### 统计代写|最优控制作业代写optimal control代考|Singularities of Semiconcave Functions

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## 统计代写|最优控制作业代写optimal control代考|Singularities of Semiconcave Functions

By a singular point, or singularity, of a semiconcave function $u$ we mean a point where $u$ is not differentiable. This chapter is devoted to the analysis of the set of all singular points for $u$, which is called singular set and is denoted here by $\Sigma(u)$. As we have already remarked, the singular set of a semiconcave function has zero measure by Rademacher’s theorem. However, we will see that much more detailed properties can be proved.

In Section $4.1$ we study the rectifiability properties of the singular set. We divide the singular points according to the dimension of the superdifferential of $u$ denoting by $\Sigma^{k}(u)$ the set of points $x$ such that $D^{+} u(x)$ has dimension $k$. Then we show that $\Sigma^{k}(u)$ is countably $(n-k)$-rectifiable for all integers $k=1, \ldots, n$. In particular, the whole singular set $\Sigma(u)$ is countably ( $n-1)$-rectifiable.

Sections $4.2$ and $4.3$ are devoted to the propagation of singularities for semiconcave functions: given a singular point $x_{0}$, we look for conditions ensuring that $x_{0}$ belongs to a connected component of dimension $v \geq 1$ of the singular set. We study first the propagation along Lipschitz arcs and then along Lipschitz manifolds of higher dimension. In general we find that a sufficient condition for the propagation of singularities from $x_{0}$ is that the inclusion $D^{*} u\left(x_{0}\right) \subset \partial D^{+} u\left(x_{0}\right)$ (see Proposition 3.3.4-(b) ) is strict.

As an application of the previous analysis, we study in Section $4.4$ some properties of the distance function from a closed set $S$. Using our propagation results, we show that the distance function has no isolated singularities except for the special case when the singularity is the center of a spherical connected component of the complement of $S$. In general, we show that a point $x_{0}$ which is singular for the distance function belongs to a connected set of singular points whose Hausdorff dimension is at least $n-k$, with $k=\operatorname{dim}\left(D^{+} d\left(x_{0}\right)\right)$.

## 统计代写|最优控制作业代写optimal control代考|Rectifiability of the singular sets

Throughout this chapter $\Omega \subset \mathbb{R}^{n}$ is an open set and $u: \Omega \rightarrow \mathbb{R}$ is a semiconcave function. We denote by $\Sigma(u)$ the set of points of $\Omega$ where $u$ is not differentiable and

we call it the singular set of $u$. In the following we use some notions from measure theory, like Hausdorff measures and rectifiable sets, which are recalled in Appendix A. 3 .

We know from Theorem 2.3.1-(ii) that $D u$ is a function with bounded variation if $u$ is semiconcave with a linear modulus. For functions of bounded variation one can introduce the jump set, whose rectifiability properties have been widely studied (see Appendix A. 6). We now show that the jump set of $D u$ coincides with the singular set $\Sigma(u)$. To this purpose we need two preliminary results. The first one is a lemma about approximate limits (see Definition A. 6.2).

Lemma 4.1.1 Let $w \in L^{1}(A)$, with $A \subset \mathbb{R}^{n}$ open, let $\bar{x} \in A$, and let ap $\lim {x \rightarrow \bar{x}} w(x)=\bar{w}$. Then for any $\theta \in \mathbb{R}^{n}$ with $|\theta|=1$ we can find a sequence $\left{x{k}\right} \subset A$ such that
$$x_{k} \rightarrow \bar{x}, \quad \frac{x_{k}-\bar{x}}{\left|x_{k}-\bar{x}\right|} \rightarrow \theta, \quad w\left(x_{k}\right) \rightarrow \bar{w} \quad \text { as } k \rightarrow \infty$$
Proof – For any $k \in \mathbb{N}$, let us define
$$A_{k}=\left{x \in A \backslash{\bar{x}}:\left|\frac{x-\bar{x}}{|x-\bar{x}|}-\theta\right|<\frac{1}{k}\right}$$ Any such set $A_{k}$ is the intersection of $A$ with an open cone of vertex $\bar{x}$. Therefore $$\lim {\rho \rightarrow 0^{+}} \frac{\text { meas }\left(B{\rho}(\bar{x}) \cap A_{k}\right)}{\rho^{n}}>0 .$$
By the definition of approximate limit we have
$$\lim {\rho \rightarrow 0^{+}} \frac{\operatorname{meas}\left(\left{x \in B{\rho}(\bar{x}) \cap A_{k}:|w(x)-\bar{w}|>1 / k\right}\right)}{\rho^{n}}=0 .$$
Comparing the above relations we see that the set
$$\left{x \in B_{\rho}(\bar{x}) \cap A_{k}:|w(x)-\bar{w}| \leq 1 / k\right}$$
is nonempty if $\rho$ is small enough. Thus, we can find $x_{k} \in A_{k}$ such that $\left|w\left(x_{k}\right)-\bar{w}\right| \leq$ $1 / k,\left|x_{k}-\bar{x}\right| \leq 1 / k$. Repeating this construction for all $k$ we obtain a sequence $\left{x_{k}\right}$ with the desired properties.

