Morphisms of tautological control systems

TL;DR

This paper explores the relationships and conditions under which morphisms between tautological control systems preserve reachability, with applications to second-order control systems, advancing the understanding of system mappings in control theory.

Contribution

It introduces conditions for the existence of morphisms between tautological control systems and relates these to reachability and lifting properties, with specific results for second-order systems.

Findings

Sufficient conditions for reachability transfer between systems.

Establishment of a correspondence between lifting control systems and morphisms.

Application of results to second-order control systems.

Abstract

In this paper, we investigate morphisms of tautological control systems. Given a tautological control system $\mathfrak{H}$ on the manifold N and a mapping $\Phi: M \to N$, we study existence of tautological control system $\mathfrak{G}$ on the manifold $M$ such that there exists a trajectory-preserving morphism $(\Phi, \Phi^ #)$ from $\mathfrak{G}$ to $\mathfrak{H}$. Sufficient conditions are given such that reachability of $\mathfrak{H}$ implies the reachability of $\mathfrak{G}$. Correspondence between the notion of lifting ordinary control systems and morphisms of tautological control systems are examined. We give an application of the above results to the class of second-order type control systems, where the special structure of second-order type leads to additional results.

Equations

$$r_{\mathcal{U},\mathcal{V}}$$

$${\mathcal{F}}_{\mathcal{X}}(\mathcal{U})=\{X|\mathcal{U}|X\in \mathcal{X}\}.$$

$$\{X|{\mathbb{T}}_1\times {\mathcal{U}}_1|X\in {\mathcal{O}}_{\mathfrak{G}}({\mathbb{T}}_2,{\mathcal{U}}_2)\}\subseteq {\mathcal{O}}_{\mathfrak{G}}({\mathbb{T}}_1,{\mathcal{U}}_1).$$

$${\mathcal{O}}_{\mathfrak{G},\mathrm{full}}(\mathbb{T},\mathcal{U})=\mathrm{LI}{\mathrm{\Gamma }}^{\nu }(\mathbb{T};\mathcal{F}(\mathcal{U})).$$

$${\mathcal{O}}_{\mathfrak{G},\infty }(\mathbb{T},\mathcal{U})=\{X\in {\mathcal{O}}_{\mathfrak{G},\mathrm{full}}(\mathbb{T},\mathcal{U})|X\in {\mathrm{LB\Gamma }}^{\nu }(\mathbb{T};T\mathcal{U})\}.$$

$${\mathcal{O}}_{\mathfrak{G},\mathrm{cpt}}(\mathbb{T},\mathcal{U})$$

$${\mathcal{O}}_{\mathfrak{G},\mathrm{pwc}}(\mathbb{T},\mathcal{U})=\{X\in {\mathcal{O}}_{\mathfrak{G},\mathrm{full}}(\mathbb{T},\mathcal{U})|t\to X(t)\text{is piecewise constant}\}.$$

$${\mathcal{F}}_{\Sigma }(\mathcal{U})=\{F^u|\mathcal{U}\in {\mathrm{\Gamma }}^{\nu }(T\mathcal{U})|u\in \mathfrak{C}\}.$$

$$F_{\mathcal{F}}$$

$$X(x)=\sum _i{g}_i{X}_{\alpha (i)}(x)$$

$$T_x\Phi (X)=\sum _i{g}_i{T}_x\Phi (X_{\alpha (i)}(x)))=\sum _i{g}_{i}Y(\Phi (x))=Y(\Phi (x)),$$

Full text

1 Introduction

Morphisms of ordinary control systems have been studied by several researchers. From a theoretical point of view, the study of morphisms is interesting since it reveals system structures that must be understood. And it can help us solve the analysis and design problems for ordinary control systems which are often difficult due to the complicated nature of the equations describing the systems. Among the existing works the notion of quotient control systems has been introduced by different authors. In Ref1, the analysis of the Lie algebra of an ordinary control systems leads to a decomposition into smaller systems. In Ref2, the problem of reduction for ordinary control systems with symmetries is studied. In Ref3, Ref4, Pappas et. al study abstractions of control systems where a constructive procedure was proposed to compute smaller systems. In Ref5, Tabuada and Pappas study quotients of ordinary control systems by the use of category theory. In Ref6, quotients of Lagrangian mechanical control system are investigated. In a certain sense all the above work focus on the notion of projection. A different research direction is taken in Ref7, where the notion of lifting ordinary control system is introduced. In this paper, we will study morphisms of tautological control systems introduced by Lewis in Ref8. It is to be seen that our research is in the same direction as in Ref7.

