### 统计代写|最优控制作业代写optimal control代考|A Model Problem

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

Before starting the analysis of our variational problem, let us introduce semiconcave functions, which are the central topic in this monograph and will play an important role later in this chapter. It is convenient to consider, first, a special class of semiconcave functions, while the general definition will be given in Chapter 2 .

Here and in what follows we write $[x, y]$ to denote the segment with endpoints $x, y$, for any $x, y \in \mathbb{R}^{n}$. Moreover, we denote by $x \cdot y$, or by $\langle x, y\rangle$, the Euclidean scalar product, and by $|x|$ the usual norm in $\mathbb{R}^{n}$. Furthermore, $B_{r}(x)$-and, at times, $B(x, r)$-stands for the open ball centered at $x$ with radius $r$. We will also use the abbreviated notation $B_{r}$ for $B_{r}(0)$.

Definition 1.1.1 Let $A \subset \mathbb{R}^{n}$ be an open set. We say that a function $u: A \rightarrow \mathbb{R}$ is semiconcave with linear modulus if it is continuous in $A$ and there exists $C \geq 0$ such that
$$u(x+h)+u(x-h)-2 u(x) \leq C|h|^{2}$$
for all $x, h \in \mathbb{R}^{n}$ such that $[x-h, x+h] \subset A$. The constant $C$ above is called $a$ semiconcavity constant for $u$ in $S$.

Remark 1.1.2 The above definition is often taken in the literature as the definition of a semiconcave function. For us, instead, it is a particular case of Definition 2.1.1, where the right-hand side of $(1.1)$ is replaced by a term of the form $|h| \omega(|h|)$ for some function $\omega(*)$ such that $\omega(\rho) \rightarrow 0$ as $\rho \rightarrow 0$. The function $\omega$ is called modulus of semiconcavity, and therefore we say that a function which satisfies (1.1) is semiconcave with a linear modulus.

Semiconcave functions with a linear modulus admit some interesting characterizations, as the next result shows.

Proposition 1.1.3 Given $u: A \rightarrow \mathbb{R}$, with $A \subset \mathbb{R}^{n}$ open convex, and given $C \geq 0$, the following properties are equivalent:
(a) $u$ is semiconcave with a linear modulus in A with semiconcavity constant $C$;
(b) u satisfies
$$\lambda u(x)+(1-\lambda) u(y)-u(\lambda x+(1-\lambda) y) \leq C \frac{\lambda(1-\lambda)}{2}|x-y|^{2},$$
for all $x, y$ such that $[x, y] \subset A$ and for all $\lambda \in[0,1]$;

(c) the function $x \rightarrow u(x)-\frac{C}{2}|x|^{2}$ is concave in $A$;
(d) there exist two functions $u_{1}, u_{2}: A \rightarrow \mathbb{R}$ such that $u=u_{1}+u_{2}, u_{1}$ is concave, $u_{2} \in C^{2}(A)$ and satisfies $\left|D^{2} u_{2}\right|_{\infty} \leq C$;
(e) for any $v \in \mathbb{R}^{n}$ such that $|v|=1$ we have $\frac{\partial^{2} u}{\partial v^{2}} \leq C$ in $A$ in the sense of distributions, that is
$$\int_{A} u(x) \frac{\partial^{2} \phi}{\partial v^{2}}(x) d x \leq C \int_{A} \phi(x) d x, \quad \forall \phi \in C_{0}^{\infty}(A), \phi \geq 0$$
(f) $u$ can be represented as $u(x)=\inf {i \in \mathcal{I}} u{i}(x)$, where $\left{u_{i}\right}_{i \in \mathcal{I}}$ is a family of functions of $C^{2}(A)$ such that $\left|D^{2} u_{i}\right|_{\infty} \leq C$ for all $i \in \mathcal{I}$.

## 统计代写|最优控制作业代写optimal control代考|A problem in the calculus of variations

We now start the analysis of our model problem. Given $0<T \leq+\infty$, we set $\left.Q_{T}=\right] 0, T\left[\times \mathbb{R}^{n}\right.$. We suppose that two continuous functions
$$L: \bar{Q}{T} \times \mathbb{R}^{n} \rightarrow \mathbb{R}, \quad u{0}: \mathbb{R}^{n} \rightarrow \mathbb{R}$$
are given. The function $L$ will be called the running cost, or lagrangian, while $u_{0}$ is called the initial cost. We assume that both functions are bounded from below.
For fixed $(t, x) \in \bar{Q}{T}$, we introduce the set of admissible arcs $$\mathcal{A}(t, x)=\left{y \in W^{1,1}\left([0, t] ; \mathbb{R}^{n}\right): y(t)=x\right}$$ and the cost functional $$J{t}[y]=\int_{0}^{t} L(s, y(s), \dot{y}(s)) d s+u_{0}(y(0)) .$$
Then we consider the following problem:
$$\text { minimize } J_{t}[y] \text { over all arcs } y \in \mathcal{A}(t, x) \text {. }$$
Problems of this form are classical in the calculus of variations. In the case we are considering the initial endpoint of the admissible trajectories is free, and the terminal

one is fixed. Cases where the endpoints are both fixed or both free are also interesting and could be studied by similar techniques, but will not be considered here.

