Suppose that \(\Delta=\Delta_{f_{1}, \ldots, f_{r}}\) has constant rank \(r\), and let \(X \in\) \(\mathbb{V}(\mathcal{O})\). Then, the following two properties are equivalent: 1\. \(\Delta\) is invariant under \(X\). 2\. Let \(\mathcal{O}_{1}\) be an open subset of \(\mathcal{O}\) and let $t \in \mathbb{R}\( be so that \)(t, x) \in \mathcal{D}_{X}\( for all \)x \in \mathcal{O}_{1}\(. Define \)\mathcal{O}_{0}:=e^{t X} \mathcal{O}_{1}$. Assume that \(Y \in \mathrm{V}\left(\mathcal{O}_{0}\right)\) is such that $Y(z) \in \Delta(z)\( for each \)z \in \mathcal{O}_{0}\(. Then, \)\operatorname{Ad}_{t X} Y(x) \in \Delta(x)\( for each \)x \in \mathcal{O}_{1}$.

A distribution on the open subset \(\mathcal{O} \subseteq \mathbb{R}^{n}\) is a map \(\Delta\) which assigns, to each \(x \in \mathcal{O}\), a subspace \(\Delta(x)\) of \(\mathbb{R}^{n}\). A vector field $f \in \mathbb{V}(\mathcal{O})\( is pointwise in \)\Delta\(, denoted \)f \in_{p} \Delta$, if \(f(x) \in \Delta(x)\) for all \(x \in \mathcal{O}\). A distribution is invariant under a vector field \(f \in \mathbb{V}(\mathcal{O})\) if $$ g \in_{p} \Delta \Rightarrow[f, g] \in_{p} \Delta, $$ and it is involutive if it is invariant under all \(f \in_{p} \Delta\), that is, it is pointwise closed under Lie brackets: $$ f \in_{p} \Delta \text { and } g \in_{p} \Delta \quad \Rightarrow \quad[f, g] \in_{p} \Delta . $$ The distribution generated by a set of vector fields $f_{1}, \ldots, f_{r} \in \mathrm{V}(\mathcal{O})\( is defined \)b y$ $$ \Delta_{f_{1}, \ldots, f_{r}}(x):=\operatorname{span}\left\\{f_{1}(x), \ldots, f_{r}(x)\right\\} $$ for each \(x \in \mathcal{O}\). A distribution has constant rank \(r\) if \(\operatorname{dim} \Delta(x)=r\) for all \(x \in \mathcal{O}\).

If the vector fields \(Y_{1}, \ldots, Y_{\ell}\) and \(X\) are analytic, then $\operatorname{span}\left\\{\operatorname{Ad}_{t X} Y_{j}\left(x^{0}\right), j=1, \ldots \ell, t \in \mathcal{I}_{X, x^{0}}\right\\}$ equals $$ \operatorname{span}\left\\{\operatorname{ad}_{X}^{k} Y_{j}\left(x^{0}\right), j=1, \ldots \ell, k \geq 0\right\\} $$ for each \(x^{0} \in \mathcal{O}\). Proof. Fix \(x^{0} \in \mathcal{O}\). Let \(S_{0}\) and \(S_{1}\) be, respectively, the sets of vectors \(\nu \in \mathbb{R}^{n}\) and \(\mu \in \mathbb{R}^{n}\) such that $$ \nu^{\prime} \operatorname{ad}_{X}^{k} Y_{j}\left(x^{0}\right)=0, \quad j=1, \ldots \ell, k \geq 0 $$ and $$ \mu^{\prime} \operatorname{Ad}_{t X} Y_{j}\left(x^{0}\right)=0, \quad j=1, \ldots \ell, t \in \mathcal{I}_{X, x^{0}} $$ Take any \(\nu \in S_{0}\). For each \(j=1, \ldots \ell\), by Lemma \(4.4 .3\), $$ \left.\nu^{\prime} \frac{\partial^{k} \mathrm{Ad}_{t X} Y_{j}\left(x^{0}\right)}{\partial t^{k}}\right|_{t=0}=0, \forall k \geq 0 . $$ Since, by Lemma 4.4.4, \(\operatorname{Ad}_{t} X Y_{j}\left(x^{0}\right)\) is analytic as a function of \(t\), this means that $\nu^{\prime} \operatorname{Ad}_{t X} Y_{j}\left(x^{0}\right) \equiv 0\(, so \)\nu \in S_{1}$. Conversely, if \(\nu \in S_{1}\), then $\nu^{\prime} \operatorname{Ad}_{\ell X} Y_{j}\left(x^{0}\right) \equiv 0$ implies that all derivatives at zero vanish, so \(\nu \in S_{0}\) (analyticity is not needed here). Thus \(S_{0}=S_{1}\), and the result is proved.

