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

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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.

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