]

and an auxiliary vector field $W:\mathbb{R}\times G\rightarrow TG$ by

[TABLE]

where

[TABLE]

Notice that $W$ is a time-dependent vector field which is not left-invariant. The main reason for introducing it is the following: if $\mathbf{w}:\mathbb{R}\rightarrow G$ is one of its integral curves and if we define

[TABLE]

then $(\mathbf{w},v_{1},\ldots,v_{m})$ is a trajectory of our modified system (2.2). Moreover, let us denote by $\mathcal{C}_{W}$ the set of critical points of $W$, that is:

[TABLE]

Recall that one such critical point $w\in\mathcal{C}_{W}$ is a local attractor if there exists $U\subset G$ a neighborhood of $w$ such that given any initial condition $(t_{0},w_{0})\in\mathbb{R}\times U$ and $\mathbf{w}:\mathbb{R}\rightarrow G$ the unique integral curve of $W$ satisfying $\mathbf{w}(t_{0})=w_{0}$ then $\lim_{t\to\infty}\mathbf{w}(t)=w$.

The next result tells us that if the identity element of $G$ is a local attractor of the auxiliary vector field $W$ then we can solve the PTTP locally near the target state $x_{\infty}=\mathbf{x}_{r}(0)$, and also provides a recipe to obtain the tracking trajectory $(\mathbf{x},u_{1},\ldots,u_{m})$.

Proposition 2.4**.**

Suppose that $e\in\mathcal{C}_{W}$ and is a local attractor for $W$. Then there exists $U_{\infty}\subset G$ a neighborhood of $x_{\infty}$ enjoying the following property: for every $x_{0}\in U_{\infty}$ there exists $(\mathbf{x},u_{1},\ldots,u_{m})$ a trajectory of (2.1) such that $\mathbf{x}(0)=x_{0}$ and $\lim_{t\to\infty}\mathbf{x}(t)\cdot\mathbf{x}_{r}(t)^{-1}=e$. The trajectory $(\mathbf{x},u_{1},\ldots,u_{m})$ can be obtained as follows: for $\mathbf{w}:\mathbb{R}\rightarrow G$ the unique integral curve of $W$ satisfying $\mathbf{w}(0)=x_{0}\cdot x_{\infty}^{-1}$, define

[TABLE]

Proof.

Let $U\subset G$ be an attractive neighborhood of $e$. Then $U_{\infty}\mathrel{\dot{=}}U\cdot x_{\infty}$ is clearly a neighborhood of $x_{\infty}$, and if $x_{0}\in U_{\infty}$ then $x_{0}\cdot x_{\infty}^{-1}\in U$, hence $\lim_{t\to\infty}\mathbf{w}(t)=e$. If $v_{1},\ldots,v_{m}$ are as in (2.7) then $(\mathbf{w},v_{1},\ldots,v_{m})$ is a trajectory of the modified system (2.2), so the conclusion follows from Proposition 2.3. ∎

3. Some results on stability

In this section we shall depart a little from the original setting for the PTTP and establish some technical results on the stability of time-dependent vector fields that will be needed in the next sections. Since the group structure here plays no role, we shall take a step back and work in the more general framework of smooth manifolds.

Remark 3.1*.*

As pointed out by H. B. Silveira in personal communication, our approach in this section (see especially Proposition 3.6) holds some connections with the periodic version of LaSalle’s Invariance Principle [4] (for its use in a similar context see [8]). The proofs we present here are, nevertheless, self-contained.

Let $M$ be a smooth manifold, which for simplicity we assume to be compact, and $X:\mathbb{R}\times M\rightarrow TM$ a time-dependent vector field. Recall that given $(t_{0},x_{0})\in\mathbb{R}\times M$ its $\omega$-limit set, $\omega_{X}(t_{0},x_{0})$, is the set of all $x\in M$ enjoying the following property: there exists an increasing sequence $t_{n}\to\infty$ such that $\mathbf{x}(t_{n})\to x$, where $\mathbf{x}:\mathbb{R}\rightarrow M$ is the unique integral curve of $X$ satisfying $\mathbf{x}(t_{0})=x_{0}$. Of course the compactness of $M$ ensures that the $\omega$-limit sets of $X$ are never empty.

Definition 3.2**.**

A continuous function $V:M\rightarrow\mathbb{R}$ is said to be non-decreasing along $X$ if for every integral curve $\mathbf{x}:\mathbb{R}\rightarrow M$ of $X$ the function $V\circ\mathbf{x}$ is non-decreasing.

For instance, if $V\in C^{\infty}(M)$ satisfies $\mathrm{d}V(X(t,x))\geq 0$ everywhere then clearly $V$ is non-decreasing along $X$.

Proposition 3.3**.**

Let $V:M\rightarrow\mathbb{R}$ be continuous and non-decreasing along $X$. Then $V$ is constant on $\omega_{X}(t_{0},x_{0})$ for any $(t_{0},x_{0})\in\mathbb{R}\times M$.

Proof.

