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

statistics-lab™ 为您的留学生涯保驾护航 在代写最优控制optimal control方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写最优控制optimal control代写方面经验极为丰富，各种代写最优控制Soptimal control相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

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

Let $\Omega \subset \mathbb{R}^{n}$ be an open set and let $H \in C\left(\Omega \times \mathbb{R} \times \mathbb{R}^{n}\right)$. Let us again consider the general nonlinear first order equation
$$H(x, u, D u)=0, \quad x \in \Omega \subset \mathbb{R}^{n},$$
in the unknown $u: \Omega \rightarrow \mathbb{R}$. As usual, evolution equations can be recast in this form by considering time as an additional space variable.

As we have already mentioned, when one considers boundary value problems or Cauchy problems for equations of the above form, one finds that in general no global smooth solutions exist even if the data are smooth. On the other hand, the property of being a Lipschitz continuous function satisfying the equation almost everywhere is usually too weak to have uniqueness results. Therefore, a crucial step in the analysis is to give a notion of a generalized solution such that global existence and uniqueness results can be obtained. In Chapter 1 we have seen a class of problems which are well posed in the class of semiconcave solutions. Here we present the notion of a viscosity solution, which has a much wider range of applicability.

Definition 5.2.1 A function $u \in C(\Omega)$ is called a viscosity subsolution of equation (5.14) if, for any $x \in \Omega$, it satisfies
$$H(x, u(x), p) \leq 0, \quad \forall p \in D^{+} u(x) .$$
Similarly, we say that $u$ is a viscosity supersolution of equation (5.14) if, for any $x \in \Omega$, we have
$$H(x, u(x), p) \geq 0, \quad \forall p \in D^{-} u(x)$$
If u satisfies both of the above properties, it is called a viscosity solution of equation (5.14).

Observe that, by virtue of Proposition 3.1.7, condition (5.15) (resp. (5.16)) can be restated in an equivalent way by requiring
$$H(x, u(x), D \phi(x)) \leq 0 \quad \text { (resp. } H(x, u(x), D \phi(x)) \geq 0 \text { ) }$$
for any $\phi \in C^{1}(\Omega)$ such that $u-\phi$ has a local maximum (resp. minimum) at $x$.
We see that if $u$ is differentiable everywhere the notion of a viscosity solution coincides with the classical one since we have at any point $D^{+} u(x)=D^{-} u(x)=$ ${D u(x)}$. On the other hand, if $u$ is not differentiable everywhere, the definition of a viscosity solution includes additional requirements at the points of nondifferentiability. The reason for taking inequalities (5.15)-(5.16) as the definition of solution might not be clear at first sight, as well as the relation with the semiconcavity property considered in Chapter 1 . However, we will see that with this definition one can obtain existence and uniqueness results for many classes of Hamilton-Jacobi equations, and that the viscosity solution usually coincides with the one which is relevant for the applications, like the value function in optimal control. The relationship with semiconcavity will be examined in detail in the next section.

## 统计代写|最优控制作业代写optimal control代考|Semiconcavity and viscosity

We now analyze the relation between the notions of a semiconcave solution and a viscosity solution to Hamilton-Jacobi equations. We will see that the two notions are strictly related when the hamiltonian is a convex function of $D u$. We begin with the following result.

Proposition $5.3 .1$ Let u be a semiconcave function satisfying equation (5.14) almost everywhere. If $H$ is convex in the third argument, then $u$ is a viscosity solution of the equation.

Proof – As a first step. we show that $u$ satisfies the equation at all points of differentiability (our assumption is a priori slightly weaker). Let $u$ be differentiable at some $x_{0} \in \Omega$. Then there exists a sequence of points $x_{k}$ converging to $x_{0}$ such that
(i) $u$ is differentiable at $x_{k}$;
(ii) $H\left(x_{k}, u\left(x_{k}\right), D u\left(x_{k}\right)\right)=0$;
(iii) $D u\left(x_{k}\right)$ has a limit for $k \rightarrow \infty$.
From Proposition 3.3.4(a) we deduce that the limit of $D u\left(x_{k}\right)$ is $D u\left(x_{0}\right)$. By the continuity of $H$, the equation holds at $x_{0}$ as well.

Let us now take an arbitrary $x \in \Omega$ and check that (5.15) is satisfied. We first observe that
$$H(x, u(x), p)=0, \quad \forall p \in D^{} u(x)$$ This follows directly from the definition of $D^{} u$, the continuity of $H$ and the property that the equation holds at the points of differentiability. Since $D^{+} u(x)$ is the convex hull of $D^{*} u(x)$ (see Theorem $3.3 .6$ ) and $H$ is convex, inequality (5.15) follows.
Now let us check inequality (5.16). For a given $x \in \Omega$, suppose that $D^{-} u(x)$ contains some vector $p$. Then, by Proposition 3.1.5(c) and Proposition 3.3.4(c), $u$ is differentiable at $x$ and $D u(x)=p$. Thus, $(5.16)$ holds as an equality by the first part of the proof.

