### 统计代写|最优控制作业代写optimal control代考|Calculus of Variations

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## 统计代写|最优控制作业代写optimal control代考|Calculus of Variations

We begin now the analysis of some optimization problems where the results about semiconcave functions and their singularities can be applied. In this chapter we consider what Fleming and Rishel [80] call “the simplest problem in the calculus of variations,” a case where the dynamic programming approach is particularly powerful, and which will serve as a guideline for the analysis of optimal control problems in the following.

The problem in the calculus of variations we study here has been introduced in Chapter 1, where some results have been given in the case where the integrand is independent of $t, x$. Here, we consider a general integrand, so that the minimizers no longer admit an explicit description as in the Hopf formula. However, the structure of minimizers is described by classical results: we give a result about existence and regularity of minimizers, and we show that they satisfy the well-known Euler-Lagrange equations. We then apply the dynamic programming approach and introduce the value function of the problem, which is semiconcave and is a viscosity solution of an associated Hamilton-Jacobi equation. The main purpose of our analysis is to study the singularities of the value function and their interpretation in the calculus of variations. For instance, we derive a correspondence between generalized gradients of the value function and the minimizing trajectories of the problem. This shows, in particular, that the singularities of the value function are exactly the endpoints for which the minimizer of the variational problem is not unique. In addition, we can bound the size of $\bar{\Sigma}$, the closure of the singular set, proving that it enjoys the same rectifiability properties of $\Sigma$ itself. This result is interesting because on the complement of $\bar{\Sigma}$ the value function has the same regularity as the data and can be computed by the method of characteristics.

The chapter is organized as follows. In Section $6.1$ we give the statement of the problem and the existence result for minimizers. In Section $6.2$ we show that minimizers are regular and derive the Euler-Lagrange equations. Starting from Section $6.3$ we focus our attention on problems with one free endpoint; we introduce the notions of irregular and conjugate point and we prove Jacobi’s necessary optimality condition. Then, in Section 6.4, we apply the dynamic programming approach to this problem. We show that the value function $u$ solves the associated Hamilton-Jacobi

equation in the viscosity sense and that it is semiconcave. In addition, the minimizers for the variational problem with a given final endpoint $(t, x)$ are in one-to-one correspondence with the elements of $D^{*} u(t, x)$; in particular, the differentiability of $u$ is equivalent to the uniqueness of the minimizer. We also show that $u$ is as regular as the data of the problem in the complement of the closure of its singular set $\Sigma$.
The rest of the chapter is devoted to study the structure of $\bar{\Sigma}$. Since the properties of $\Sigma$ are well known from the general analysis of Chapter 4 , we focus our attention on the set $\bar{\Sigma} \backslash \Sigma$. The starting point is given in Section $6.5$, where we show that $\bar{\Sigma}=\Sigma \cup \Gamma$, where $\Gamma$ is the set of conjugate points. In addition, we prove some results about conjugate points showing, roughly speaking, that these are the points at which the singularities of $u$ are generated. Then, in Section $6.6$, we prove that $\Sigma$ has the same rectifiability property as $\Sigma$, i.e., it is a countably $\mathcal{H}^{\prime n}$-rectifiable subset of $\mathbb{R} \times \mathbb{R}^{n}$. By the previous remarks, this is equivalent to proving the rectifiability of $\Gamma \backslash \Sigma$. Combining a careful analysis of the hamiltonian system satisfied by the minimizing arcs with some tools from geometric measure theory, we obtain the finer estimate
$$\mathcal{H}^{n-1+\frac{2}{R-1}}(\Gamma \backslash \Sigma)=0,$$
where $k \geq 3$ is the differentiability class of the data. This yields in particular the desired Hausdorff estimate on $\bar{\Sigma}$, which shows that $u$ is as smooth as the data in the complement of a closed rectifiable set of codimension one.

