### 统计代写|最优控制作业代写optimal control代考|Application to the distance function

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## 统计代写|最优控制作业代写optimal control代考|Application to the distance function

In this section we examine some properties of the singular set of the distance function $d_{S}$ associated to a nonempty closed subset $S$ of $\mathbb{R}^{n}$. As in Section $3.4$, we denote by $\operatorname{proj}{S}(x)$ the set of closest points in $S$ to $x$, i.e., $$\operatorname{proj}{S}(x)=\left{y \in S: d_{S}(x)=|x-y|\right} \quad x \in \mathbb{R}^{n} .$$
Our first result characterizes the isolated singularities of $d_{S}$.
Theorem 4.4.1 Let $S$ be a nonempty closed subset of $\mathbb{R}^{n}$ and $x \notin S$ a singular point of $d_{5}$. Then the following properties are equivalent:
(a) $x$ is an isolated point of $\Sigma\left(d_{S}\right)$.
(b) $\partial D^{+} d_{S}(x)=D^{} d_{S}(x)$. (c) $\operatorname{proj}{S}(x)=\partial B{r}(x)$ where $r:=d_{S}(x)$.
Proof – The implication (a) $\Rightarrow$ (b) is an immediate corollary of the propagation result of Section 4.2. Indeed, if $\partial D^{+} d_{S}(x) \backslash D^{} d S(x)$ is nonempty, then Theorem 4.2.2 ensures the existence of a nonconstant singular arc with initial point $x$. In particular, $x$ could not be isolated.

Let us now show that (b) implies (c). First, we claim that, if (b) holds, then $x$ must be a singular point of magnitude $\kappa(x)=n$, i.e., $\operatorname{dim} D^{+} d s(x)=n$. For suppose the strict inequality $\kappa(x)} d_{S}(x)$. Therefore, $D^{+} d_{S}(x) \subset \partial B_{1}$ as all reachable gradients of $d_{S}$ are unit vectors. But the last inclusion contradicts the fact that $D^{+} d S(x)$ is a convex set of dimension at least 1. Our claim is thus proved. Now, we use the fact that $D^{+} d_{S}(x)$ is an $n$-dimensional convex set with $$\partial D^{+} d_{S}(x)=D^{} d_{S}(x) \subset \partial B_{1}$$ to conclude that $D^{+} d_{S}(x)=\bar{B}{1}$ and $D^{} d{S}(x)=\partial B_{1}$. Then, we invoke formula (3.40) to discover $$\operatorname{proj}{S}(x)=x-d{S}(x) D^{} d_{S}(x)=\partial B_{r}(x),$$
which proves (c).
Finally, let us show that (c) implies (a). From Corollary 3.4.5 (iii) we know that $d s$ is differentiable along each segment $] x, y\left[\right.$ with $y \in \operatorname{proj}{S}(x)=\partial B{r}(x)$. So, $d_{S} \in C^{1}\left(B_{r}(x) \backslash{x}\right)$ and the proof is complete.

In other words, the previous result shows that a point $x_{0}$ is an isolated singularity for the distance function from a set $S$ only if there exists an open sphere $B$ centered at $x_{0}$, such that $B \cap S=\emptyset$ and $\partial B \subset S$. In particular, if $S$ is a simply connected set in $\mathbb{R}^{2}$, or a set in $\mathbb{R}^{n}$ with trivial $n-1$ homotopy group, then the distance from $S$ has no isolated singularities in the complement of $S$.

## 统计代写|最优控制作业代写optimal control代考|Hamilton–Jacobi Equations

Hamilton-Jacobi equations are nonlinear first order equations which have been first introduced in classical mechanics, but find application in many other fields of mathematics. Our interest in these equations lies mainly in the connection with calculus of variations and optimal control. We have seen in Chapter 1 how the dynamic programming approach leads to the analysis of a Hamilton-Jacobi equation and other examples will be considered in the remainder of the book. However, our point of view in this chapter will be to study Hamilton-Jacobi equations for their intrinsic interest without referring to specific applications.

We begin by giving, in Section 5.1, a fairly general exposition of the method of characteristics. This method allows us to construct smooth solutions of first order equations, and in general can be applied only locally. However, this method is interesting also for the study of solutions that are not smooth. As we will see in the following, characteristic curves (or suitable generalizations) often play an important role for generalized solutions and are related to the optimal trajectories of the associated control problem.