Let $\mathbf{x}:M\rightarrow\mathbb{R}$ be the unique integral curve of $X$ satisfying $\mathbf{x}(t_{0})=x_{0}$. For $j=1,2$ let $x_{j}\in\omega_{X}(t_{0},x_{0})$ and take increasing sequences $\{t_{n}^{j}\}_{n\in\mathbb{N}}$, $t_{n}^{j}\to\infty$, such that $\mathbf{x}(t_{n}^{j})\to x_{j}$ as $n\to\infty$. By continuity, $V(\mathbf{x}(t_{n}^{j}))\to V(x_{j})$ and since $V\circ\mathbf{x}$ is non-decreasing we must have

[TABLE]

We first extract a subsequence $\{t_{n_{k}}^{2}\}_{k\in\mathbb{N}}$ of $\{t_{n}^{2}\}_{n\in\mathbb{N}}$ with the property that $t_{k}^{1}\leq t_{n_{k}}^{2}$ for every $k\in\mathbb{N}$: again, since $V$ is non-decreasing along $X$ one gets

[TABLE]

By letting $k\to\infty$ in the left-hand side of the inequality above we conclude that $V(x_{1})\leq V(x_{2})$, and hence the equality holds. ∎

Corollary 3.4**.**

If $V:M\rightarrow\mathbb{R}$ be continuous and non-decreasing along $X$ then

[TABLE]

where $\mathbf{x}:M\rightarrow\mathbb{R}$ be the unique integral curve of $X$ satisfying $\mathbf{x}(t_{0})=x_{0}$.

Proof.

It suffices to prove that any increasing sequence $\{t_{n}\}_{n\in\mathbb{N}}$, $t_{n}\to\infty$, admits a subsequence $\{t_{n_{k}}\}_{k\in\mathbb{N}}$ such that $V(\mathbf{x}(t_{n_{k}}))\to V(x)$ as $k\to\infty$. And indeed, by compactness of $M$ there exist $\{t_{n_{k}}\}_{k\in\mathbb{N}}$ subsequence of $\{t_{n}\}_{n\in\mathbb{N}}$ and $y\in M$ such that $\mathbf{x}(t_{n_{k}})\to y$, and by continuity $V(\mathbf{x}(t_{n_{k}}))\to V(y)$, as $k\to\infty$. Since obviously $y\in\omega_{X}(t_{0},x_{0})$ we have $V(x)=V(y)$ by the previous proposition, and the conclusion follows. ∎

The last two results in this section do not assume compactness of $M$. We do, however, endow it with a Riemannian metric: below we denote by $\|\cdot\|$ the induced norm on each tangent space.

Lemma 3.5**.**

Let $\mathbf{x}:\mathbb{R}\rightarrow M$ be a smooth curve such that

[TABLE]

and $\{t_{n}\}_{n\in\mathbb{N}}$ be an increasing sequence such that $t_{n}\to\infty$ and $\mathbf{x}(t_{n})\to x\in M$ as $n\to\infty$. Then

[TABLE]

Proof.

We may assume w.l.o.g. that $M$ is connected, and let $d:M\times M\rightarrow\mathbb{R}$ be the distance function on $M$ induced by the Riemannian metric444I.e. the distance between two given points is the infimum of the lengths of all piecewise smooth curves connecting them.: we then must prove that

[TABLE]

whatever $\epsilon\in\mathbb{R}$. If we denote by $I(a,b)\subset\mathbb{R}$ the closed interval with endpoints $a,b\in\mathbb{R}$ then by definition of $d$ we have

[TABLE]

which, we claim, goes to zero as $n\to\infty$. Indeed, given $\delta>0$ there exists $R>0$ such that $\|\mathbf{x}^{\prime}(t)\|<\delta$ for every $t>R$. Moreover, since $t_{n}\to\infty$ there exists $n_{0}\in\mathbb{N}$ such that

[TABLE]

thus proving our claim. Now for every $n\in\mathbb{N}$

[TABLE]

since both terms go to zero. ∎

Proposition 3.6**.**

Let $\mathbf{x}:\mathbb{R}\rightarrow M$ be a smooth curve, $\{t_{n}\}_{n\in\mathbb{N}}$ an increasing sequence and $x\in M$ as in Lemma 3.5. Let also $f:\mathbb{R}\times M\rightarrow\mathbb{R}$ be continuous, $T$-periodic (for some $T>0$) and such that $\lim_{t\to\infty}f(t,\mathbf{x}(t))=0$. Then

[TABLE]

Proof.

Let $s\in\mathbb{R}$. For each $n\in\mathbb{N}$ select $l_{n}\in\mathbb{Z}$ such that

[TABLE]

hence the sequence $\{s_{n}\}_{n\in\mathbb{N}}$ admits a convergent subsequence, say

[TABLE]

We define

[TABLE]

for each $k\in\mathbb{N}$, so clearly $s_{n_{k}}^{*}\to s$. Applying Lemma 3.5 with $\epsilon\mathrel{\dot{=}}-\theta+s$ one gets

[TABLE]

Since $f$ is continuous and $T$-periodic we have

[TABLE]

which is zero thanks to our last hypothesis on $f$ and the fact that $t_{n_{k}}^{*}\to\infty$. ∎

Now back to the setup of Section 2, we use our results above to prove:

Theorem 3.7**.**

Let $(t_{0},w_{0})\in\mathbb{R}\times G$. If $w\in\omega_{W}(t_{0},w_{0})$ then

[TABLE]

In particular, every $\omega$-limit point is a critical point of $W$.

