Assume that \(\Sigma\) is a time-invariant discrete-time system of class \(\mathcal{C}^{1}\), $$ x^{+}=\mathcal{P}(x, u), $$ with \(X \subseteq \mathbb{R}^{n}\) and \(\mathcal{U} \subseteq \mathbb{R}^{m}\) open, and let \(\left(x^{0}, u^{0}\right)\) be an equilibrium pair, i.e. $$ \mathcal{P}\left(x^{0}, u^{0}\right)=x^{0} . $$ Assume that the linearization of \(\Sigma\) at \(\left(x^{0}, u^{0}\right)\) is asymptotically controllable. Then \(\Sigma\) is locally asymptotically controllable (to \(x^{0}\) ). Moreover, there exists in that case a matrix $F \in \mathbb{R}^{m \times n}$ such that the closed-loop system $$ x^{+}=\mathcal{P}_{c l}(x):=\mathcal{P}\left(x, u^{0}+F\left(x-x^{0}\right)\right) $$ is locally asymptotically stable.

\(\diamond\) A far more restrictive problem is that of asking that \(\Sigma\) be linearizable by means of coordinate changes alone, i.e., that there be some diffeomorphism defined in a neighborhood of \(x^{0}\), and a controllable pair \((A, b)\), so that \(T_{*}(x) f(x)=A T(x)\) and \(T_{*}(x) g(x)=b\). This can be seen as feedback linearization with \(\alpha \equiv 0\) and \(\beta \equiv 1\). Show that such a linearization is possible if and only if $g\left(x^{0}\right), \operatorname{ad}_{f} g\left(x^{0}\right), \ldots, \operatorname{ad}_{f}^{n-1} g\left(x^{0}\right)$ are linearly independent and \(\left[\operatorname{ad}_{f}^{i} g, \operatorname{ad}_{f}^{j} g\right]=0\) for all \(i, j \geq 0\).

Let \(\Sigma\) be a linear (time-invariant) continuous-time system with no controls, \(\dot{x}=A x\), and let \(P>0, P \in \mathbb{R}^{n \times n}\). Prove that the condition $$ A^{\prime} P+P A<0 $$ is sufficient for \(V(x):=x^{\prime} P x\) to be a Lyapunov function for \(\Sigma\).

This problem is a variation of Proposition 5.9.1. Suppose given a smooth control-affine system, and \(V\) as there, so that \((5.43)\) holds, and consider, for each \(x \in X\), the following vector subspace of \(\mathbf{R}^{n}\) $$ \Delta(x):=\operatorname{span}\left\\{f(x), \operatorname{ad}_{f}^{k} g_{i}(x), i=1, \ldots, m, k=0,1,2, \ldots\right\\} $$ (in the differential-geometric terminology of Chapter \(4, \Delta\) defines a "distribution"). Assume that \(\nabla V(x)=0\) implies \(x=0\) and that \(\operatorname{dim} \Delta(x)=n\) for all \(x \neq 0 .\) Show that the feedback law \((5.44)\) stabilizes the system. (Hint: Prove by induction on \(k\) that, if \(\xi\) is a solution of \(\dot{x}=f(x)\) so that $L_{f} V(\xi(t)) \equiv 0\( and \)L_{g i} V(\xi(t)) \equiv 0\( for all \)i$, then also \(L_{\text {ad } j_{g_{i}}} V(\xi(t)) \equiv 0\), for all \(k\) and all \(i\). You will need to use the facts that \(L_{\text {ad }_{f}^{k+1} g_{i}}\) can be expressed in terms of \(L_{f} L_{\text {ad }_{f}^{k} g_{i}}\) and $L_{\text {ad }_{f}^{k_{i}}} L_{f}\(, and that \)\nabla\left(L_{f} V\right)$ vanishes on the set \(\left\\{x \mid L_{f} V(x)=0\right\\}\) (why?).)

\diamond\( Let \)\mathcal{S}_{n, m}^{c, \mathrm{dtH}}$, respectively \(\mathcal{S}_{n, m}^{c, \mathrm{H}}\), denote the set of all controllable pairs \((A, B) \in \mathcal{S}_{n, m}^{c}\) for which \(A\) is discrete-time Hurwitz (convergent), respectively Hurwitz, and consider the map $$ \beta:(A, B) \rightarrow\left((A-I)(A+I)^{-1},(A+I)^{-1} B\right) \text {. } $$ Prove that this map induces a bijection between $\mathcal{S}_{n, m}^{c, \mathrm{~d} t \mathrm{H}}\( and \)\mathcal{S}_{n, m}^{c, \mathrm{H}}$. Hint: Use the Hautus criterion to check controllability of \(\beta(A, B)\).

Suppose that the pairs \((A, b)\) and $(\widetilde{A}, \widetilde{b}) \in \mathcal{S}_{n, 1}$ are feedback equivalent in the sense of Definition 5.2.1. That is, there exist \(T \in\) \(G L(n), F \in \mathbb{R}^{1 \times n}\), and a nonzero scalar \(V\) such that \(T^{-1}(A+b F) T=\widetilde{A}\) and $T^{-1} b V=\widetilde{b}\(. Consider the systems \)\Sigma\( and \)\widetilde{\Sigma}$ given respectively by \(\dot{x}=A x+u b\) and $\dot{x}=\widetilde{A} x+u \widetilde{b}\(. Show that, for each open subset \)\mathcal{O} \subseteq \mathbb{R}^{n},(\Sigma, \mathcal{O})$ is feedback equivalent to \(\left(\tilde{\Sigma}, T^{-1} \mathcal{O}\right)\).