## 统计代写|最优控制作业代写optimal control代考|Existence of minimizers

Let us consider the problem in the calculus of variations of Chapter 1 in a more general setting. We fix $T>0$, a connected open set $\Omega \subset \mathbb{R}^{n}$ and two closed subsets $S_{0}, S_{T} \subset \bar{\Omega}$. We denote by $\mathrm{AC}\left([0, T], \mathbb{R}^{n}\right)$ the class of all absolutely continuous arcs $\xi:[0, T] \rightarrow \mathbb{R}^{n}$ and define the set of admissible arcs by
$$\mathcal{A}=\left{\xi \in \mathrm{AC}\left([0, T], \mathbb{R}^{n}\right): \xi(t) \in \bar{\Omega} \text { for all } t \in[0, T], \xi(0) \in S_{0}, \xi(T) \in S_{T}\right}$$
Moreover, we define the functionals $\Lambda, J$ on $\mathcal{A}$ by
$$\Lambda(\xi)=\int_{0}^{T} L(s, \xi(s), \dot{\xi}(s)) d s$$
and
$$J(\xi)=\Lambda(\xi)+u_{0}(\xi(0))+u_{T}(\xi(T))$$
Here $L:[0, T] \times \bar{\Omega} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$ and $u_{0}, u_{T}: \bar{\Omega} \rightarrow \mathbb{R}$ are given continuous functions called running cost, initial cost and final cost, respectively, and $\Lambda$ is the action functional. We then consider the following minimization problem:
(CV) Find $\xi_{} \in \mathcal{A}$ such that $J\left(\xi_{}\right)=\min {J(\xi): \xi \in \mathcal{A}}$.

## 统计代写|最优控制作业代写optimal control代考|Necessary conditions and regularity

We show in this section that the minimizing arcs for problem (CV) are regular and solve a system of equations called the Euler-Lagrange equations. Our analysis is restricted to those minimizers which are contained in $\Omega$; minimizers touching the boundary of $\Omega$ would require a longer analysis. We need some further assumptions on the data. Namely, we assume that the lagrangian $L$ is of class $C^{1}$ and that for all $r>0$ there exists $\mathcal{C}(r)>0$ such that
\begin{aligned} &\left|L_{x}(t, x, v)\right|+\left|L_{v}(t, x, v)\right| \leq \tilde{C}(r) \theta(|v|) \ &\forall t \in[0, T], x \in \bar{\Omega} \cap B_{r}, v \in \mathbb{R}^{n} \end{aligned}
where $\theta$ is the Nagumo function appearing in hypothesis (ii) of Theorem 6.1.2. Observe that property (iii) of the same theorem is implied by (6.6). In addition, we assume that $\theta$ satisfies
$$\theta(q+m) \leq K_{M}[1+\theta(q)] \quad \forall m \in[0, M], q \geq 0$$
It is easily checked that assumption (6.7) is satisfied for many classes of superlinear functions $\theta$, such as powers or exponentials. It is violated in cases where $\theta$ grows “very fast”, e.g., $\theta(q)=e^{e q}, q \in \mathbb{R}$.

## 统计代写|最优控制作业代写optimal control代考|Calculus of Variations

Hn−1+2R−1(Γ∖Σ)=0,

## 统计代写|最优控制作业代写optimal control代考|Existence of minimizers

\mathcal{A}=\left{\xi \in \mathrm{AC}\left([0, T], \mathbb{R}^{n}\right): \xi(t) \in \bar{ \Omega} \text { 对于所有 } t \in[0, T], \xi(0) \in S_{0}, \xi(T) \in S_{T}\right}\mathcal{A}=\left{\xi \in \mathrm{AC}\left([0, T], \mathbb{R}^{n}\right): \xi(t) \in \bar{ \Omega} \text { 对于所有 } t \in[0, T], \xi(0) \in S_{0}, \xi(T) \in S_{T}\right}

Λ(X)=∫0吨大号(s,X(s),X˙(s))ds

Ĵ(X)=Λ(X)+在0(X(0))+在吨(X(吨))

(CV) FindX∈一种这样Ĵ(X)=分钟Ĵ(X):X∈一种.

## 统计代写|最优控制作业代写optimal control代考|Necessary conditions and regularity

|大号X(吨,X,在)|+|大号在(吨,X,在)|≤C~(r)θ(|在|) ∀吨∈[0,吨],X∈Ω¯∩乙r,在∈Rn

θ(q+米)≤ķ米[1+θ(q)]∀米∈[0,米],q≥0

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