Consider a model for the "shopping cart" shown in Figure 4.2 ("knife-edge" or "unicycle" are other names for this example). The state is given by the orientation \(\theta\), together with the coordinates \(x_{1}, x_{2}\) of the midpoint between the back wheels. Figure 4.2: Shopping cart. The front wheel is a castor, free to rotate. There is a non-slipping constraint on movement: the velocity $\left(\dot{x}_{1}, \dot{x}_{2}\right)^{\prime}\( must be parallel to the vector \)(\cos \theta, \sin \theta)^{\prime} .$ This leads to the following equations: $$ \begin{aligned} \dot{x}_{1} &=u_{1} \cos \theta \\ \dot{x}_{2} &=u_{1} \sin \theta \\ \dot{\theta} &=u_{2} \end{aligned} $$ where we may view \(u_{1}\) as a "drive" command and \(u_{2}\) as a steering control (in practice, we implement these controls by means of differential forces on the two back corners of the cart). We view the system as having state space \(\mathbb{R}^{3}\) (a more accurate state space would be the manifold \(\mathbb{R}^{2} \times \mathbb{S}^{1}\) ). (a) Show that the system is completely controllable. (b) Consider these new variables: $z_{1}:=\theta, z_{2}:=x_{1} \cos \theta+x_{2} \sin \theta, z_{3}:=\( \)x_{1} \sin \theta-x_{2} \cos \theta, v_{1}:=u_{2}\(, and \)v_{2}:=u_{1}-u_{2} z_{3}$. (Such a change of variables is called a "feedback transformation".) Write the system in these variables, as \(\dot{z}=\widetilde{f}(z, v) .\) Note that this is one of the systems \(\Sigma_{i}\) in Exercise 4.3.14. Explain why controllability can then be deduced from what you already concluded in that previous exercise.

Suppose that \(\Delta=\Delta_{f_{1}, \ldots, f_{r}}\) has constant rank \(r\) and is invariant under each of \(X_{1}, \ldots, X_{k} \in \mathrm{V}(\mathcal{O})\). Let \(\mathcal{O}_{k}\) be an open subset of \(\mathcal{O}\) and let $s_{1}, \ldots, s_{k}$ be real numbers with the following property: $$ \left(s_{i}, x\right) \in \mathcal{D}_{X_{i}}, \quad \forall x \in \mathcal{O}_{i}, i=1, \ldots, k, $$ where we define $$ \mathcal{O}_{k-1}:=e^{s_{k} X_{k}} \mathcal{O}_{k}, \ldots, \mathcal{O}_{1}:=e^{s_{2} X_{2}} \mathcal{O}_{2}, \mathcal{O}_{0}:=e^{s_{1} X_{1}} \mathcal{O}_{1} . $$ Assume that \(Y \in \mathbb{V}\left(\mathcal{O}_{0}\right)\) is such that $Y(z) \in \Delta(z)\( for each \)z \in \mathcal{O}_{0}$. Then, $\operatorname{Ad}_{s_{k} X_{k}} \ldots \operatorname{Ad}_{s_{1} X_{1}} Y(x) \in \Delta(x)\( for all \)x \in \mathcal{O}_{k}$.

Suppose that \(\Delta=\Delta_{f_{1}, \ldots, f_{r}}\) is a distribution of constant rank \(r\). Then, 1\. The following two properties are equivalent, for any $f \in \mathbb{V}(\mathcal{O})$ : (a) \(f \in_{p} \Delta\) (b) For each \(x^{0} \in \mathcal{O}\), there are a neighborhood \(\mathcal{O}_{0}\) of \(x^{0}\) and \(r\) smooth functions $\alpha_{i}: \mathcal{O}_{0} \rightarrow \mathbb{R}_{1} i=1, \ldots, r$, so that $$ f(x)=\sum_{i=1}^{r} \alpha_{i}(x) f_{i}(x) \text { for all } x \in \mathcal{O}_{0} $$ 2\. The following two properties are equivalent, for any $f \in \mathbb{V}(\mathcal{O})$ : (a) \(\Delta\) is invariant under \(f\). (b) \(\left[f, f_{j}\right] \in_{p} \Delta\) for each \(j \in\\{1, \ldots, r\\}\). 3\. Finally, the following two properties are equivalent: (a) \(\Delta\) is involutive. (b) \(\left[f_{i}, f_{j}\right] \in_{p} \Delta\) for all $i, j \in\\{1, \ldots, r\\}$.