### 统计代写|最优控制作业代写optimal control代考|Special properties of SCL

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## 统计代写|最优控制作业代写optimal control代考|Special properties of SCL

While many properties of semiconcave functions are valid in the case of an arbitrary modulus of semiconcavity, there are some results which are peculiar to the case of a linear modulus; we collect in this section some important ones, in addition to those already given in Proposition 1.1.3.

We have seen in Proposition 1.1.3 that semiconcave functions with a linear modulus can be regarded as $C^{2}$ perturbations of concave functions. This allows to extend immediately some well-known properties of concave functions, such as the following.

Theorem 2.3.1 Let $u \in \mathrm{SCL}(A)$, with $A \subset \mathbb{R}^{n}$ open. Then the following properties hold.
(i) (Alexandroff’s Theorem) $u$ is twice differentiable a.e, that is, for a.e. every $x_{0} \in A$, there exist a vector $p \in \mathbb{R}^{n}$ and a symmetric matrix $B$ such that
$$\lim {x \rightarrow x{0}} \frac{u(x)-u\left(x_{0}\right)-\left\langle p, x-x_{0}\right)+\left\langle B\left(x-x_{0}\right), x-x_{0}\right\rangle}{\left|x-x_{0}\right|^{2}}=0 .$$
(ii) The gradient of u, defined almost everywhere in A, belongs to the class $\mathrm{BV}_{\text {loc }}\left(A, \mathbb{R}^{n}\right)$.
Proof – Properties (i) and (ii) hold for a convex function (see e.g., $[72$, Ch. 6.3]). Since $u$ is the difference of a smooth function and a convex one, $u$ also satisfies these properties.

The following result shows that semiconcave functions with linear modulus exhibit a behavior similar to $C^{2}$ functions near a minimum point.

Theorem 2.3.2 Let $u \in \mathrm{SCL}(A)$, with $A \subset \mathbb{R}^{n}$ open, and let $x_{0} \in A$ be a point of local minimum for $u$. Then there exist a sequence $\left{x_{h}\right} \subset A$ and an infinitesimal sequence $\left{\varepsilon_{h}\right} \subset \mathbb{R}+$ such that $u$ is $t$ wice differentiable in $x_{h}$ and that
$$\lim {h \rightarrow \infty} x{h}=x_{0}, \quad \lim {h \rightarrow \infty} D u\left(x{h}\right)=0, \quad D^{2} u\left(x_{h}\right) \geq-\varepsilon_{h} I \quad \forall h .$$
The proof of this theorem is based on the following result.

## 统计代写|最优控制作业代写optimal control代考|A differential Harnack inequality

Let us consider the parabolic Hamilton-Jacobi equation
$$\partial_{f} u(t, x)+|\nabla u(t, x)|^{2}=\Delta u(t, x), \quad t \geq 0, x \in \mathbb{R}^{n} .$$
We have seen in Proposition 2.2.6 that the solutions to this equation are semiconcave. We now show how such a semiconcavity result is related to the classical Harnack inequality for the heat equation.

A remarkable feature of equation $(2.15)$ is that it can be reduced to the heat equation by a change of unknown called the Cole-Hopf transformation, or logarithmic transformation. In fact, if we set $w(t, x)=\exp (-u(t, x))$, a direct computation shows that $u$ satisfies $(2.15)$ if and only if $\partial_{t} w=\Delta w$. Let us investigate the properties of $w$ which follow from the semiconcavity of $u$.