Its proof depends on some auxiliary results that we prove below. First of all, we must obtain a more convenient expression for the functions $a_{k}$ defined in (2.6).

Lemma 3.8**.**

For $(x,v)\in TG$ we have555Given $X\in\mathfrak{g}$ the adjoint map $\operatorname{\mathrm{ad}}(X):\mathfrak{g}\rightarrow\mathfrak{g}$ is defined by $Y\in\mathfrak{g}\mapsto[X,Y]\in\mathfrak{g}$. By means of the canonical isomorphism $\mathfrak{g}\cong T_{e}G$ it makes perfect sense to write $\operatorname{\mathrm{ad}}(v)$ for $v\in T_{e}G$ – as we do often throughout the text – which we regard as a linear map $T_{e}G\rightarrow T_{e}G$. In that sense, $\operatorname{\mathrm{ad}}:T_{e}G\rightarrow\operatorname{\mathfrak{gl}}(T_{e}G)$ is precisely the differential of the adjoint map $\operatorname{\mathrm{Ad}}:G\rightarrow\operatorname{\mathrm{GL}}(T_{e}G)$ at $e\in G$ [3, Proposition 1.91].

[TABLE]

Also, for each $k\in\{1,\ldots,m\}$:

[TABLE]

for $(t,w)\in\mathbb{R}\times G$.

Proof.

We start by showing that

[TABLE]

Indeed, notice that $\mathrm{d}L_{x^{-1}}v\in T_{e}G$, which we then identify with an element of $\mathfrak{g}$, thus making sense of (3.3). We consider the map $F\mathrel{\dot{=}}\operatorname{\mathrm{Ad}}\circ L_{x}:G\rightarrow\operatorname{\mathrm{GL}}(\mathfrak{g})$: by the chain rule we have, on the one hand,

[TABLE]

On the other hand, we can write

[TABLE]

i.e. $F=\Lambda\circ\operatorname{\mathrm{Ad}}$ where

[TABLE]

is a linear map: for $\xi\in T_{e}G$ we have again by the chain rule

[TABLE]

which for $\xi\mathrel{\dot{=}}\mathrm{d}L_{x^{-1}}v$ proves (3.3) thanks to our previous conclusions.

Now, recalling that the map $\operatorname{\mathrm{trace}}:\operatorname{\mathfrak{gl}}(\mathfrak{g})\rightarrow\mathbb{R}$ is linear and by definition $V=\operatorname{\mathrm{trace}}\circ\operatorname{\mathrm{Ad}}$, identity (3.1) follows immediately from (3.3) after a third application of the chain rule.

To conclude, it follows from the definition of $a_{k}$ and from (3.1) that

[TABLE]

where we used that $\operatorname{\mathrm{Ad}}\left(\mathbf{x}_{r}(t)\right)X_{k}$ is left-invariant for all $t\in\mathbb{R}$. ∎

We can now elucidate a couple of questions raised by Proposition 2.4.

Corollary 3.9**.**

Every666$Z(G)$: the center of $G$ i.e. the subgroup of all $x\in G$ such that $x\cdot y=y\cdot x$ for every $y\in G$. When $G$ is semisimple $Z(G)$ is discrete. $w\in Z(G)$ is a critical point of the auxiliary vector field $W$. However, if the identity element is a local attractor of $W$ then $G$ must be semisimple.

Proof.

Since $Z(G)=\ker\operatorname{\mathrm{Ad}}$ we have $\operatorname{\mathrm{Ad}}(w)=\text{id}_{\mathfrak{g}}$, and then for each $k\in\{1,\ldots,m\}$

[TABLE]

for every $t\in\mathbb{R}$ since $\operatorname{\mathrm{ad}}(X)$ is traceless777See footnote 8. for all $X\in\mathfrak{g}$. By definition of $W$ we have then $W(t,w)=0$ for all $t\in\mathbb{R}$ i.e. $w$ is a critical point.

In particular $e\in\mathcal{C}_{W}$. If $G$ were not semisimple then $Z(G)$ would be a Lie subgroup of $G$ of positive dimension, hence any neighborhood of $e$ would contain infinitely many points in $Z(G)$. Since $Z(G)\subset\mathcal{C}_{W}$ this proves that $e$ would not be an isolated point of $\mathcal{C}_{W}$, even less a local attractor. ∎

The next technical remark will also be needed in Section 5. We define

[TABLE]

where $u_{1}^{r},\ldots,u_{m}^{r}$ are the controls of our reference trajectory of system (2.1). We will consider $X_{r}$ both as a time-dependent vector field on $G$ – of which $\mathbf{x}_{r}$ is an integral curve – and as a smooth curve in $\mathfrak{g}$.

Lemma 3.10**.**

Let $\Lambda^{0}:\mathbb{R}\rightarrow\mathfrak{g}$ be any smooth curve and define $\lambda:\mathbb{R}\rightarrow\mathfrak{g}$ by

[TABLE]

Then its $n$-th derivative is given by

[TABLE]

where $\Lambda^{n}:\mathbb{R}\rightarrow\mathfrak{g}$ is defined inductively by

[TABLE]

Proof.

