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Problem 4

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 $\left.-1 .\right)$

Expert verified

The given nonlinear system with state variables \(x_1\) and \(x_2\) and control variable \(u\) is complete, as solutions exist for every initial state and every control, due to the continuous functions in the system. However, \(\mathcal{R}^{1}(0)\) is not a closed set when using a control input \(\omega_k\) that switches fast between \(+1\) and \(-1\), which leads to states with no limit points inside the reachable set. This doesn't contradict Remark 10.1.10 because the control input used is not piecewise continuous, and therefore, the result from Remark 10.1.10 doesn't apply to this control input.

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Chapter 10

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.)

Chapter 10

(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 \(k \rightarrow \infty\).)

Chapter 10

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 \(\mathbb{K}\left(\mathbb{R}^{n}\right)\).

Chapter 10

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?

Chapter 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.)

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