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Problem 2

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?

Expert verified

The reachable set \(\mathcal{R}^T(x^0)\) is not compact for the initial state \(x^0 = -1\) and some time duration \(T\), as the system will enter an unbounded reachable set when the control input is at its maximum value (\(u=1\)). This does not contradict Remark 10.1.10 because the example given does not satisfy all the conditions mentioned in the remark, which assumes that the reachable set is compact for all \(x^0\) and \(T\). The fact that there exists a specific case where the reachable set is not compact shows that the mentioned conditions in Remark 10.1.10 do not hold for this particular example.

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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

Consider the undamped harmonic oscillator \(\dot{x}_{1}=x_{2}, \dot{x}_{2}=\) \(-x_{1}+u\), with control-value set \(\mathcal{U}=[-1,1]\). Show, for each two states \(x^{0}, x^{\mathrm{f}}\) : 1\. there is a unique control \(\tilde{\omega}\) steering \(x^{0}\) to \(x^{\mathrm{f}}\) in minimal time, 2\. this control has \(|u(t)|=1\) for almost all \(t\), and 3 . the intervals between sign changes have length \(\pi\). Finally, provide a complete description of the optimal controls for the problem of transferring any state to \(x^{f}=0\), including a diagram. (Hint: For the controllability part, recall Exercise 3.6.8.)

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 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 $\left.-1 .\right)$

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