Using the identity $\operatorname{\mathrm{Ad}}(\mathbf{x}_{r})^{\prime}=\operatorname{\mathrm{Ad}}(\mathbf{x}_{r})\cdot\operatorname{\mathrm{ad}}(X_{r})$, which in turn follows easily from (3.3), we have

[TABLE]

∎

Proposition 3.11**.**

Let $\mathbf{w}:\mathbb{R}\rightarrow G$ be an integral curve of $W$. For each $k\in\{1,\ldots,m\}$ the function $b_{k}:\mathbb{R}\rightarrow\mathbb{R}$ defined by

[TABLE]

is bounded.

Proof.

We shall write down an explicit expression for $b_{k}$, which boils down to computing the partial derivatives of $a_{k}$ since

[TABLE]

First, since $\operatorname{\mathrm{Ad}}(x):\mathfrak{g}\rightarrow\mathfrak{g}$ is a Lie algebra homomorphism for each $x\in G$, it follows easily that

[TABLE]

for every $X\in\mathfrak{g}$ (just apply both sides on an arbitrary $Y\in\mathfrak{g}$). Now, by (3.2) we have that $a_{k}(t,w)=\operatorname{\mathrm{trace}}\left\{\operatorname{\mathrm{Ad}}(w)\cdot\operatorname{\mathrm{ad}}\left(\lambda\right)\right\}$ where $\lambda\mathrel{\dot{=}}\operatorname{\mathrm{Ad}}(\mathbf{x}_{r})X_{k}$, hence

[TABLE]

by (3.5), since $\lambda^{\prime}=\operatorname{\mathrm{Ad}}(\mathbf{x}_{r})\operatorname{\mathrm{ad}}(X_{r})X_{k}$ thanks to Lemma 3.10.

Moreover, using (3.3) and taking into account that $\mathbf{w}$ is an integral curve of $W$

[TABLE]

from which it follows that

[TABLE]

thanks again to a double application of (3.5)

Summing both derivatives evaluated at $(t,\mathbf{w}(t))$, we conclude that

[TABLE]

where $B_{k}:\mathbb{R}\rightarrow\operatorname{\mathfrak{gl}}(\mathfrak{g})$ is defined given by

[TABLE]

Denoting by $\|\cdot\|$ any norm in $\operatorname{\mathfrak{gl}}(\mathfrak{g})$, it follows from the compactness of $G$ the existence of $M>0$ such that $\|\operatorname{\mathrm{Ad}}(x)\|\leq M$ for every $x\in G$, hence for every $t\in\mathbb{R}$ we have

[TABLE]

where $\|\operatorname{\mathrm{trace}}\|$ stands for the norm of the linear functional $\operatorname{\mathrm{trace}}:\operatorname{\mathfrak{gl}}(\mathfrak{g})\rightarrow\mathbb{R}$: in order to finish the proof, it suffices to show that $B_{k}$ is bounded. But

[TABLE]

and while the first term is clearly bounded for the map

[TABLE]

is $T$-periodic, the second term is bounded because $a_{j}:\mathbb{R}\times G\rightarrow\mathbb{R}$ is $T$-periodic for each $j\in\{1,\ldots,n\}$ and hence $a_{j}(\mathbb{R}\times G)=a_{j}([0,T]\times G)$ is a compact set. ∎

Corollary 3.12**.**

If $\mathbf{w}:\mathbb{R}\rightarrow G$ is an integral curve of $W$ then

[TABLE]

In particular

[TABLE]

where $\|\cdot\|$ is the norm associated to any left-invariant Riemannian metric on $G$.

Proof.

Let $\alpha\mathrel{\dot{=}}V\circ\mathbf{w}$ where $V$ is our Lyapunov-like function (2.5). Its first derivative is

[TABLE]

by definition of $a_{k}$ (2.6), and thus non-negative. Differentiating once again yields

[TABLE]

with $b_{k}$ is as in Proposition 3.11, hence bounded, which implies boundedness of $\alpha^{\prime\prime}$. In turn, this ensures that $\alpha^{\prime}$ is uniformly continuous. Now

[TABLE]

so $V$ is non-decreasing along $W$, thus thanks to Corollary 3.4 we have

[TABLE]

where $w\in\omega_{W}(0,\mathbf{w}(0))$ is arbitrary (recall that the latter set is never empty). We have proved that

[TABLE]

which brings us into position to apply Barbalat’s Lemma [2, Lemma 8.2] to $\alpha^{\prime}$ and conclude that

[TABLE]

which clearly proves (3.6) thanks to our previous computations. Our second statement now follows:

[TABLE]

∎

Proof of Theorem 3.7.