Next we give a result showing, roughly speaking, that for the gradient of a semiconcave function the notions of limit and of approximate limit coincide.

## 统计代写|最优控制作业代写optimal control代考|Propagation along Lipschitz arcs

Let $u$ be a semiconcave function in an open domain $\Omega \subseteq \mathbb{R}^{n}$. The rectiflability properties of $\Sigma(u)$, obtained in the previous section, can be regarded as “upper bounds” for $\Sigma(u)$. From now on, we shall study the singular set of $u$ trying to obtain “lower bounds” for such a set. In the rest of the chapter, we restrict our attention to semiconcave functions with a linear modulus.

Given a point $x_{0} \in \Sigma(u)$, we are interested in conditions ensuring the existence of other singular points approaching $x_{0}$. The following example explains the nature of such conditions.
Example 4.2.1 The functions
$$u_{1}\left(x_{1}, x_{2}\right)=-\sqrt{x_{1}^{2}+x_{2}^{2}}, \quad u_{2}\left(x_{1}, x_{2}\right)=-\left|x_{1}\right|-\left|x_{2}\right|$$
are concave in $\mathbb{R}^{2}$, and $(0,0)$ is a singular point for both of them. Moreover, $(0,0)$ is the only singularity for $u_{1}$ while
$$\Sigma\left(u_{2}\right)=\left{\left(x_{1}, x_{2}\right): x_{1} x_{2}=0\right}$$
So, $(0,0)$ is the intersection point of two singular lines of $u_{2}$. Notice that $(0,0)$ has magnitude 2 with respect to both functions as
$$\begin{gathered} D^{+} u_{1}(0,0)=\left{\left(p_{1}, p_{2}\right): p_{1}^{2}+p_{2}^{2} \leq 1\right} \ D^{+} u_{2}(0,0)=\left{\left(p_{1}, p_{2}\right):\left|p_{1}\right| \leq 1,\left|p_{2}\right| \leq 1\right} \end{gathered}$$
The different structure of $\Sigma\left(u_{1}\right)$ and $\Sigma\left(u_{2}\right)$ in a neighborhood of $x_{0}$ is captured by the reachable gradients. In fact,
$$\begin{gathered} D^{} u_{1}(0,0)=\left{\left(p_{1}, p_{2}\right): p_{1}^{2}+p_{2}^{2}=1\right}=\partial D^{+} u_{1}(0,0) \ D^{} u_{2}(0,0)=\left{\left(p_{1}, p_{2}\right):\left|p_{1}\right|=1,\left|p_{2}\right|=1\right} \neq \partial D^{+} u_{2}(0,0) \end{gathered}$$
In other words, the inclusion $D^{*} u(x) \subset \partial D^{+} u(x)$ (see Proposition 3.3.4(b)) is an equality for $u_{1}$ and a proper inclusion for $u_{2}$.

The above example suggests that a sufficient condition to exclude that $x_{0}$ is an isolated point of $\Sigma(u)$ should be that $D^{*} u\left(x_{0}\right)$ fails to cover the whole boundary of $D^{+} u\left(x_{0}\right)$. As we shall see, such a condition implies a much stronger property, namely that $x_{0}$ is the initial point of a Lipschitz singular arc.

