Consider this nonlinear system, with \(X=\mathbb{R}^{2}\) and \(\mathcal{U}=[-1,1]\) : $$ \begin{aligned} &\dot{x}_{1}=\left(1-x_{2}^{2}\right) u^{2} \\ &\dot{x}_{2}=u . \end{aligned} $$ Show that the system is complete, that is, solutions exist for every initial state and every control. Show, however, that \(\mathcal{R}^{1}(0)\) is not a closed set. Why does this not contradict the discussion in Remark 10.1.10? (Hint: Consider the states \(x_{k}^{f}=\alpha\left(\omega_{k}\right)\), where \(\omega_{k}\) is a control that switches fast between \(+1\) and $\left.-1 .\right)$

Let $\mathcal{R}^{\leq T}\left(x^{0}\right):=\bigcup_{0 \leq t \leq T} \mathcal{R}^{t}\left(x^{0}\right)\(. Show that, for any \)x\( and \)T, \mathcal{R}^{\leq T}\left(x^{0}\right)$ is: \- connected, \- compact, \- but not necessarily convex. (Hint: (For compactness.) If $x_{k}^{\mathrm{f}} \in \mathcal{R}^{t_{k}}\left(x^{0}\right)\(, you may assume that \)t_{k} \backslash T\( or \)t_{k} \backslash T\( for some \)T$ (why?). Then, restrict or extend controls \(\omega_{k}\) to \([0, T]\). Finally, use a compactness argument as in the proof that \(\mathcal{R}^{T}\left(x^{0}\right)\) is compact.)

(a) Consider a controllable single-input system \(\dot{x}=A x+b u\). Show that there is some \(\delta>0\) with the following property: for each $\gamma \neq 0\(, the function \)t \mapsto \gamma^{\prime} e^{-t A} b\( has at most \)n-1$ zeros in the interval \([0, \delta]\). (b) Conclude that, for controllable single-input systems, with \(\mathcal{U}=[\underline{u}, \bar{u}]\), there is some \(\delta>0\) so that, whenever \(x^{0}\) and \(x^{\mathrm{f}}\) are so that \(x^{\mathrm{f}}\) can be reached in time at most \(\delta\) from \(x^{0}\), the time optimal control steering \(x^{0}\) to \(x^{f}\) is piecewise constant, taking values \(\underline{u}\) or \(\bar{u}\) in at most \(n\) intervals. (c) Conclude further that, if \(T\) is the minimum time for steering \(x^{0}\) to \(x^{\mathrm{f}}\), then the time optimal control steering a state \(x^{0}\) to a state \(x^{\mathrm{f}}\) is piecewise constant, with at most \(T n / \delta\) switches. (Hint: (For (a).) Assume the property is false on \([0,1 / k]\). Note that one may restrict attention to \(\gamma\) 's so that \(\|\gamma\|=1\). What can be said about the zeros of derivatives of \(\gamma^{\prime} e^{-t A} b\) ? Take limits \(k \rightarrow \infty\).)

For any metric space \(M\), we use \(K(M)\) to denote the family of all nonempty compact subsets of \(M\), and define $$ D\left(C_{1}, C_{2}\right):=\max \left\\{\max _{x \in C_{1}} d\left(x, C_{2}\right), \max _{x \in C_{2}} d\left(x, C_{1}\right)\right\\} . $$ Show that \(D\) defines a metric on \(\mathbb{K}\) (usually called the Hausdorff metric). Now consider a linear system \(\dot{x}=A x+B u\) and a fixed initial state \(x^{0} \in \mathbb{R}^{n}\). Show that the mapping $T \mapsto \mathcal{R}^{T}\left(x^{0}\right)\( is continuous as a map from \)\mathbb{R}$ into \(\mathbb{K}\left(\mathbb{R}^{n}\right)\).

Consider the nonlinear, affine in controls, system \(\dot{x}=x^{2}+u\), with \(\mathcal{U}=[-1,1]\) (any other compact convex set \(\mathcal{U}\) could be used). Show that there are states \(x^{0}\) and times \(T\) for which \(\mathcal{R}^{T}\left(x^{0}\right)\) is not compact. Why does this not contradict the discussion in Remark 10.1.10?

Let \(\mathcal{V} \subseteq \mathcal{U}\) be so that the convex hull co \((\mathcal{V})=\mathcal{U} .\) Show that for each $\omega \in \mathcal{L}_{\mathrm{u}}(0, T)\( there is some sequence \)\nu_{k} \stackrel{\mathrm{w}}{\rightarrow} \omega\( so that \)\nu_{k}(t) \in \mathcal{V}\( for all \)t \in[0, T]$. (Suggestion: you may want to argue as follows. First, show that \(\omega\) can be weakly approximated by piecewise constant controls, i.e., of the form \(\sum_{\text {finite }} I_{J_{i}} u_{i}\), for intervals \(J_{i} \subseteq[0, T]\) and elements \(u_{i} \in \mathcal{U} .\) Next argue that, on each interval \(J\), a constant control with value \(u=\sum_{i=1}^{r} \rho_{i} v_{i}\), with \(\sum \rho_{i}=1\), $v_{i} \in \mathcal{V}, \rho_{i} \geq 0\( for all \)i$, can be in turn weakly approximated by controls with values in \(\mathcal{V}\). For this last approximation, you may think first of the special case \(r=2\) : in that case, the sequence \(\omega_{k}\left(v_{1}-v_{2}\right)+v_{2}\) converges to $\rho v_{1}+(1-\rho) v_{2}\(, if \)\omega_{k}$ is the sequence constructed in Example 10.1.7.)