Let $\|\cdot\|$ stand for the norm associated to some left-invariant metric on $G$. For $\mathbf{w}:\mathbb{R}\rightarrow G$ the unique integral curve of $W$ satisfying $\mathbf{w}(t_{0})=w_{0}$ we have, thanks to Corollary 3.12, that $\|\mathbf{w}^{\prime}(t)\|\to 0$ as $t\to\infty$. Moreover, the function $f:\mathbb{R}\times G\rightarrow\mathbb{R}$ defined by $f(t,w)\mathrel{\dot{=}}a_{k}(t,w)$ is $T$-periodic and satisfies, again by Corollary 3.12,

[TABLE]

But since $w\in\Omega_{W}(t_{0},w_{0})$ there exists a sequence $\{t_{n}\}_{n\in\mathbb{N}}$ increasing to infinity such that $\mathbf{w}(t_{n})\to w$ as $n\to\infty$. The conclusion follows from Proposition 3.6. ∎

4. Regular trajectories

Up to this point, all the results obtained are valid for arbitrary periodic reference trajectories of (2.1). In this section we introduce a special class of trajectories such that the set of critical points of their associated auxiliary vector fields $W$ admit a nice algebraic description: it coincides with the set of critical points of our Lyapunov-like function $V$. This characterization allows us show that the identity element is a local attractor for $W$ provided $G$ is semisimple, hence solving the PTTP in a neighborhood of the target state $x_{\infty}$ by Proposition 2.4.

Definition 4.1**.**

A trajectory $(\mathbf{x},u_{1},\ldots,u_{m})$ of (2.1) (not necessarily periodic) is said to be regular if

[TABLE]

In Section 5 we prove the existence of regular periodic trajectories through any initial state $x_{0}\in G$ under some extra assumptions on system (2.1). Theorem 3.7 admits the following:

Corollary 4.2**.**

Assume $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ is a regular periodic trajectory of system (2.1) and let $W$ be its auxiliary vector field. If $w\in G$ is any $\omega$-limit point of $W$ then

[TABLE]

Or, by Lemma 3.8: $\mathrm{d}VX(w)=0$ for all $X\in\mathfrak{g}$.

Proof.

By Theorem 3.7 we have $a_{k}(t,w)=0$ for every $t\in\mathbb{R}$ and $k\in\{1,\ldots,m\}$, so by Lemma 3.8

[TABLE]

But since $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ is regular the linear functional $\operatorname{\mathrm{trace}}\left\{\operatorname{\mathrm{Ad}}(w)\cdot\operatorname{\mathrm{ad}}(\cdot)\right\}$ vanishes on $\mathfrak{g}$. ∎

In particular, for a regular periodic trajectory $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ the set of critical points of its auxiliary field $W$ can be expressed as

[TABLE]

Indeed, if $w$ belongs to the set in the right-hand side of (4.1) then by (3.2)) we have $a_{k}(t,w)=0$ for all $t\in\mathbb{R}$ and every $k\in\{1,\ldots,m\}$, hence clearly $w\in\mathcal{C}_{W}$. We have thus equated, in this case:

• •

the set of critical points of the vector field $W$,

• •

the set of $\omega$-limit points of $W$ and

• •

the set of critical points of the Lyapunov-like function $V$.

The next result gathers some interesting consequences of (4.1), which, however, we will not use in what follows.

Proposition 4.3**.**

Let $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ be a regular periodic trajectory of (2.1) and $W$ be its auxiliary vector field. Then for $x,y\in G$:

1. (1)

$x\in\mathcal{C}_{W}\Longrightarrow y\cdot x\cdot y^{-1}\in\mathcal{C}_{W}$. 2. (2)

$x\cdot y\in\mathcal{C}_{W}\Longrightarrow y\cdot x\in\mathcal{C}_{W}$. 3. (3)

$x\in\mathcal{C}_{W}\Longrightarrow x^{-1}\in\mathcal{C}_{W}$.

Moreover, if $G$ is semisimple and $\mathcal{C}_{W}$ is a subgroup of $G$ then $\mathcal{C}_{W}$ is finite.

Proof.

1. (1)

For $X\in\mathfrak{g}$ a simple computation shows that

[TABLE]

since $x$ belongs to $\mathcal{C}_{W}$, hence so does $y\cdot x\cdot y^{-1}$. 2. (2)

Follows from the previous item since $y\cdot(x\cdot y)\cdot y^{-1}=y\cdot x$. 3. (3)

Pick $Y_{1},\ldots,Y_{n}$ an orthonormal basis of $\mathfrak{g}$ w.r.t. some $\operatorname{\mathrm{Ad}}$-invariant888An inner product on $\mathfrak{g}$ is said to be $\operatorname{\mathrm{Ad}}$-invariant if $\operatorname{\mathrm{Ad}}(x)$ is orthogonal w.r.t. it for every $x\in G$. Such an inner product always exists when $G$ is compact, and thanks to the relationship between the adjoint maps (see footnote 5) one also has that $\operatorname{\mathrm{ad}}(X)$ is skew-symmetric w.r.t. it for every $X\in\mathfrak{g}$ [3, Proposition 4.24]. In particular, $\operatorname{\mathrm{trace}}\operatorname{\mathrm{ad}}(X)=0$ for every $X\in\mathfrak{g}$. inner product $\langle\cdot,\cdot\rangle$. Then:

[TABLE]

which equals [math] if $x\in\mathcal{C}_{W}$. Since $X\in\mathfrak{g}$ is arbitrary, $x^{-1}\in\mathcal{C}_{W}$.

