### 统计代写|最优控制作业代写optimal control代考|Superdifferential of a semiconcave function

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## 统计代写|最优控制作业代写optimal control代考|Superdifferential of a semiconcave function

The superdifferential of a semiconcave function enjoys many properties that are not valid for a general Lipschitz continuous function, and that can be regarded as extensions of analogous properties of concave functions. We start with the following basic estimate. Throughout the section $A \subset \mathbb{R}^{n}$ is an open set.

Proposition 3.3.1 Let $u: A \rightarrow \mathbb{R}$ be a semiconcave function with modulus $\omega$ and let $x \in A$. Then, a vector $p \in \mathbb{R}^{n}$ belongs to $D^{+} u(x)$ if and only if
$$u(y)-u(x)-\langle p, y-x\rangle \leq|y-x| \omega(|y-x|)$$
Jor any pont y EA such that $[y, r\rfloor$ s. $_{-}$

Proof – If $p \in \mathbb{R}^{n}$ satisfies (3.18), then, by the very definition of superdifferential, $p \in D^{+} u(x)$. In order to prove the converse, let $p \in D^{+} u(x)$. Then, dividing the semiconcavity inequality $(2.1)$ by $(1-\lambda)|x-y|$, we have
$$\left.\left.\frac{u(y)-u(x)}{|y-x|} \leq \frac{u(x+(1-\lambda)(y-x))-u(x)}{(1-\lambda)|y-x|}+\lambda \omega(|x-y|), \quad \forall \lambda \in\right] 0,1\right] .$$
Hence, taking the limit as $\lambda \rightarrow 1^{-}$, we obtain
$$\frac{u(y)-u(x)}{|y-x|} \leq \frac{\langle p, y-x\rangle}{|y-x|}+\omega(|x-y|),$$
since $p \in D^{+} u(x)$. Estimate (3.18) follows.
Remark 3.3.2 In particular, if $u$ is concave on a convex set $A$. we find that $p \in$ $D^{+} u(x)$ if and only if
$$u(y) \geq u(x)+\langle p, y-x\rangle, \quad \forall y \in A .$$
In convex analysis (see Appendix A. 1) this property is usually taken as the definition of the superdifferential. Therefore, the Fréchet super- and subdifferential coincide with the classical semidifferentials of convex analysis in the case of a concave (resp. convex) function.

Before investigating further properties of the superdifferential, let us show how Proposition 3.3.1 easily yields a compactness property for semiconcave functions.

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

A function $u: A \rightarrow \mathbb{R}$ is called a marginal function if it can be written in the form
$$u(x)=\inf _{s \in S} F(s, x),$$
where $S$ is some topological space and the function $F: S \times A \rightarrow \mathbb{R}$ depends smoothly on $x$. Functions of this kind appear often in the literature, sometimes with different names (see e.g., the lower $C^{k}$-functions in [123]).

Under suitable regularity assumptions for $F$, a marginal function is semiconcave.
For instance, Corollary $2.1 .6$ immediately implies the following.
Proposition 3.4.1 Let $A \subset \mathbb{R}^{n}$ be open and let $S \subset \mathbb{R}^{m}$ be compact. If $F=F(s, x)$ is continuous in $C(S \times A)$ together with its partial derivatives $D_{x} F$, then the function u defined in (3.34) belongs to $\mathrm{SC}{l o c}(A)$. If $D{x x}^{2} F$ also exists and is continuous in $S \times A$, then $u \in \mathrm{SCL}{l o c}(A)$. We now show that the converse also holds. Theorem 3.4.2 Let $u: A \rightarrow \mathbb{R}$ be a semiconcave function. Then $u$ can be locally written as the minimum of functions of class $C^{1}$. More precisely, for any $K \subset A$ compact, there exists a compact set $S \subset \mathbb{R}^{2 n}$ and a continuous function $F: S \times K \rightarrow$ $\mathbb{R}$ such that $F(s, \cdot)$ is $C^{1}$ for any $s \in S$, the gradients $D{x} F(s, \cdot)$ are equicontinuous, and
$$u(x)=\min _{s \in S} F(s, x), \quad \forall x \in K .$$
If the modulus of semiconcavity of $u$ is linear, then $F$ can be chosen such that $F(s,-)$ is $C^{2}$ for any $s$, with uniformly bounded $C^{2}$ norm.