In the following we call an arc a continuous map $\mathbf{x}:[0, \rho] \rightarrow \mathbb{R}^{n}, \rho>0$. We shall say that the arc $\mathbf{x}$ is singular for $u$ if the support of $\mathbf{x}$ is contained in $\Omega$ and $\mathbf{x}(s) \in \Sigma(u)$ for every $s \in[0, \rho]$. The following result describes the “arc structure” of the singular set $\Sigma(u)$.

## 统计代写|最优控制作业代写optimal control代考|Rectifiability of the singular sets

Xķ→X¯,Xķ−X¯|Xķ−X¯|→θ,在(Xķ)→在¯ 作为 ķ→∞

A_{k}=\left{x \in A \backslash{\bar{x}}:\left|\frac{x-\bar{x}}{|x-\bar{x}|}-\theta \right|<\frac{1}{k}\right}A_{k}=\left{x \in A \backslash{\bar{x}}:\left|\frac{x-\bar{x}}{|x-\bar{x}|}-\theta \right|<\frac{1}{k}\right}任何这样的集合一种ķ是的交集一种有一个开放的顶点圆锥X¯. 所以林ρ→0+ 测量 (乙ρ(X¯)∩一种ķ)ρn>0.

\lim {\rho \rightarrow 0^{+}} \frac{\operatorname{meas}\left(\left{x \in B{\rho}(\bar{x}) \cap A_{k}:| w(x)-\bar{w}|>1 / k\right}\right)}{\rho^{n}}=0 。\lim {\rho \rightarrow 0^{+}} \frac{\operatorname{meas}\left(\left{x \in B{\rho}(\bar{x}) \cap A_{k}:| w(x)-\bar{w}|>1 / k\right}\right)}{\rho^{n}}=0 。

\left{x \in B_{\rho}(\bar{x}) \cap A_{k}:|w(x)-\bar{w}| \leq 1 / k\right}\left{x \in B_{\rho}(\bar{x}) \cap A_{k}:|w(x)-\bar{w}| \leq 1 / k\right}

## 统计代写|最优控制作业代写optimal control代考|Propagation along Lipschitz arcs

\Sigma\left(u_{2}\right)=\left{\left(x_{1}, x_{2}\right): x_{1} x_{2}=0\right}\Sigma\left(u_{2}\right)=\left{\left(x_{1}, x_{2}\right): x_{1} x_{2}=0\right}

\begin{聚集} D^{+} u_{1}(0,0)=\left{\left(p_{1}, p_{2}\right): p_{1}^{2}+p_{ 2}^{2} \leq 1\right} \ D^{+} u_{2}(0,0)=\left{\left(p_{1}, p_{2}\right):\left| p_{1}\右| \leq 1,\left|p_{2}\right| \leq 1\right} \end{聚集}\begin{聚集} D^{+} u_{1}(0,0)=\left{\left(p_{1}, p_{2}\right): p_{1}^{2}+p_{ 2}^{2} \leq 1\right} \ D^{+} u_{2}(0,0)=\left{\left(p_{1}, p_{2}\right):\left| p_{1}\右| \leq 1,\left|p_{2}\right| \leq 1\right} \end{聚集}

\begin{聚集} D^{} u_{1}(0,0)=\left{\left(p_{1}, p_{2}\right): p_{1}^{2}+p_{2 }^{2}=1\right}=\partial D^{+} u_{1}(0,0) \ D^{} u_{2}(0,0)=\left{\left(p_{ 1}, p_{2}\right):\left|p_{1}\right|=1,\left|p_{2}\right|=1\right} \neq \partial D^{+} u_{ 2}(0,0) \end{聚集}\begin{聚集} D^{} u_{1}(0,0)=\left{\left(p_{1}, p_{2}\right): p_{1}^{2}+p_{2 }^{2}=1\right}=\partial D^{+} u_{1}(0,0) \ D^{} u_{2}(0,0)=\left{\left(p_{ 1}, p_{2}\right):\left|p_{1}\right|=1,\left|p_{2}\right|=1\right} \neq \partial D^{+} u_{ 2}(0,0) \end{聚集}

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