The first step in the dynamic programming approach to the above problem is the introduction of the value function.

## 统计代写|最优控制作业代写optimal control代考|The Hopf formula

From now on we consider the special case of $L(t, x, q)=L(q)$ and $T=+\infty$. We assume that
$\left{\begin{array}{l}\text { (i) } L \text { is convex and } \lim {|q| \rightarrow \infty} \frac{L(q)}{|q|}=+\infty \ \text { (ii) } u{0} \in \operatorname{Lip}\left(\mathbb{R}^{n}\right) .\end{array}\right.$
Then we can show that the value function of our problem admits a simple representation formula called Hopf’s formula.
Theorem 1.3.1 Under hypotheses (1.9) the value function u satisfies
$$u(t, x)=\min {z \in \mathbb{R}^{x}}\left[t L\left(\frac{x-z}{t}\right)+u{0}(z)\right]$$
for all $(t, x) \in Q_{T}$.

Proof – Observe that the minimum in (1.10) exists thanks to hypotheses (1.9). Let us denote by $v(t, x)$ the left-hand side of $(1.10)$.
For fixed $(t, x) \in Q_{T}$ and $z \in \mathbb{R}^{n}$, let us set
$$y(s)=z+\frac{s}{t}(x-z), \quad 0 \leq s \leq t .$$
Then $y \in \mathcal{A}(t, x)$ and therefore
$$u(t, x) \leq J_{t}[y]=t L\left(\frac{x-z}{t}\right)+u_{0}(z) .$$
Taking the infimum over $z$ we obtain that $u(t, x) \leq v(t, x)$.
To prove the opposite inequality, let us take $\zeta \in \mathcal{A}(t, x)$. From Jensen’s inequality it follows that
$$L\left(\frac{x-\zeta(0)}{t}\right)=L\left(\frac{1}{t} \int_{0}^{t} \dot{\zeta}(s) d s\right) \leq \frac{1}{t} \int_{0}^{t} L(\zeta(s)) d s$$
and therefore
$$v(t, x) \leq u_{0}(\zeta(0))+t L\left(\frac{x-\zeta(0)}{t}\right) \leq J_{t}[\zeta] .$$
Taking the infimum over $\zeta \in \mathcal{A}(t, x)$ we conclude that $v(t, x) \leq u(t, x)$.
Using Hopf’s formula we can prove a first regularity property of $u$.

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

(a)在是半凹的，在 A 中具有线性模量，具有半凹常数C;
(b) 你满足
λ在(X)+(1−λ)在(是的)−在(λX+(1−λ)是的)≤Cλ(1−λ)2|X−是的|2,

(c) 职能X→在(X)−C2|X|2是凹进去的一种;
(d) 存在两个功能在1,在2:一种→R这样在=在1+在2,在1是凹的，在2∈C2(一种)并满足|D2在2|∞≤C;
(e) 对于任何在∈Rn这样|在|=1我们有∂2在∂在2≤C在一种在分布的意义上，即
∫一种在(X)∂2φ∂在2(X)dX≤C∫一种φ(X)dX,∀φ∈C0∞(一种),φ≥0
（F）在可以表示为在(X)=信息一世∈一世在一世(X)， 在哪里\left{u_{i}\right}_{i \in \mathcal{I}}\left{u_{i}\right}_{i \in \mathcal{I}}是一个函数族C2(一种)这样|D2在一世|∞≤C对全部一世∈一世.

## 统计代写|最优控制作业代写optimal control代考|A problem in the calculus of variations

最小化 Ĵ吨[是的] 在所有弧上 是的∈一种(吨,X).

## 统计代写|最优控制作业代写optimal control代考|The Hopf formula

$\left{ （一世） 大号 是凸的并且 林|q|→∞大号(q)|q|=+∞ (二) 在0∈唇⁡(Rn).\对。吨H和n在和C一种nsH这在吨H一种吨吨H和在一种l在和F在nC吨一世这n这F这在rpr这bl和米一种d米一世吨s一种s一世米pl和r和pr和s和n吨一种吨一世这nF这r米在l一种C一种ll和dH这pF′sF这r米在l一种.吨H和这r和米1.3.1在nd和rH是的p这吨H和s和s(1.9)吨H和在一种l在和F在nC吨一世这n在s一种吨一世sF一世和s在(吨,X)=分钟和∈RX[吨大号(X−和吨)+在0(和)]F这r一种ll(t, x) \in Q_{T}$。

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## MATLAB代写

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