Let \(V: \mathcal{O} \rightarrow \mathbb{R}\) be a continuous and positive definite function defined on some neighborhood \(\mathcal{O}\) of \(x^{0}\), where \(X=\mathbb{R}^{n}\). Assume that \(V\) satisfies property (3) in Definition 5.7.1. Show that, then, there exists a local controlLyapunov function \(W\) for the same system, with \(W=V\) on some neighborhood of \(x^{0}\). (Hint: For \(\delta>0\) small enough, let \(W=V\) on the ball \(\mathcal{B}_{\delta}\left(x^{0}\right)\) and \(W(x):=V\left(\delta\left(x-x^{0}\right) /\left\|x-x^{0}\right\|\right)\) outside. Verify the definition of properness for \(\varepsilon\) less than \(\left.\inf \left\\{V(x),\left\|x-x^{0}\right\|=\delta\right\\} .\right)\)

The function \(\varphi\) can be obtained as the solution of an optimization problem. For each fixed \(x \neq 0\), thought of as a parameter, not as a state, we may consider the pair \((a(x), B(x))\) as a \((1, m)\) pair describing a linear system of dimension \(\mathbf{1}\), with \(m\) inputs. The equations for this system are as follows, where we prefer to use " \(z\) " to denote its state, so as not to confuse with \(x\), which is now a fixed element of $\mathbb{R}^{n} \backslash\\{0\\}$ : $$ \dot{z}=a z+\sum_{i=1}^{m} b_{i} u_{i}=a z+B u \text {. } $$ The control-Lyapunov function condition guarantees that this system is asymptotically controllable. In fact, the condition " $B=0 \Rightarrow a<0^{\prime \prime}$ means precisely that this system is asymptotically controllable. A stabilizing feedback law \(k=k(x)\) for the original nonlinear system must have, for each fixed \(x\), the property that $$ a+\sum_{i=1}^{m} b_{i} k_{i}<0 \text {. } $$ This means that \(u=k z\) must be a stabilizing feedback for the linear system (5.59). Consider for this system the infinite-horizon linear-quadratic problem of minimizing (cf. Theorem 41 (p. 384)) $$ \int_{0}^{\infty} u^{2}(t)+\beta(x) z^{2}(t) d t $$ (For motivation, observe that the term \(u^{2}\) has greater relative weight when \(\beta\) is small, making controls small if \(x\) is small.) Prove that solving this optimization problem leads to our formula (5.56).

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

Suppose that \(x^{0}\) is an equilibrium state: \(f\left(x^{0}\right)=0\). Show that, if the system is feedback linearizable about \(x^{0}\), then it is possible to choose \(T\left(x^{0}\right)=0\) and \(\alpha\left(x^{0}\right)=0\).

Consider a continuous-time system \(\dot{x}=f(x)\) with no controls and \(x=\mathbb{R}^{n}\). Suppose that \(V: \mathbb{R}^{n} \rightarrow \mathbb{R}\) is proper and positive definite, and satisfies \(\dot{V}(x)=L_{f} V(x)<0\) for all \(x \neq 0\) (this is the Lyapunov condition in Lemma 5.7.4). Show that there exists a continuous function \(\alpha:[0, \infty) \rightarrow[0, \infty)\) which is positive definite (that is, \(\alpha(0)=0\) and \(\alpha(r)>0\) for all \(r>0\) ) such that the following differential inequality holds: \(\nabla V(x) \cdot f(x)=\dot{V}(x) \leq-\alpha(V(x))\) for all $x \in \mathbb{R}^{n} .$ (Hint: Study the maximum of \(L_{f} V(x)\) on the set where \(V(x)=r_{.}\))

\(\diamond\) 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 \(i, j \geq 0\).