Remark 5.3.2 A more careful analysis shows that the convexity of $H$ and the semiconcavity of $u$ play independent roles in the viscosity property. In fact, in the previous proof, the deduction that $u$ is a supersolution uses only semiconcavity and is valid also if $H$ is not convex. On the other hand, it is possible to prove (see see [110, p. 96] or [20, Prop. II.5.1]) that, if $H$ is convex, any Lipschitz continuous $u$ (not necessarily semiconcave) satisfying the equation almost everywhere is a viscosity subsolution of the equation.

## 统计代写|最优控制作业代写optimal control代考|Propagation of singularities

We now turn to the analysis of the singular set of semiconcave solutions to HamiltonJacobi equations. In Chapter 4 we have obtained results which are valid for any semiconcave function; here we focus our attention on semiconcave functions which are solutions to Hamilton-Jacobi equations and we obtain stronger results on the propagation of singularities in this case. As in the previous chapter, our discussion is restricted to semiconcave functions with linear modulus (some results in the case of a general modulus can be found in [1]).
We consider again an equation of the form
$$F(x, u(x), D u(x))=0 \quad \text { a.e. in } \quad \Omega$$
Throughout the rest of this chapter, $F: \bar{\Omega} \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ is a continuous function satisfying the following assumptions:
(Al) $p \mapsto F(x, u, p)$ is convex;
(A2) for any $(x, u) \in \Omega \times \mathbb{R}$ and any $p_{0}, p_{1} \in \mathbb{R}^{n}$,
$$\left\lfloor p_{0}, p_{1}\right] \subset\left{p \in \mathbb{R}^{n}: F(x, u, p)=0\right} \quad \Longrightarrow \quad p_{0}=p_{1} .$$
Condition (A2) requires that the 0 -level set $\left{p \in \mathbb{R}^{n}: F(x, u, p)=0\right}$ contains no straight line. Clearly, such a property holds, in particular, if $F$ is strictly convex with respect to $p$. Observe, however, that it also holds for functions like $F(x, u, p)=|p|$ which are not strictly convex.

Remark 5.4.1 In Proposition 5.3.1 we have seen that, under the convexity assumption (A1), any semiconcave function $u$ which solves equation (5.44) almost everywhere is also a viscosity solution of the equation. Thus, $u$ satisfies for all $x \in \Omega$
$$\begin{array}{ll} F(x, u(x), p)=0 & \forall p \in D^{*} u(x) \ F(x, u(x), p) \leq 0 & \forall p \in D^{+} u(x) \end{array}$$
The first result we prove is that condition (4.8) is necessary and sufficient for the propagation of a singularity at $x_{0}$, if $u$ is a semiconcave solution of (5.44). Notice that, for a general semiconcave function, (4.8) is only a sufficient condition.

Theorem 5.4.2 Let (A1) and (A2) be satisfied. Let $u \in \mathrm{SCL}{l o c}(\Omega)$ be a solution of (5.44) and let $x{0} \in \Sigma(u)$. Then, the following properties are equivalent:
(i) $\partial D^{+} u\left(x_{0}\right) \backslash D^{*} u\left(x_{0}\right) \neq \emptyset$;
(ii) there exist a sequence $x_{i} \in \Sigma(u) \backslash\left{x_{0}\right}$, converging to $x_{0}$, and a number $\delta>0$ such that diam $D^{+} u\left(x_{i}\right) \geq \delta$, for any $i \in \mathbb{N}$.

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

H(X,在,D在)=0,X∈Ω⊂Rn,

H(X,在(X),p)≤0,∀p∈D+在(X).

H(X,在(X),p)≥0,∀p∈D−在(X)

H(X,在(X),Dφ(X))≤0 （分别。 H(X,在(X),Dφ(X))≥0 )

## 统计代写|最优控制作业代写optimal control代考|Semiconcavity and viscosity

(i)在可微分于Xķ;
(二)H(Xķ,在(Xķ),D在(Xķ))=0;
㈢D在(Xķ)有一个限制ķ→∞.

H(X,在(X),p)=0,∀p∈D在(X)这直接来自于D在, 的连续性H以及等式在可微点处所具有的性质。自从D+在(X)是凸包D∗在(X)（见定理3.3.6） 和H是凸的，不等式（5.15）如下。

## 统计代写|最优控制作业代写optimal control代考|Propagation of singularities

F(X,在(X),D在(X))=0 一个和在 Ω

(Al)p↦F(X,在,p)是凸的；
(A2) 对于任何(X,在)∈Ω×R和任何p0,p1∈Rn,
\left\lfloor p_{0}, p_{1}\right] \subset\left{p \in \mathbb{R}^{n}: F(x, u, p)=0\right} \quad \ Longrightarrow \quad p_{0}=p_{1} 。\left\lfloor p_{0}, p_{1}\right] \subset\left{p \in \mathbb{R}^{n}: F(x, u, p)=0\right} \quad \ Longrightarrow \quad p_{0}=p_{1} 。

F(X,在(X),p)=0∀p∈D∗在(X) F(X,在(X),p)≤0∀p∈D+在(X)

(i)∂D+在(X0)∖D∗在(X0)≠∅;
(ii) 存在一个序列x_{i} \in \Sigma(u) \backslash\left{x_{0}\right}x_{i} \in \Sigma(u) \backslash\left{x_{0}\right}, 收敛到X0, 和一个数字d>0这样直径D+在(X一世)≥d, 对于任何一世∈ñ.

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。