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

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

This chapter and the following two are devoted to the general properties of semiconcave functions. We begin here by studying the direct consequences of the definition and some basic examples, while the next chapters deal with generalized differentials and singularities. At this stage we study semiconcave functions without referring to specific applications; later in the book we show how the results obtained here can be applied to Hamilton-Jacobi equations and optimization problems.

The chapter is structured as follows. In Section $2.1$ we define semiconcave functions in full generality, and study some direct consequences of the definition, like the Lipschitz continuity and the relationship with the differentiability. Then we consider some examples in Section 2.2, like the distance function from a set, or the solutions to certain partial differential equations. We give an account of the vanishing viscosity method for Hamilton-Jacobi equations, where semiconcavity estimates play an important role. In Section $2.3$ we recall some properties which are peculiar to semiconcave functions with a linear modulus, like Alexandroff’s theorem or Jensen’s lemma. In Section $2.4$ we investigate the relation between viscous Hamilton-Jacobi equations and the heat equation induced by the Cole-Hopf transformation, showing that semiconcavity corresponds to the Li-Yau differential Harnack inequality for the heat equation. Finally, in Section $2.5$ we analyze the relation between semiconcavity and a generalized one-sided estimate, a property which will be applied later in the book to prove semiconcavity of viscosity solutions.

## 统计代写|最优控制作业代写optimal control代考|Definition and basic properties

Throughout the section $S$ will be a subset of $\mathbb{R}^{n}$.
Definition 2.1.1 We say that a function $u: S \rightarrow \mathbb{R}$ is semiconcave if there exists a nondecreasing upper semicontinuous function $\omega: \mathbb{R}{+} \rightarrow \mathbb{R}{+}$such that $\lim _{\rho \rightarrow 0^{+}} \omega(\rho)=0$ and
$$\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \leq \lambda(1-\lambda)|x-y| \omega(|x-y|)$$

for any pair $x, y \in S$, such that the segment $[x, y]$ is contained in $S$ and for any $\lambda \in[0,1]$. We call $\omega a$ modulus of semiconcavity for $u$ in $S$. A function $v$ is called semiconvex in $S$ if $-v$ is semiconcave.

In the case of $\omega$ linear, we recover the class of semiconcave functions introduced in the previous chapter (see Definition 1.1.1 and Proposition 1.1.3). We recall that, if $\omega(\rho)=\frac{C}{2} \rho$, for some $C \geq 0$, then $C$ is called a semiconcavity constant for $u$ in $S$.
We denote by $\mathrm{SC}(S)$ the space of all semiconcave functions in $S$ and by $\mathrm{SCL}(S)$ the functions which are semiconcave in $S$ with a linear modulus. A usual, we use the notation $S C_{l o c}(S)$ or $S C L_{l o c}(S)$ for the functions which are semiconcave (with a linear modulus) locally in $S$, i.e., on every compact subset of $S$.

As we have remarked in Chapter 1 , semiconcave functions with a linear modulus are the most common in the literature. Although they are a smaller class, they are sufficient for many applications; in addition, they enjoy stronger properties than general semiconcave functions and are easier to analyze, since they are more closely related to concave functions. Nevertheless, it is interesting to consider semiconcave functions with a general modulus, since they are a larger class, sharing many of the properties of the case of a linear modulus.

An interesting consequence of the general definition of semiconcavity given above is that any $C^{1}$ function is semiconcave, without any assumption on its second derivatives, as the next result shows.

## 统计代写|最优控制作业代写optimal control代考|Examples

A first interesting example of a semiconcave function is provided by the distance function. We recall that the distance function from a given nonempty closed set $C \subset$ $\mathbb{R}^{n}$ is defined by
$$d_{C}(x)=\min {y \in C}|y-x|, \quad\left(x \in \mathbb{R}^{n}\right)$$ As we show below, $d{C}$ is not semiconcave in the whole space $\mathbb{R}^{n}$, but is semiconcave on the complement of $C$, at least locally. On the other hand, the square of the distance function is semiconcave in $\mathbb{R}^{\pi}$. Before proving this result, let us introduce a property of sets which is useful for the analysis of the semiconcavity of $d_{C}$.