Proposition 2.4.1 Let $w$ be a positive solution of the heat equation in $[0, T] \times \mathbb{R}^{n}$ whose first and second derivatives are bounded. Then w satisfies
$$\nabla^{2} w+\frac{w}{2 t} I-\frac{\nabla w \otimes \nabla w}{w} \geq 0$$
Here $\nabla^{2} w$ denotes the hessian matrix of $w$ with respect to the space variables; inequality (2.16) means that the matrix on the left-hand side is positive semidefinite.
Proof – It is not restrictive to assume that $w$ is greater than some positive constant; if this is not the case, we can replace $w$ by $w+\varepsilon$ and then let $\varepsilon \rightarrow 0^{+}$. Let us set $u(t, x)=-\ln (w(t, x))$. Then $u$ is a solution of equation (2.15). In addition, $u$ is bounded together with its first and second derivatives. Therefore, by Proposition $2.2 .6, u\left(t,{ }^{-}\right)$is semiconcave with modulus $\omega(\rho)=\rho /(4 t)$. Using the equivalent formulations of Proposition 1.1.3, we can restate this property as
$$\nabla^{2} u \leq \frac{1}{2 t} I$$
On the other hand, an easy computation shows that
$$\nabla^{2} u=-\frac{\nabla^{2} w}{w}+\frac{\nabla w \otimes \nabla w}{w^{2}}$$
and this proves (2.16). Taking the trace of the left-hand side of (2.16), we obtain
$$\Delta w+\frac{n w}{2 t}-\frac{|\nabla w|^{2}}{w} \geq 0$$
which implies $(2.17)$, since $w$ solves the heat equation.
Inequality (2.17) is called a differential Harnack estimate. The connection with the classical Harnack inequality is explained by the following result.

## 统计代写|最优控制作业代写optimal control代考|A generalized semiconcavity estimate

In this section we compare the semiconcavity estimate with another one-sided estimate, a priori weaker, which was introduced in [46]. We prove here that the two estimates are in some sense equivalent, and this has applications for the study of certain Hamilton-Jacobi equations, as we will see in the following (see Theorem $5.3 .7)$.

Let us consider a function $u: A \rightarrow \mathbb{R}$, with $A \subset \mathbb{R}^{n}$ open. Given $x 0 \in A$, we set, for $0<\delta<\operatorname{dist}\left(x_{0}, \partial A\right), x \in B_{1}$,
$$u_{x_{0}, \delta}(x)=\frac{u\left(x_{0}+\delta x\right)-u\left(x_{0}\right)}{\delta}$$

Definition 2.5.1 Let $C \subset A$ be a compact set. We say that u satisfies a generalized one-sided estimate in $C$ if there exist $\left.K \geq 0, \delta_{0} \in\right] 0$, $\operatorname{dist}(C, \partial A)[$ and a nondecreasing upper semicontinuous function $\tilde{\omega}:[0,1] \rightarrow \mathbb{R}{+}$, such that $\lim {h \rightarrow 0} \tilde{\omega}(h)=0$ and
\begin{aligned} &\lambda u_{x_{0}, \delta}(x)+(1-\lambda) u_{x_{0}, \delta}(y)-u_{x_{0}, \delta}(\lambda x+(1-\lambda) y) \ &\leq \lambda(1-\lambda)|x-y|{K \delta+\widetilde{\omega}(|x-y|)} \end{aligned}
for all $\left.x_{0} \in C, \delta \in\right] 0, \delta_{0}\left[, x, y \in B_{1}, \lambda \in[0,1]\right.$.
It is easily seen that, if $u$ is semiconcave in $A$, then the above property is satisfied taking $\tilde{\omega}$ equal to a modulus of semiconcavity of $u$ in $A$ and $K=0$. Conversely, semiconcavity can be deduced from the one-sided estimate above, as the next result shows.

Theorem 2.5.2 Let $u: A \rightarrow \mathbb{R}$, with A open and let $C$ be a compact subset of $A$. If u satisfies a generalized one-sided estimate in $C$, then $u$ is semiconcave in $C$.