Tautological control systems proposed by Lewis provide us with a framework that models systems in a manner that does not make any choice of parameterisation by control. This framework is able to handle a variety of degrees of regularity: finitely differentiable, Lipschitz, smooth and especially real analytic. Due to the use of sheaf theory, the framework can deal with distinctions between local and global, which are unnoticed in current control theory. The problem to be explored in this paper goes as follows.

Given a $C^{\nu}$ tautological control system $\mathfrak{H}=(N,\mathscr{G})$ on the $C^{r}$ manifold $N$ and a mapping $\Phi\in C^{r}(M,N)$ where $M$ is a $C^{r}$ manifold. We will be interested in answering the main questions:

(i) Does there exist a tautological control system $\mathfrak{G}=(M,\mathscr{F})$ such that there exists a trajectory-preserving morphism $(\Phi,\Phi^{\#})$ from $\mathfrak{G}$ to $\mathfrak{H}$? (ii) When does the reachability of $\mathfrak{H}$ imply the reachability of $\mathfrak{G}$?

See Section 2 for the definitions of tautological control system and trajectory-preserving morphism, and the precise information about $\nu$ and $r$ above.

Similar question has been studied in Ref7 in the context of ordinary control systems, where the systems are assumed to be only $C^{1}$ control systems. Here we deal with tautological control systems. It is know in Ref8 that given an ordinary control system, we can associate a tautological control system to it. With the correspondence between trajectories of an ordinary control system and its associated tautological control system as claimed in Ref8, then the useful system theoretic properties can be found in the framework of tautological control system without having to go back to the ordinary control system framework. See Section 2 for detailed interpretations of the correspondence between ordinary control system and tautological control system.

Our work here is different from the work in Ref7. First, the answer to the question in Ref7 is localised, and distinctions between local and global are not well dealt with. Our work tries to understand the passage from local to global and patch together local constructions to give global constructions. It is to be seen that with the aid of sheaf language, globally defined control constructions can be got. Second, our work also investigate the case of real analytic systems. Real analyticity plays an important role in geometric control theory (e.g. Orbit Theorem Ref9). However, real analyticity hasn’t been well understood in control theory. One reason is that the underlying geometry is a lot different from smooth differential geometry, and the techniques of real analytic differential geometry are difficult to learn and to learn to apply. In this paper we try to make some discussions on real analytic system with the aid of sheaf theory.

Our work is in the framework of tautological control system introduced by Lewis and Jafarpour in Ref8 and Ref11. Here we focus on the structure of trajectory-preserving morphism of tautological control systems. The main result (see Theorem 3) provides a sufficient condition for the reachability of $\mathfrak{H}$ to imply the reachability of $\mathfrak{G}$. Compared with the framework of ordinary control system, it is to be seen that there is something different in the framework of tautological control system, where there is no choice of parameterisation by control.

The paper is organized as follows. In Section 2 we provide some basic notions and present some known results that will be needed for the subsequent work. Section 3 contains the main results. Question (i) is investigated for systems with different degrees of regularity. For question (ii) we give sufficient conditions for the reachability of $\mathfrak{H}$ to imply the reachability of $\mathfrak{G}$. An extended result for small-time local controllability is presented. Correspondence between the notion of lifting ordinary control systems and morphisms of tautological control systems are examined. In Section 4 an example of second-order type control systems is studied. Compared with the main theorem for general tautological control systems in Section 3, we get supplementary results for this class of systems due to the special structure properties of second-order type.