As for the second part of the statement, since $\mathcal{C}_{W}$ is a closed set it is a Lie subgroup of $G$: let $\mathfrak{h}\subset\mathfrak{g}$ be its Lie algebra. Let $X\in\mathfrak{h}$ and for each $Y\in\mathfrak{g}$ define $f_{Y}:\mathbb{R}\rightarrow\mathbb{R}$ by999Here and below we denote by $\mathbf{e}^{X}\in G$ the exponential of $X\in\mathfrak{g}$.

[TABLE]

Since then $\mathbf{e}^{tX}\in\mathcal{C}_{W}$ we have $f_{Y}(t)=0$ for every $t\in\mathbb{R}$, and thus

[TABLE]

also equals [math] for all $t\in\mathbb{R}$, in particular for $t=0$: we have thus proved that $\operatorname{\mathrm{trace}}\left\{\operatorname{\mathrm{ad}}\left(X\right)\cdot\operatorname{\mathrm{ad}}(Y)\right\}=0$ for every $Y\in\mathfrak{g}$. But this is the Killing form of $\mathfrak{g}$, which is non-degenerate since we are assuming $G$ semisimple101010This is Cartan’s Criterion for Semisimplicity [3, Theorem 1.45]: the Killing form of $\mathfrak{g}$ is the bilinear form $B:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ defined by $B(X,Y)\mathrel{\dot{=}}\operatorname{\mathrm{trace}}\{\operatorname{\mathrm{ad}}(X)\cdot\operatorname{\mathrm{ad}}(Y)\}$. It is always negative semidefinite, while non-degenerate precisely when $G$ is semisimple., from which we conclude that $X=0$. Since $X\in\mathfrak{h}$ is arbitrary we have $\mathfrak{h}=\{0\}$ i.e. $\mathcal{C}_{W}$ is a discrete subgroup of $G$. Since $G$ is compact and $\mathcal{C}_{W}$ is closed the latter must be finite. ∎

As we have seen, $e\in\mathcal{C}_{W}$ and if this set is finite then $e$ is an isolated point, which is a necessary condition for $e$ to be a local attractor. Next we shall focus on proving the latter property without relying on the assumption of $\mathcal{C}_{W}$ being a group.

Proposition 4.4**.**

If $G$ is semisimple then every $x\in Z(G)$ is a non-degenerate critical point of $V$, and, in particular, an isolated point in $\mathcal{C}_{W}$.

Proof.

By Corollary 3.9, every $x\in Z(G)$ belongs to $\mathcal{C}_{W}$, hence is a critical point of $V$ by our characterization of the latter set following Corollary 4.2 (since $(\mathbf{x_{r}},u_{1}^{r},\ldots,u_{m}^{r})$ is regular). As such, we check its non-degeneracy by computing the Hessian matrix of $V$ in convenient coordinates around $x$.

We denote by $B$ the Killing form of $\mathfrak{g}$. Since $G$ is assumed semisimple, $-B$ is an inner product on $\mathfrak{g}$ and we denote by $Y_{1},\ldots,Y_{n}\in\mathfrak{g}$ an orthonormal basis w.r.t. it to introduce the so-called coordinates of second kind: let $\varphi:\mathbb{R}^{n}\rightarrow G$ be defined by

[TABLE]

Simple computations show that

[TABLE]

hence $\varphi$ is a local diffeomorphism near $s=0$. Moreover

[TABLE]

where we used two well-known facts: that $Z(G)$ is precisely the kernel of the $\operatorname{\mathrm{Ad}}$ homomorphism; and the identity [3, Proposition 1.91]:

[TABLE]

Another simple computation then shows that

[TABLE]

thus showing that the Hessian matrix of $V$ at $x=\varphi(0)$ is non-degenerate. The last claim follows from Morse Lemma. ∎

Next we characterize the center of $G$ in terms of $V$ regardless of semisimplicity. Let $\mathfrak{g}_{\mathbb{C}}$ denote the complexification of $\mathfrak{g}$ and let $\operatorname{\mathrm{Ad}}_{\mathbb{C}}(x):\mathfrak{g}_{\mathbb{C}}\rightarrow\mathfrak{g}_{\mathbb{C}}$ be the complexification of $\operatorname{\mathrm{Ad}}(x)$. Therefore

[TABLE]

where $\lambda_{1}(x),\ldots,\lambda_{n}(x)\in\mathbb{C}$ are the eigenvalues of $\operatorname{\mathrm{Ad}}_{\mathbb{C}}(x)$, repeated according to their multiplicities. Let $\langle\cdot,\cdot\rangle$ be any $\operatorname{\mathrm{Ad}}$-invariant inner product on $\mathfrak{g}$ (recall that $G$ is compact) and let $\langle\cdot,\cdot\rangle_{\mathbb{C}}$ stand for its sesquilinear extension to $\mathfrak{g}_{\mathbb{C}}$, which is then a Hermitian inner product on $\mathfrak{g}_{\mathbb{C}}$. Clearly $\operatorname{\mathrm{Ad}}_{\mathbb{C}}(x)$ is unitary w.r.t. $\langle\cdot,\cdot\rangle_{\mathbb{C}}$, in particular it is a diagonalizable map and

[TABLE]

It follows then that for every $x\in G$ we have

[TABLE]

Therefore, if $x\in Z(G)$ then $\operatorname{\mathrm{Ad}}(x)=\mathrm{id}_{\mathfrak{g}}$ and hence $V(x)=n$, and thus is a global maximum of $V$.