Proof – Let $\omega$ be the modulus of semiconcavity of $u$ and let $\omega_{1}$ be a function such that $\omega_{1}(0)=0$, that $\omega_{1}(r) \geq \omega(r)$ and that the function $x \rightarrow|x| \omega_{1}(|x|)$ belongs to $C^{1}\left(\mathbb{R}^{n}\right)$. The existence of such an $\omega_{1}$ has been proved in Lemma 3.1.8. If $\omega$ is linear we simply take $\omega_{1} \equiv \omega$.

Let us set $S=\left{(y, p): y \in K, p \in D^{+} u(y)\right}$. By Proposition 3.3.4(a) and the local Lipschitz continuity of $u, S$ is a compact set. Then we define
$$F(y, p, x)=u(y)+\langle p, x-y\rangle+|y-x| \omega_{1}(|y-x|)$$
Then $F$ has the required regularity properties. In addition $F(y, p, x) \geq u(x)$ for all $(y, p, x) \in S \times K$ by Proposition 3.3.1. On the other hand, if $x \in K$, then $D+u(x)$ is nonempty and so thêré exists at lesast a vectō $p$ such that $(x, p) \in S$. Since $F(x, p, x)=u(x)$, we obtain $(3.35)$.

If $u$ is semiconcave with a linear modulus, then it admits another representation as the infimum of regular functions by a procedure that is very similar to the Legendre transformation.

## 统计代写|最优控制作业代写optimal control代考|Inf-convolutions

Given $g: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and $\varepsilon>0$, the functions
$$x \rightarrow \inf {y \in \mathbb{R}^{n}}\left(g(y)+\frac{|x-y|^{2}}{2 \varepsilon}\right) \quad x \rightarrow \sup {y \in \mathbb{R}^{n}}\left(g(y)-\frac{|x-y|^{2}}{2 \varepsilon}\right)$$
are called inf- and sup-convolutions of $g$ respectively, due to the formal analogy with the usual convolution. They have been used in various contexts as a way to approximate $g$; one example is the uniqueness theory for viscosity solutions of HamiltonJacobi equations. In some cases it is useful to consider more general expressions, where the quadratic term above is replaced by some other coercive function. In this section we analyze such general convolutions, showing that their regularity properties are strictly related with the properties of semiconcave functions studied in the previous sections.
Definition 3.5.1 Let $g \in C\left(\mathbb{R}^{n}\right)$ satisfy
$$|g(x)| \leq K(1+|x|)$$
for some $K>0$ and let $\phi \in C\left(\mathbb{R}^{n}\right)$ be such that

$$\lim {|q| \rightarrow+\infty} \frac{\phi(q)}{|q|}=+\infty .$$ The inf-convolution of $g$ with kernel $\phi$ is the function $$g \phi(x)=\inf {y \in \mathbb{R}^{a}}[g(y)+\phi(x-y)],$$
while the sup-convolution of $g$ with kernel $\phi$ is defined by
$$g^{\phi}(x)=\sup {y \in \mathbb{R}^{n}}[g(y)-\phi(x-y)] .$$ We observe that the function $u$ given by Hopf’s formula (1.10) is an infconvolution with respect to the $x$ variable for any fixed $t$. In addition, inf-convolutions are a particular case of the marginal functions introduced in the previous section. We give below some regularity properties of the inf-convolutions. The corresponding statements about the sup-convolutions are easily obtained observing that $g^{\phi}=-\left((-g){\phi}\right)$.

## 统计代写|最优控制作业代写optimal control代考|Superdifferential of a semiconcave function

Jor 任何 pont y EA 使得[是的,r⌋s。−

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

F(是的,p,X)=在(是的)+⟨p,X−是的⟩+|是的−X|ω1(|是的−X|)

## 统计代写|最优控制作业代写optimal control代考|Inf-convolutions

X→信息是的∈Rn(G(是的)+|X−是的|22e)X→支持是的∈Rn(G(是的)−|X−是的|22e)

|G(X)|≤ķ(1+|X|)

Gφ(X)=支持是的∈Rn[G(是的)−φ(X−是的)].我们观察到函数在Hopf 的公式 (1.10) 给出的是关于X任何固定的变量吨. 此外，inf-convolutions 是上一节介绍的边缘函数的一个特例。我们在下面给出了 inf 卷积的一些规律性属性。观察到关于上卷积的相应陈述很容易获得Gφ=−((−G)φ).

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