2 Preliminaries

Throughout the paper, manifolds are supposed to be second-countable Hausdorff manifolds. We will use the letter $n$ to denote the dimension of the manifold $M$. The tangent bundle of a manifold $M$ is denoted by $\pi_{\mathrm{TM}}:TM\rightarrow M$ and the cotangent bundle by $\pi_{\mathrm{T^{*}M}}:T^{*}M\rightarrow M$. The derivative of a differentiable map $\Phi:M\to N$ is denoted by $T\Phi:TM\rightarrow TN$, with $T_{x}\Phi=T\Phi|T_{x}M$. If $I\in\mathbb{R}$ is an interval and if $\xi:I\rightarrow M$ is a curve that is differentiable at $t\in I$, we denote the tangent vector field to the curve at $t$ by $\xi{{}^{\prime}}(t)=T_{t}\xi(1)$. Let $r\in\mathbb{Z}_{\geq 0}\cup\{\infty,\omega\}$. The set of sections of a vector bundle $E\rightarrow M$ of class $C^{r}$ is denoted by $\mathrm{\Gamma}^{r}(E)$. We denote by $\mathrm{C}^{r}(M,N)$ the set of mappings of class $C^{r}$ from $M$ to $N$.

For an interval $I$ and a topological space $\mathcal{X}$, a curve $\gamma:I\rightarrow\mathcal{X}$ is measurable if $\gamma^{-1}(\mathcal{O})$ is Lebesgue measurable for every open $\mathcal{O}\subseteq\mathcal{X}$. By $\mathrm{L}^{\infty}(I;\mathcal{X})$ we denote the measurable curves $\gamma:I\rightarrow\mathcal{X}$ for which there exists a compact set $K\subseteq\mathcal{X}$ with $\lambda(\{t\in I|\gamma(t)\notin K\})=0$, where $\lambda$ is a Lebesgue measure, i.e., $\mathrm{L}^{\infty}(I;\mathcal{X})$ is the set of essentially bounded curves. By $\mathrm{L}^{\infty}_{\mathrm{loc}}(I;\mathcal{X})$ we denote the locally essentially bounded curves, meaning those measurable curves whose restrictions to compact subintervals are essentially bounded.

Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, depending from the context. For a manifold $M$ of class $C^{r}$ and an interval $I\subseteq\mathbb{R}$, we denote the set of Caratheodory time-varying vector field of class $C^{\nu}$ by $\mathrm{CF\Gamma}^{\nu}(I;TM)$, the set of locally integrally $C^{\nu}$-bounded time-varying vector fields by $\mathrm{LI\Gamma}^{\nu}(I;TM)$, and the set of locally essentially $C^{\nu}$-bounded time-varying vector fields by $\mathrm{LB\Gamma}^{\nu}(I;TM)$. For a manifold $M$ of class $C^{r}$ and a topological space $\mathcal{P}$, we denote the set of separately parameterised vector fields of class $C^{\nu}$ by $\mathrm{SP\Gamma}^{\nu}(\mathcal{P};TM)$, the set of jointly parameterised vector fields of class $C^{\nu}$ by $\mathrm{JP\Gamma}^{\nu}(\mathcal{P};TM)$. For detailed explanations for these concepts, we refer to Ref8.

Definition 1

Ref8 Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $M$ be a manifold of class $C^{r}$. A presheaf of sets of $C^{\nu}$-vector fields is an assignment to each open set $\mathcal{U}\subseteq M$ a subset $\mathscr{F}(\mathcal{U})$ of $\Gamma^{\nu}(T\mathcal{U})$ with the property that, for open sets $\mathcal{U},\mathcal{V}\subseteq M$ with $\mathcal{V}\subseteq\mathcal{U}$, the map

[TABLE]

takes values in $\mathscr{F}(\mathcal{V})$. Elements of $\mathscr{F}(\mathcal{U})$ are called local sections of $\mathscr{F}$ over $\mathcal{U}$.

Example 1

Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $M$ be a manifold of class $C^{r}$. If $\mathscr{X}\subseteq\Gamma^{\nu}(TM)$ is any family of vector fields on $M$, then we can define an associated presheaf $\mathscr{F}_{\mathscr{X}}$ of sets of vector fields by

[TABLE]

A presheaf of this sort will be called globally generated.

With the above definition of presheaf in mind, we have the following definition of tautological control system.

Definition 2

Ref8 Let $m\in\mathbb{Z}_{\geq 0}$ and $m^{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m^{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. A $C^{\nu}$-tautological control system is a pair $\mathfrak{G}=(M,\mathscr{F})$ where $M$ is a manifold of class $C^{r}$ whose elements are called states and where $\mathscr{F}$ is a presheaf of sets of $C^{\nu}$ vector fields on $M$. A tautological control system $\mathfrak{G}=(M,\mathscr{F})$ is globally generated if $\mathscr{F}$ is globally generated.