We claim that the converse is also true i.e. if $V(x)=n$ then $x\in Z(G)$. We must prove that $\lambda_{j}(x)=1$ for every $j\in\{1,\ldots,n\}$. Indeed we have

[TABLE]

which implies that

[TABLE]

Moreover

[TABLE]

so if $\mathsf{Re}\lambda_{k}(x)<1$ for some $k\in\{1,\ldots,n\}$ then

[TABLE]

which would lead us to a contradiction, hence $\mathsf{Re}\lambda_{k}(x)=1$ for every $k\in\{1,\ldots,n\}$, and since $|\lambda_{k}(x)|=1$ we must also have $\mathsf{Im}\lambda_{k}(x)=0$ for every $k\in\{1,\ldots,n\}$. In particular, we have proved:

Lemma 4.5**.**

Any $x\in G$ belongs to $Z(G)$ if and only if $V(x)=\dim\mathfrak{g}$.

Now we can prove one of the main results of the present work. Recall that $W$ denotes the auxiliary vector fields associated to a regular periodic reference trajectory.

Theorem 4.6**.**

If $G$ is semisimple then every $x\in Z(G)$ is a local attractor of $W$.

Proof.

By Proposition 4.4 we may find a neighborhood $U\subset G$ of $x$ such that $\overline{U}$ is compact and contains no points in $\mathcal{C}_{W}$ other than $x$. We may further assume $U$ connected, and define

[TABLE]

Clearly $M<n$ for $\partial U\cap Z(G)=\emptyset$. Also, since $V(x)=n$ we have by continuity that there exists $U^{\prime}\subset U$ another neighborhood of $x$ such that $V(y)>M$ for every $y\in U^{\prime}$.

Now let $(t_{0},w_{0})\in\mathbb{R}\times U^{\prime}$ and $\mathbf{w}:\mathbb{R}\rightarrow G$ be the unique integral curve of $W$ satisfying $\mathbf{w}(t_{0})=w_{0}$. Since $V\circ\mathbf{w}$ is non-decreasing (for $V$ is non-decreasing along $W$, as pointed out earlier) we have

[TABLE]

for every $t\geq t_{0}$ since $w_{0}\in U^{\prime}$ by hypothesis. Thus $\mathbf{w}(t)\notin\partial U$ and by continuity we have that $\mathbf{w}(t)\in U$ for every $t\geq t_{0}$.

It remains to show that $\mathbf{w}(t)\to x$ as $t\to\infty$. Indeed, we will show that any sequence $\{t_{n}\}_{n\in\mathbb{N}}$ increasing to infinity admits a subsequence $\{t_{n_{k}}\}_{k\in\mathbb{N}}$ such that $\mathbf{w}(t_{n_{k}})\to x$ as $k\to\infty$. We may assume w.l.o.g. that $t_{n}\geq t_{0}$ for every $n\in\mathbb{N}$, hence $\mathbf{w}(t_{n})\in U$. Because $\overline{U}$ is compact there exists $w\in\overline{U}$ and a subsequence $\{t_{n_{k}}\}_{k\in\mathbb{N}}$ of $\{t_{n}\}_{n\in\mathbb{N}}$ such that $\mathbf{w}(t_{n_{k}})\to w$ as $k\to\infty$. In particular $w\in\omega_{W}(t_{0},w_{0})\subset\mathcal{C}_{W}$, but as we have seen $\overline{U}\cap\mathcal{C}_{W}=\{x\}$. ∎

In particular $e$ is a local attractor of $W$, hence Proposition 2.4 solves the PTTP locally.

5. Existence of regular periodic trajectories: sufficient conditions

Recall that we are always assuming that our left-invariant system (2.1) is bracket-generating i.e. the Lie algebra spanned by $X_{1},\ldots,X_{m}$ is $\mathfrak{g}$. In this section, we will prove that if moreover

[TABLE]

then given any $x_{\infty}\in G$ and $T>0$ there exists $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ a smooth $T$-periodic trajectory of (2.1) satisfying $\mathbf{x}_{r}(0)=x_{\infty}$ and which is regular in the sense of Definition 4.1.

For $j\in\{1,\ldots,m\}$ and $p\in\mathbb{Z}$ let

[TABLE]

Clearly $J_{j}^{p}\subset(pT,(p+1)T)=(0,T)+pT$ and

[TABLE]

Take $\chi_{j}\in C_{c}^{\infty}(J_{j}^{0})$ equal to $1$ in some open interval $I_{j}\subset J_{j}^{0}$ and with zero integral, and let $u_{j}^{r}:\mathbb{R}\rightarrow\mathbb{R}$ be the unique $T$-periodic function which is equal to $\chi_{j}$ on $[0,T]$: this is clearly smooth.

Since $\operatorname{\mathrm{supp}}\chi_{j}\subset J_{j}^{0}$ it is easy to check that

[TABLE]

which, together with (5.2), easily ensures the following:

Proposition 5.1**.**

Given $t\in\mathbb{R}$, if $u_{j}^{r}(t)\neq 0$ for some $j\in\{1,\ldots,m\}$ then $u_{k}^{r}(t)=0$ for every $k\neq j$.

