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

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

In this section we introduce a partial differential equation which is solved by the value function of our variational problem. We assume throughout that hypotheses (1.9) are satisfied. We use the notation
$$u_{t}=\frac{\partial u}{\partial t}, \quad \nabla u=\left(\frac{\partial u}{\partial x_{1}}, \ldots, \frac{\partial u}{\partial x_{n}}\right)$$
Theorem 1.4.1 Let $u$ be differentiable at a point $(t, x) \in Q_{T}$. Then
$$u_{t}(t, x)+H(\nabla u(t, x))=0$$
where
$$H(p)=\sup _{q \in \mathbb{R}^{n}}[p \cdot q-L(q)] .$$
Equation (1.14) is called the Hamilton-Jacobi equation of our problem in the calculus of variations. In the terminology of control theory, such an equation is also called Bellman’s equation or dynamic programming equation. The function $H$ is called the hamiltonian. In general, a function defined as in (1.15) is called the Legendre transform of $L$ (see Appendix A.1).

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

We describe in this section the method of characteristics, which is a classical approach to the study of first order partial differential equations like the HamiltonJacobi equation (1.16). This method explains why such equations do not possess in general smooth solutions for all times, and has some interesting connections with the variational problem associated to the equation. A more general treatment of these topics will be given in Section 5.1.

Suppose that $H, u_{0}$ are in $C^{2}\left(\mathbb{R}^{n}\right)$, and suppose that we already know that problem (1.16) has a solution $u$ of class $C^{2}$ in some strip $Q_{T}$. For fixed $z \in \mathbb{R}^{n}$, let us denote by $X(t ; z)$ the solution of the ordinary differential equation (here the dot denotes differentiation with respect to $t$ )
$$\dot{X}=D H(\nabla u(t, X)), \quad X(0)=z$$
Such a solution is defined in some maximal interval $\left[0, T_{z}[\right.$ (although it will later turn out that $T_{z}=T$ for all $\left.z\right)$. The curve $t \rightarrow(t, X(t ; z))$ is called the characteristic curve associated with $u$ and starting from the point $(0, z)$. Let us now set
$$U(t ; z)=u(t, X(t ; z)), \quad P(t ; z)=\nabla u(t, X(t ; z)) .$$
Then, using the fact that $u$ solves problem (1.16) we find that
$$\begin{gathered} \dot{U}=u_{t}(t, X)+\nabla u(t, X) \cdot \dot{X}=-H(P)+D H(P) \cdot P \ \dot{P}=\nabla u_{t}(t, X)+\nabla^{2} u(t, X) \dot{X}=\nabla\left(u_{t}+H(\nabla u)\right)(t, X)=0 \end{gathered}$$
Therefore $P$ is constant, and so the right-hand side of (1.19) is also constant. Thus, $X$ is defined in $[0, T$ [ and we can compute explicitly $X, U, P$ obtaining
$$\left{\begin{array}{l} P(t ; z)=D u_{0}(z) \ X(t ; z)=z+t D H\left(D u_{0}(z)\right) \ U(t ; z)=u_{0}(z)+t\left[D H\left(D u_{0}(z)\right) \cdot D u_{0}(z)-H\left(D u_{0}(z)\right)\right] \end{array}\right.$$
Observe that the right-hand side of $(1.21)$ is no longer defined in terms of the solution $u$, but only depends on the initial value $u_{0}$. This suggests that, even without assuming in advance the existence of a solution, one can use these formulas to define one. As we are now going to show, such a construction can be in general carried out only locally in time.

We need the following classical result about the global invertibility of maps (see e.g., [11, Th. 3.1.8]).

## 统计代写|最优控制作业代写optimal control代考|Semiconcavity of Hopf’s solution

In this section we show that the semiconcavity property characterizes the value function among all possible Lipschitz continuous solutions of the Hamilton-Jacobi equation (1.16).
Theorem 1.6.1 Let $L, u_{0}$ satisfy assumptions (1.9). Suppose in addition that
(i) $L \in C^{2}\left(\mathbb{R}^{n}\right), D^{2} L(q) \leq \frac{2}{\alpha} I \quad \forall q \in \mathbb{R}^{n}$
(ii) $u_{0}(x+h)+u_{0}(x-h)-2 u_{0}(x) \leq C_{0}|h|^{2}, \quad \forall x, h \in \mathbb{R}^{n}$

for suitable constants $\alpha>0, C_{0} \geq 0$. Then there exists a constant $C_{1} \geq 0$ such that
\begin{aligned} &u(t+s, x+h)+u(t-s, x-h)-2 u(t, x) \ &\leq \frac{2 t C_{0}}{2 t+\alpha\left(t^{2}-s^{2}\right) C_{0}}\left(|h|+C_{1}|s|\right)^{2} \end{aligned}
for all $t>0, s \in]-t, t\left[, x, h \in \mathbb{R}^{n}\right.$.
Proof – For fixed $t, s, x, h$ as in the statement of the theorem, let us choose $\hat{x} \in \mathbb{R}^{n}$ such that
$$u(t, x)=t L\left(\frac{x-\hat{x}}{t}\right)+u_{0}(\hat{x}) .$$
Such a $\hat{x}$ exists by Hopf’s formula; in addition, by (1.13), there exists $C_{1}$, depending only on $L$, such that
$$\frac{|x-\hat{x}|}{t} \leq C_{1} .$$
We set, for $\lambda \geq 0$,
$$x_{\lambda}^{+}=\hat{x}+\lambda\left(h-s \frac{x-\hat{x}}{t}\right), \quad x_{\lambda}^{-}=\hat{x}-\lambda\left(h-s \frac{x-\hat{x}}{t}\right) .$$
Then we have
$$\frac{x_{\lambda}^{+}+x_{\lambda}^{-}}{2}=\hat{x}, \quad \frac{x_{\lambda}^{+}-x_{\lambda}^{-}}{2}=\lambda\left(h-s \frac{x-\hat{x}}{t}\right) .$$
By (1.29) we have
$$\frac{\left|x_{\lambda}^{+}-x_{\lambda}^{-}\right|}{2} \leq \lambda\left(|h|+C_{1}|s|\right) .$$
By Hopf’s formula (1.10) we have
$$u(t \pm s, x \pm h) \leq(t \pm s) L\left(\frac{x \pm h-x_{\lambda}^{\pm}}{t \pm s}\right)+u_{0}\left(x_{\lambda}^{\pm}\right) .$$

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

H(p)=支持q∈Rn[p⋅q−大号(q)].

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

X˙=DH(∇在(吨,X)),X(0)=和

$$\left{磷(吨;和)=D在0(和) X(吨;和)=和+吨DH(D在0(和)) 在(吨;和)=在0(和)+吨[DH(D在0(和))⋅D在0(和)−H(D在0(和))]\对。$$

## 统计代写|最优控制作业代写optimal control代考|Semiconcavity of Hopf’s solution

(i)大号∈C2(Rn),D2大号(q)≤2一种一世∀q∈Rn
(二)在0(X+H)+在0(X−H)−2在0(X)≤C0|H|2,∀X,H∈Rn

|X−X^|吨≤C1.

Xλ+=X^+λ(H−sX−X^吨),Xλ−=X^−λ(H−sX−X^吨).

Xλ++Xλ−2=X^,Xλ+−Xλ−2=λ(H−sX−X^吨).

|Xλ+−Xλ−|2≤λ(|H|+C1|s|).

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