Now we introduce a notion of a trajectory for a tautological control system. Since trajectories are associated to ”open-loop systems”, we first discuss ”open-loop systems”. Let $\mathfrak{G}=(M,\mathscr{F})$ be a $C^{\nu}$-tautological control system. An open-loop system for $\mathfrak{G}$ is a triple $\mathfrak{G}_{ol}=(X,\mathbb{T},\mathcal{U})$ where $\mathbb{T}\subseteq\mathbb{R}$ is an interval called the time-domain, $\mathcal{U}\subseteq M$ is open and $X\in\mathrm{LI\Gamma^{\nu}}(\mathbb{T};\mathscr{F}(\mathcal{U}))$. An open-loop subfamily for $\mathfrak{G}$ is an assignment, to each interval $\mathbb{T}\subseteq\mathbb{R}$ and each open set $\mathcal{U}\subseteq M$, a subset $\mathscr{O}_{\mathfrak{G}}(\mathbb{T},\mathcal{U})\subseteq\mathrm{LI\Gamma^{\nu}}(\mathbb{T};\mathscr{F}(\mathcal{U}))$ with the property that, if $(\mathbb{T}_{1},\mathcal{U}_{1})$ and $(\mathbb{T}_{2},\mathcal{U}_{2})$ are such that $\mathbb{T}_{1}\subseteq\mathbb{T}_{2},\mathcal{U}_{1}\subseteq\mathcal{U}_{2}$, then

[TABLE]

The full subfamily for $\mathfrak{G}$ is the open-loop subfamily $\mathscr{O}_{\mathfrak{G},\mathrm{full}}$ defined by

[TABLE]

The locally essentially bounded subfamily for $\mathfrak{G}$ is the open-loop subfamily $\mathscr{O}_{\mathfrak{G},\infty}$ defined by

[TABLE]

The locally essentially compact subfamily for $\mathfrak{G}$ is the open-loop subfamily $\mathscr{O}_{\mathfrak{G},\mathrm{cpt}}$ defined by

[TABLE]

The piecewise constant subfamily for $\mathfrak{G}$ is the open-loop subfamily $\mathscr{O}_{\mathfrak{G},\mathrm{pwc}}$ defined by

[TABLE]

A $(\mathbb{T},\mathcal{U})$-trajectory for $\mathscr{O}_{\mathfrak{G}}$ is a curve $\xi:\mathbb{T}\rightarrow\mathcal{U}$ such that $\xi{{}^{\prime}}(t)=X(t,\xi(t))$ for some $X\in\mathscr{O}_{\mathfrak{G}}(\mathbb{T},\mathcal{U})$. A plain trajectory for $\mathscr{O}_{\mathfrak{G}}$ is a curve that is a $(\mathbb{T},\mathcal{U})$-trajectory for $\mathscr{O}_{\mathfrak{G}}$ for some time domain $\mathbb{T}$ and some open set $\mathcal{U}\subseteq M$. We denote the set of $(\mathbb{T},\mathcal{U})$-trajectories for $\mathscr{O}_{\mathfrak{G}}$ by $\mathrm{Traj}(\mathbb{T},\mathcal{U},\mathscr{O}_{\mathfrak{G}})$ and the set of trajectories for $\mathscr{O}_{\mathfrak{G}}$ by $\mathrm{Traj}(\mathscr{O}_{\mathfrak{G}})$. Then according to Ref11, we can assert that for every $x\in M,s\in\mathbb{R}$ and $X\in\mathscr{O}_{\mathfrak{G},\mathrm{full}}(\mathbb{T},\mathcal{U})$, there exists an open interval $J_{X}(s,x)\subseteq\mathbb{T}$ containing $s$ such that $\xi{{}^{\prime}}(t)=X(t,\xi(t)),\xi(s)=x$ for almost all $t\in J_{X}(s,x)$.