Definition 2.2.1 We say that a set $C \subset \mathbb{R}^{n}$ satisfies an interior sphere condition for some $r>0$ if $C$ is the union of closed spheres of radius $r$, i.e., for any $x \in C$ there exists y such that $x \in \overline{B_{r}(y)} \subset C$.

Proposition 2.2.2 Let $C \subset \mathbb{R}^{n}$ be a closed set, $C \neq \emptyset, \mathbb{R}^{n}$. Then the distance funcrion $d_{C}$ satisfies the following properties:
(i) $d_{C}^{2} \in \mathrm{SCL}\left(\mathbb{R}^{n}\right)$ with semiconcavity constant 2 .
(ii) $d_{C} \in \mathrm{SCL}{\text {loc }}\left(\mathbb{R}^{n} \backslash C\right.$ ). More precisely, given a set $S$ (not necessarily compact) such that dist $(S, C)>0, d{C}$ is semiconcave in $S$ with semiconcavity constant equal to $\operatorname{dist}(S, C)^{-1}$.
(iii) If C satisfies an interior sphere condition for some $r>0$, then $d c \in \mathrm{SCL}\left(\overline{\mathbb{R}^{n} \backslash C}\right)$ with semiconcavity constant equal to $r^{-1}$.
(iv) $d_{C}$ is not locally semiconcave in the whole space $\mathbb{R}^{n}$.
Proof – (i) For any $x \in \mathbb{R}^{n}$ we have
$$d_{C}^{2}(x)-|x|^{2}=\inf {y \in C}|x-y|^{2}-|x|^{2}=\inf {y \in C}|y|^{2}-2\langle x, y\rangle$$
Since the infimum of linear functions is concave we deduce, by Proposition 1.1.3, that property (i) holds.
(ii) Let us first observe that, given $z, h \in \mathbb{R}^{n}, z \neq 0$, we have
\begin{aligned} &(|z+h|+|z-h|)^{2} \ &\leq 2\left(|z+h|^{2}+|z-h|^{2}\right)=4\left(|z|^{2}+|h|^{2}\right) \leq\left(2|z|+\frac{|h|^{2}}{|z|}\right)^{2} \end{aligned}

## 统计代写|最优控制作业代写optimal control代考|Examples

dC(X)=分钟是的∈C|是的−X|,(X∈Rn)正如我们在下面展示的，dC整个空间都不是半凹的Rn, 但在的补码上是半凹的C，至少在本地。另一方面，距离函数的平方是半凹的R圆周率. 在证明这个结果之前，让我们介绍一个集合的性质，它有助于分析dC.

(i)dC2∈小号C大号(Rn)半凹常数为 2 。
(二)dC∈小号C大号地方 (Rn∖C)。更准确地说，给定一个集合小号（不一定紧凑）使得 dist(小号,C)>0,dC是半凹的小号半凹常数等于距离⁡(小号,C)−1.
(iii) 如果 C 满足某个内部球体条件r>0， 然后dC∈小号C大号(Rn∖C¯)半凹常数等于r−1.
(四)dC在整个空间中不是局部半凹的Rn.

$$d_{C}^{2}(x)-|x|^{2}=\inf {y \in C}|xy|^ {2}-|x|^{2}=\inf {y \in C}|y|^{2}-2\langle x, y\rangle 小号一世nC和吨H和一世nF一世米在米这Fl一世n和一种rF在nC吨一世这ns一世sC这nC一种在和在和d和d在C和,b是的磷r这p这s一世吨一世这n1.1.3,吨H一种吨pr这p和r吨是的(一世)H这lds.(一世一世)大号和吨在sF一世rs吨这bs和r在和吨H一种吨,G一世在和n和,H∈Rn,和≠0,在和H一种在和 (|和+H|+|和−H|)2 ≤2(|和+H|2+|和−H|2)=4(|和|2+|H|2)≤(2|和|+|H|2|和|)2$$

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