In Section $5.2$ we recall the basic definitions and results from the theory of viscosity solutions for Hamilton-Jacobi equations. In this theory solutions are defined by means of inequalities satisfied by the generalized differentials or by test functions. With such a definition it is possible to obtain existence and uniqueness theorems under quite general hypotheses. In addition, in most cases where the equation is associated to a control problem, the viscosity solution coincides with the value function of the problem. Although this section is meant to be a collection of results whose proof can be found in specialized monographs, we have included the proofs of some simple statements in order to give to the reader the flavor of the techniques of the theory.

In Section $5.3$ we analyze the relation between semiconcavity and the viscosity property. Roughly speaking, it turns out that the two properties are equivalent when the hamiltonian is a convex function of the gradient of the solution. However, it is also possible to obtain semiconcavity results under different assumptions on the hamiltonian.

## 统计代写|最优控制作业代写optimal control代考|Method of characteristics

In Section $1.5$ we have introduced the method of characteristics to construct a local classical solution of the Cauchy problem for equations of the form $\partial_{t} u+H(\nabla u)=0$. We now show how this method can be extended to study general first order equations.
As a first step, let us show how the procedure of Section $1.5$ can be generalized to Cauchy problems where the hamiltonian depends also on $t, x$. Let us consider the problem
$$\begin{gathered} \partial_{t} u(t, x)+H(t, x, \nabla u(t, x))=0, \quad(t, x) \in\left[0, \infty\left[\times \mathbb{R}^{n}\right.\right. \ u(0, x)=u_{0}(x), \quad x \in \mathbb{R}^{n}, \end{gathered}$$
with $H$ and $u_{0}$ of class $C^{2}$.
Suppose, first, we have a solution $u \in C^{2}\left([0, T] \times \mathbb{R}^{n}\right)$ of the above problem. Given $z \in \mathbb{R}^{n}$, we call characteristic curve associated to $u$ starting from $z$ the curve $t \rightarrow(t, X(t ; z))$, where $X(* ; z)$ solves
$$\dot{X}=H_{p}(t, X, \nabla u(t, X)), \quad X(0)=z .$$
Here and in the following the dot denotes differentiation with respect to $t$. Now, if we set
$$U(t ; z)=u(t, X(t ; z)), \quad P(t ; z)=\nabla u(t, X(t ; z))$$
we find that
$$\begin{gathered} \dot{U}=u_{t}(t, X)+\nabla u(t, X) \cdot \dot{X}=-H(t, X, P)+P \cdot H_{p}(t, X, P) \ \dot{P}=\nabla u_{t}(t, X)+\nabla^{2} u(t, X) H_{p}(t, X, \nabla u(t, X)) \end{gathered}$$
Taking into account that
\begin{aligned} 0 &=\nabla\left(u_{t}(t, x)+H(t, x, \nabla u(t, x))\right) \ &=\nabla u_{t}(t, x)+H_{x}(t, x, \nabla u(t, x))+\nabla^{2} u(t, x) H_{p}(t, x, \nabla u(t, x)) \end{aligned}
we obtain that
$$\dot{P}=-H_{x}(t, X, \nabla u(t, X))=-H_{x}(t, X, P)$$

## 统计代写|最优控制作业代写optimal control代考|Application to the distance function

(a)X是一个孤立点Σ(d小号).
(二)∂D+d小号(X)=Dd小号(X). （C）项目⁡小号(X)=∂乙r(X)在哪里r:=d小号(X).

## 统计代写|最优控制作业代写optimal control代考|Hamilton–Jacobi Equations

Hamilton-Jacobi 方程是非线性一阶方程，首次引入经典力学，但在许多其他数学领域都有应用。我们对这些方程的兴趣主要在于与变分法和最优控制的联系。我们在第 1 章中已经看到动态规划方法是如何导致分析 Hamilton-Jacobi 方程的，本书的其余部分将考虑其他示例。然而，我们在本章中的观点将是研究 Hamilton-Jacobi 方程的内在兴趣，而不涉及具体的应用。

## 统计代写|最优控制作业代写optimal control代考|Method of characteristics

∂吨在(吨,X)+H(吨,X,∇在(吨,X))=0,(吨,X)∈[0,∞[×Rn 在(0,X)=在0(X),X∈Rn,

X˙=Hp(吨,X,∇在(吨,X)),X(0)=和.

0=∇(在吨(吨,X)+H(吨,X,∇在(吨,X))) =∇在吨(吨,X)+HX(吨,X,∇在(吨,X))+∇2在(吨,X)Hp(吨,X,∇在(吨,X))

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