In the following, we introduce the concept of morphism of tautological control systems. Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $\mathfrak{G}=(M,\mathscr{F})$ be a $C^{\nu}$-tautological control system, let $N$ be a $C^{r}$-manifold, and let $\Phi\in C^{r}(M,N).$ The direct image of $\mathfrak{G}$ by $\Phi$ is the tautological control system $\Phi_{*}\mathfrak{G}=(N,\Phi_{*}\mathscr{F})$ defined by $\Phi_{*}\mathscr{F}(\mathcal{V})=\mathscr{F}(\Phi^{-1}(\mathcal{V}))$ for $\mathcal{V}\subseteq N$ open.

Definition 3

Ref8 Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $\mathfrak{G}=(M,\mathscr{F})$ and $\mathfrak{H}=(N,\mathscr{G})$ be $C^{\nu}$-tautological control systems. A morphism from $\mathfrak{G}$ to $\mathfrak{H}$ is a pair $(\Phi,\Phi^{\#})$ such that

(i) $\Phi\in C^{r}(M,N)$ and

(ii) $\Phi^{\#}=(\Phi^{\#}_{\mathcal{V}})_{{\mathcal{V}\mathrm{open}}}$ is a family of mappings $\Phi^{\#}_{\mathcal{V}}:\mathscr{G}(\mathcal{V})\rightarrow\Phi_{*}\mathscr{F}(\mathcal{V}),\mathcal{V}\subseteq N$ defined as follows:

(a) there exists a family $L_{\mathcal{V}}\in\mathrm{L}(\mathrm{\Gamma^{\mathcal{\nu}}}(T\mathcal{V});\mathrm{\Gamma^{\mathcal{\nu}}}(T(\Phi^{-1}(\mathcal{V}))))$ of continuous linear mappings satisfying $L_{{\mathcal{V}{{}^{\prime}}}}=L_{\mathcal{V}}|\mathrm{\Gamma^{\mathcal{\nu}}}(T\mathcal{V}{{}^{\prime}})$ if $\mathcal{V},\mathcal{V}{{}^{\prime}}\subseteq N$ are open with $\mathcal{V}{{}^{\prime}}\subseteq\mathcal{V}$;

(b) $\Phi^{\#}_{\mathcal{V}}=L_{\mathcal{V}}|\mathscr{G}(\mathcal{V})$.

From the point of view of control theory, one wishes to restrict the above definition further to account for the fact that morphisms ought to preserve trajectories.

Definition 4

Ref8 Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $\mathfrak{G}=(M,\mathscr{F})$ and $\mathfrak{H}=(N,\mathscr{G})$ be $C^{\nu}$-tautological control systems. A morphism $(\Phi,\Phi^{\#})$ from $\mathfrak{G}$ to $\mathfrak{H}$ is trajectory-preserving if, for each time-domain $\mathbb{T}$, each open $\mathcal{V}\subseteq N$, and each $Y\in\mathrm{LI\Gamma}^{\nu}(\mathbb{T};\mathscr{G}(\mathcal{V}))$, any integral curve $\xi:\mathbb{T}{{}^{\prime}}\rightarrow\Phi^{-1}(\mathcal{V})$ for the time-varying vector field $t\rightarrow\Phi^{\#}(Y_{t})$ defined on $\mathbb{T}{{}^{\prime}}\subseteq\mathbb{T}$ has the property that $\Phi\circ\xi$ is an integral curve for Y.

We have the following characterisation of trajectory-preserving morphisms.

Proposition 1

Ref8* Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $\mathfrak{G}=(M,\mathscr{F})$ and $\mathfrak{H}=(N,\mathscr{G})$ be $C^{\nu}$-tautological control systems. A morphism $(\Phi,\Phi^{\#})$ from $\mathfrak{G}$ to $\mathfrak{H}$ is trajectory-preserving if and only if, for each open $\mathcal{V}\subseteq N$, each $Y\in\mathscr{G}(\mathcal{V})$, each $y\in\mathcal{V}$, and each $x\in\Phi^{-1}(y)$, we have $T_{x}\Phi(\Phi^{\#}(Y)(x))=Y(y)$.*

Based on the above proposition, we then introduce the following definition, which will be essential in our main reachability result to be proved in Section 3.