Proof – By hypothesis inequality $(2.20)$ holds for some $K, \delta_{0}, \tilde{\omega}$. Let us take $x, y \in$ $C$ such that $[x, y] \subset C$ and $\lambda \in[0,1]$. It is not restrictive to assume $|x-y|<\delta_{0} / 2$. For any $\delta$ with $|x-y|<\delta<\delta_{0}$, we set
$$x_{0}=\lambda x+(1-\lambda) y, x^{\prime}=\delta^{-1}(1-\lambda)(x-y), y^{\prime}=\delta^{-1} \lambda(y-x) .$$
From $(2.19)$ and $(2.20)$ we obtain
\begin{aligned} &\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \ &=\delta\left{\lambda u_{x_{0}, \delta}\left(x^{\prime}\right)+(1-\lambda) u_{x_{0}, \delta}\left(y^{\prime}\right)-u_{x_{0}, \delta}\left(\lambda x^{\prime}+(1-\lambda) y^{\prime}\right)\right} \ &\leq \delta \lambda(1-\lambda)\left|x^{\prime}-y^{\prime}\right|\left{K \delta+\widetilde{\omega}\left(\left|x^{\prime}-y^{\prime}\right|\right)\right} \ &=\lambda(1-\lambda)|x-y|\left{K \delta+\widetilde{\omega}\left(\delta^{-1}|x-y|\right)\right} . \end{aligned}
Therefore
$$\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \leq \lambda(1-\lambda)|x-y| \omega(|x-y|)$$
where $\omega(\rho):=\inf {\delta \in\rfloor \rho, \delta{0}[}\left{K \delta+\tilde{\omega}\left(\delta^{-1} \rho\right)\right}$. The function $\omega$ is upper semicontinuous and nondecreasing. The conclusion will follow if we show that $\lim {h \rightarrow 0} \omega(h)=0$. Given $\varepsilon \in 10.2 K \delta$ o $[$. we choose $\eta \in] 0$. 1[ such that $\tilde{\omega}(s)<\varepsilon / 2$ for $0{0}[$; therefore
$$\omega(\rho) \leq\left{K \frac{\varepsilon}{2 K}+\tilde{\omega}\left(\frac{2 K}{\varepsilon} \rho\right)\right}<\varepsilon .$$
This shows that $\lim _{\rho \rightarrow 0} \omega(\rho)=0$ and concludes the proof.

## 统计代写|最优控制作业代写optimal control代考|Special properties of SCL

(i) (Alexandroff 定理)在是二次可微的ae，也就是说，对于ae，每X0∈一种, 存在一个向量p∈Rn和一个对称矩阵乙这样

(ii) u 的梯度，在 A 中几乎处处定义，属于类乙在地方 (一种,Rn).

## 统计代写|最优控制作业代写optimal control代考|A differential Harnack inequality

∂F在(吨,X)+|∇在(吨,X)|2=Δ在(吨,X),吨≥0,X∈Rn.

∇2在+在2吨一世−∇在⊗∇在在≥0

∇2在≤12吨一世

∇2在=−∇2在在+∇在⊗∇在在2

Δ在+n在2吨−|∇在|2在≥0

## 统计代写|最优控制作业代写optimal control代考|A generalized semiconcavity estimate

X0=λX+(1−λ)是的,X′=d−1(1−λ)(X−是的),是的′=d−1λ(是的−X).

\begin{对齐} &\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \ &=\delta\left{\lambda u_{x_ {0}, \delta}\left(x^{\prime}\right)+(1-\lambda) u_{x_{0}, \delta}\left(y^{\prime}\right)-u_ {x_{0}, \delta}\left(\lambda x^{\prime}+(1-\lambda) y^{\prime}\right)\right} \ &\leq \delta \lambda(1- \lambda)\left|x^{\prime}-y^{\prime}\right|\left{K \delta+\widetilde{\omega}\left(\left|x^{\prime}-y^{ \prime}\right|\right)\right} \ &=\lambda(1-\lambda)|xy|\left{K \delta+\widetilde{\omega}\left(\delta^{-1}|xy |\right)\right} 。\结束{对齐}\begin{对齐} &\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \ &=\delta\left{\lambda u_{x_ {0}, \delta}\left(x^{\prime}\right)+(1-\lambda) u_{x_{0}, \delta}\left(y^{\prime}\right)-u_ {x_{0}, \delta}\left(\lambda x^{\prime}+(1-\lambda) y^{\prime}\right)\right} \ &\leq \delta \lambda(1- \lambda)\left|x^{\prime}-y^{\prime}\right|\left{K \delta+\widetilde{\omega}\left(\left|x^{\prime}-y^{ \prime}\right|\right)\right} \ &=\lambda(1-\lambda)|xy|\left{K \delta+\widetilde{\omega}\left(\delta^{-1}|xy |\right)\right} 。\结束{对齐}

λ在(X)+(1−λ)在(是的)−在(λX+(1−λ)是的)≤λ(1−λ)|X−是的|ω(|X−是的|)

\omega(\rho) \leq\left{K \frac{\varepsilon}{2 K}+\波浪号{\omega}\left(\frac{2 K}{\varepsilon} \rho\right)\right} <\伐普西隆。\omega(\rho) \leq\left{K \frac{\varepsilon}{2 K}+\波浪号{\omega}\left(\frac{2 K}{\varepsilon} \rho\right)\right} <\伐普西隆。

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