Thus on $I_{j}^{p}\mathrel{\dot{=}}I_{j}+pT$ we have $u_{j}^{r}=1$, while $u_{k}^{r}=0$ for $k\neq j$, identically. Next, define $\xi_{j}:\mathbb{R}\rightarrow\mathbb{R}$ by

[TABLE]

which is obviously smooth and also $T$-periodic since $u_{j}^{r}$ is $T$-periodic and its integral over $[0,T]$ is zero. Moreover, one may check that

[TABLE]

We finally define $\mathbf{x}_{r}:\mathbb{R}\rightarrow G$ by

[TABLE]

This is well-defined thanks to (5.2), and moreover smooth by (5.4). Also, on $I_{j}^{p}$ we have

[TABLE]

where the last identity follows from Proposition 5.1. On the other hand, if $t\notin J_{j}^{p}$ for any $j\in\{1,\ldots,m\}$ and $p\in\mathbb{Z}$ then near $t$ we have $\mathbf{x}_{r}=x_{\infty}$ identically, hence $\mathbf{x}_{r}^{\prime}(t)=0$, which also agrees with (2.1) thanks to (5.3): we have proved that $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ is a trajectory of (2.1), which is $T$-periodic by construction.

Theorem 5.2**.**

If (5.1) holds then the trajectory $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ above is regular.

Proof.

We denote by

[TABLE]

which we must prove that is equal to $\mathfrak{g}$. For each $k\in\{1,\ldots,m\}$ let $\Lambda^{0}_{k}\mathrel{\dot{=}}X_{k}$ and $\lambda_{k}:\mathbb{R}\rightarrow\mathfrak{g}$ be defined by $\lambda_{k}\mathrel{\dot{=}}\operatorname{\mathrm{Ad}}(\mathbf{x}_{r})\Lambda^{0}_{k}$: this is a smooth curve that lies in $\mathcal{V}$, and since the latter is a linear subspace of $\mathfrak{g}$ the same is true for all of its derivatives. By Lemma 3.10 we have, for every $n\in\mathbb{N}$,

[TABLE]

where $X_{r}$ is given by (3.4).

We need the following technical lemma, which does not depend on the construction of $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$.

Lemma 5.3**.**

For each $n\in\mathbb{N}$ we may write

[TABLE]

where $\Delta_{0}=\Delta_{1}=0$ and, for $n\geq 2$, $\Delta_{n}$ is a sum of products enjoying the following property: in each summand there is at least one factor that is a derivative of some order of $\operatorname{\mathrm{ad}}(X_{r})$.

Proof of Lemma 5.3.

By induction on $n$. We start calculating recursively by Lemma 3.10

[TABLE]

from which we identify $\Delta_{0}=0$, $\Delta_{1}=0$ and $\Delta_{2}=\operatorname{\mathrm{ad}}(X_{r}^{\prime})$, thus proving our claim for $n=0,1,2$, which we use as basis for induction. Assuming our conclusion for some $n\geq 2$ we have

[TABLE]

where obviously

[TABLE]

Since $\Delta_{n}$ is a sum of products in which each summand there is at least one factor that is a derivative of some order of $\operatorname{\mathrm{ad}}(X_{r})$ then of course the same property holds true for both $\left(\Delta_{n}\right)^{\prime}$ and $\operatorname{\mathrm{ad}}(X_{r})\Delta_{n}$. Moreover

[TABLE]

also enjoys the aforementioned property, hence so does $\Delta_{n+1}$. ∎

Now, if $t_{1}\in I_{1}^{0}$ then $u_{1}^{r}(t_{1})=1$, while for $j\neq 1$ we have by Proposition 5.1 that $u_{j}^{r}$ vanishes identically near $t_{1}$, and therefore

[TABLE]

while

[TABLE]

It follows from Lemma 5.3 that $\Delta_{n}(t_{1})=0$ (for $\operatorname{\mathrm{ad}}:\mathfrak{g}\rightarrow\mathfrak{g}$ is linear, hence $\operatorname{\mathrm{ad}}(X_{r})^{(k)}=\operatorname{\mathrm{ad}}\left(X_{r}^{(k)}\right)$ for every $k\in\mathbb{N}$) for every $n\in\mathbb{N}$, so

[TABLE]

for every $n\in\mathbb{N}$. We conclude that

[TABLE]

belongs to $\mathcal{V}$ for every $n\in\mathbb{N}$ and $k\in\{1,\ldots,m\}$; in other words, if we denote

[TABLE]

then $\operatorname{\mathrm{Ad}}(\mathbf{x}_{r}(t_{1}))\mathcal{W}\subset\mathcal{V}$. But we assumed in (5.1) that $\mathcal{W}=\mathfrak{g}$ and $\operatorname{\mathrm{Ad}}(\mathbf{x}_{r}(t_{1}))$ is invertible, hence also $\mathcal{V}=\mathfrak{g}$ i.e. $(\mathbf{x}_{r},u_{1}^{r},\ldots,u_{m}^{r})$ is regular. ∎