Definition 5

Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $\mathfrak{G}=(M,\mathscr{F})$ and $\mathfrak{H}=(N,\mathscr{G})$ be $C^{\nu}$-tautological control systems. We say that the morphism $(\Phi,\Phi^{\#})$ from $\mathfrak{G}$ to $\mathfrak{H}$ is global in time if for each $s\in\mathbb{R}$, each open $\mathcal{V}\subseteq N$, each $Y\in\mathscr{G}(\mathcal{V})$, and each $x\in\Phi^{-1}(\mathcal{V})$, we have $J_{\Phi^{\#}(Y)}(s,x)=J_{Y}(s,\Phi(x))$.

In the remaining part of this section, we will discuss correspondences between ordinary control systems and tautological control systems.

Definition 6

Ref8 Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. A $C^{\nu}$-control system is a triple $\Sigma=(M,F,\mathfrak{C})$, where

(i) M is a $C^{r}$-manifold whose elements are called states.

(ii) $\mathfrak{C}$ is a topological space called the control set and

(iii) $F\in\mathrm{JP\Gamma}^{\nu}(\mathfrak{C};TM)$.

Proposition 2

Ref8* Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $\Sigma=(M,F,\mathfrak{C})$ be a $C^{\nu}$-control system. If $\mu\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{T};\mathfrak{C})$ then $F^{\mu}\in\mathrm{LB\Gamma}^{\nu}(\mathbb{T};TM)$, where $F^{\mu}:\mathbb{T}\times M\rightarrow TM$ is defined by $F^{\mu}(t,x)=F(x,\mu(t)).$*

Let $\Sigma=(M,F,\mathfrak{C})$ be a $C^{\nu}$-control system. For an interval $\mathbb{T}\subseteq\mathbb{R}$, a $\mathbb{T}$-trajectory is a locally absolutely continuous curve $\xi:\mathbb{T}\rightarrow M$ for which there exists $\mu\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{T};\mathfrak{C})$ such that $\xi{{}^{\prime}}(t)=F(\xi(t),\mu(t)),a.e.\ t\in\mathbb{T}.$ The set of $\mathbb{T}$-trajectories we denote by $\mathrm{Traj}(\mathbb{T},\Sigma)$. If $\mathcal{U}$ is open, we denote by $\mathrm{Traj}(\mathbb{T},\mathcal{U},\Sigma)$ those trajectories taking value in $\mathcal{U}$. To this control system we associate the $C^{\nu}$ tautological control system $\mathfrak{G}_{\Sigma}=(M,\mathscr{F}_{\Sigma})$ by

[TABLE]

The presheaf of sets of vector fields in this case is of the globally generated variety.

Suppose we have a $C^{\nu}$-tautological control system $\mathfrak{G}=(M,\mathscr{F})$ where $\mathscr{F}$ is globally generated. We define a $C^{\nu}$-control system $\Sigma_{\mathfrak{G}}=(M,F_{\mathscr{F}},\mathfrak{C}_{\mathscr{F}})$ as follows. We take $\mathfrak{C}_{\mathscr{F}}=\mathscr{F}(M)$, i.e. the control set is our family of globally defined vector fields and the topology is that induced from $\mathrm{\Gamma}^{\nu}(TM)$. We define

[TABLE]

Proposition 3

Ref8* Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $\mathfrak{G}=(M,\mathscr{F})$ be a $C^{\nu}$-tautological control system. Let $\Sigma=(M,F,\mathfrak{C})$ be a $C^{\nu}$-control system. Then the following statements hold:

(i) if $\mathfrak{G}$ is globally generated, then $\mathfrak{G}_{\Sigma_{\mathfrak{G}}}=\mathfrak{G}$;

(ii) if the map $u\rightarrow F^{u}$ from $\mathfrak{C}$ to $\mathrm{\Gamma}^{\nu}(TM)$ is injective and open onto its image, then $\Sigma_{\mathfrak{G}_{\Sigma}}=\Sigma.$*

Proposition 4

Ref8* Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty,\omega\}$, and let $r\in\{\infty,\omega\}$, as required. Let $\Sigma=(M,F,\mathfrak{C})$ be a $C^{\nu}$-control system with $\mathfrak{G}_{\Sigma}$ the associated $C^{\nu}$-tautological control system. Then the following statements hold:

(i) $\mathrm{Traj}(\mathbb{T},\mathcal{U},\Sigma)\subseteq\mathrm{Traj}(\mathbb{T},\mathcal{U},\mathscr{O}_{\mathfrak{G}_{\Sigma},\text{cpt}})$;

(ii) if the map $u\rightarrow F^{u}$ is injective and proper, then $\mathrm{Traj}(\mathbb{T},\mathcal{U},\mathscr{O}_{\mathfrak{G}_{\Sigma},\text{cpt}})\subseteq\mathrm{Traj}(\mathbb{T},\mathcal{U},\Sigma)$;

(iii) if $\mathfrak{C}$ is a Suslin topology space and if F is proper, then $\mathrm{Traj}(\mathbb{T},\mathcal{U},\mathscr{O}_{\mathfrak{G}_{\Sigma},\infty})\subseteq\mathrm{Traj}(\mathbb{T},\mathcal{U},\Sigma)$;

(iv) if, in addition, $\nu\in\{\infty,\omega\}$, then we may replace $\mathrm{Traj}(\mathbb{T},\mathcal{U},\mathscr{O}_{\mathfrak{G}_{\Sigma},\text{cpt}})$ with $\mathrm{Traj}(\mathbb{T},\mathcal{U},\mathscr{O}_{\mathfrak{G}_{\Sigma},\infty})$ in statements (i) and (ii).*

3 Morphisms of tautological control systems

3.1 Existence of tautological control system

Theorem 3.1

Let $m\in\mathbb{Z}_{\geq 0}$ and $m{{}^{\prime}}\in\{0,\mathrm{lip}\}$, let $\nu\in\{m+m{{}^{\prime}},\infty\}$, and let $r=\infty$, as required. Let $\mathfrak{H}=(N,\mathscr{G})$ be a $C^{\nu}$-tautological control system that is globally generated by a family of linearly independent vector fields, i.e. pointwise-independent everywhere. Let $M$ be a $C^{r}$ manifold and $\Phi:M\to N$ be a submersion. Then there exists a globally generated $C^{\nu}$-tautological control system $\mathfrak{G}=(M,\mathscr{F})$ such that there exists a trajectory-preserving morphism $(\Phi,\Phi^{\#})$ from $\mathfrak{G}$ to $\mathfrak{H}$.

Proof

Due to the local representatives for the submersion $\Phi$, for each $Y\in\mathscr{G}(N)$, and each $x_{\alpha}\in M$, there exist a neighbourhood $\mathcal{U}_{\alpha}\subseteq M$ of $x_{\alpha}$ and a $C^{\nu}$-vector field $X_{\alpha}$ defined on $\mathcal{U}_{\alpha}$, such that $T_{x}\Phi(X_{\alpha})=Y(\Phi(x))$ for each $x\in\mathcal{U}_{\alpha}$. Then we get an open covering $\{\mathcal{U}_{\alpha}\}$ of $M$. Since $M$ is a second-countable Hausdorff manifold, let $\{(\mathcal{W}_{i},g_{i})\}$ be a partition of unity subordinated to the open covering $\{\mathcal{U}_{\alpha}\}$ such that each open set $\mathcal{W}_{i}$ is a subset of an open set $\mathcal{U}_{\alpha(i)}$. Let $X$ be defined by

[TABLE]

a finite sum at each $x\in M$, then $X$ is a $C^{\nu}$-vector field on $M$ such that

[TABLE]

for every $x\in M$. Let $\Phi^{\#}(Y)=X$. Then for all $Y\in\mathscr{G}(N)$, this gives a family of $C^{\nu}$-vector fields on $M$ denoted by $\mathscr{X}$. Let $\mathscr{F}$ be globally generated such that $\mathscr{F}(M)=\mathscr{X}$. It is easy to verify that $(\Phi,\Phi^{\#})$ is a trajectory-preserving morphism from $\mathfrak{G}$ to $\mathfrak{H}$.

Remark 1

In the above theorem the assumption on $\mathfrak{H}=(N,\mathscr{G})$ that is globally generated by a family of linearly independent vector fields is to make sure condition (ii) in Definition 3 can be satisfied. In Ref7, existence of a lifted system is discussed in the context of ordinary control system, where a transpose mapping $(d\Phi)^{T}$ of the differential mapping $d\Phi$ is used to construct the vector fields of the control system. We should point out that the transpose mapping $(d\Phi)^{T}$ defined in a coordinate chart can not be defined globally as claimed in Ref7. When applicable, partition of unity is unavoidable to patch local constructions to give global constructions.