### 统计代写|最优控制作业代写optimal control代考|Generalized Gradients and Semiconcavity

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## 统计代写|最优控制作业代写optimal control代考|Generalized Gradients and Semiconcavity

In the last decades a branch of mathematics has developed called nonsmooth analysis, whose object is to generalize the basic tools of calculus to functions that are not differentiable in the classical sense. For this purpose, one introduces suitable notions of generalized differentials, which are extensions of the usual gradient; the best known example is the subdifferential of convex analysis. The motivation for this study is that in more and more fields of analysis, like the optimization problems considered in this book, the functions that come into play are often nondifferentiable.
For semiconcave functions, the analysis of generalized gradients is important in view of applications to control theory. As we have already seen in a special case (Corollary 1.5.10), if the value function of a control problem is smooth, then one can design the optimal trajectories knowing the differential of the value function. In the general case, where the value function is not smooth but only semiconcave, one can try to follow a similar procedure starting from its generalized gradient.

In Section $3.1$ we define the generalized differentials which are relevant for our purposes and recall basic properties and equivalent characterizations of these objects. Then, we restrict ourselves to semiconcave functions. In Section $3.2$ we show that semiconcave functions possess one-sided directional derivatives everywhere, while in Section $3.3$ we describe the special properties of the superdifferential of a semiconcave function; in particular, we show that it is nonempty at every point and that it is a singleton exactly at the points of differentiability. These properties are classical in the case of concave functions; here we prove that they hold for semiconcave functions with arbitrary modulus.

Section $3.4$ is devoted to the so-called marginal functions, which are obtained as the infimum of smooth functions. We show that semiconcave functions can be characterized as suitable classes of marginal functions. In addition, we describe the semi-differentials of a marginal function using the general results of the previous sections. In Section $3.5$ we study the so-called inf-convolutions. They are marginal functions defined by a process which is a generalization of Hopf’s formula, and provide approximations to a given function which enjoy useful properties. Finally, in Section $3.6$ we introduce proximal gradients and proximally smooth sets, and we analyze how these notions are related to semiconcavity.

## 统计代写|最优控制作业代写optimal control代考|Generalized differentials

We begin with the definitions of some generalized differentials and derivatives from nonsmooth analysis. In this section $u$ is a real-valued function defined on an open set $A \subset \mathbb{R}^{n}$.
Definition 3.1.1 For any $x \in A$, the sets
\begin{aligned} D^{-} u(x) &=\left{p \in \mathbb{R}^{n}: \liminf {y \rightarrow x} \frac{u(y)-u(x)-\langle p, y-x\rangle}{|y-x|} \geq 0\right} \ D^{+} u(x) &=\left{p \in \mathbb{R}^{n}: \limsup {y \rightarrow x} \frac{u(y)-u(x)-\langle p, y-x\rangle}{|y-x|} \leq 0\right} \end{aligned}
are called, respectively, the (Fréchet) superdifferential and subdifferential of $u$ at $x$.
From the definition it follows that, for any $x \in A$,
$$D^{-}(-u)(x)=-D^{+} u(x) .$$
Example 3.1.2
Let $A=\mathbb{R}$ and let $u(x)=|x|$. Then it is easily seen that $D^{+} u(0)=\emptyset$ whereas $D^{-} u(0)=[-1,1] .$
Let $A=\mathbb{R}$ and let $u(x)=\sqrt{|x|}$. Then, $D^{+} u(0)=\emptyset$ whereas $D^{-} u(0)=\mathbb{R}$.
Let $A=\mathbb{R}^{2}$ and $u(x, y)=|x|-|y|$. Then, $D^{+} u(0,0)=D^{-} u(0,0)=\emptyset$.
Definition 3.1.3 Let $x \in A$ and $\theta \in \mathbb{R}^{n}$. The upper and lower Dini derivatives of $u$ at $x$ in the direction $\theta$ are defined as
$$\partial^{+} u(x, \theta)=\lim {h \rightarrow 0^{+}, \theta^{\prime} \rightarrow \theta} \frac{u\left(x+h \theta^{\prime}\right)-u(x)}{h}$$ and $$\partial^{-} u(x, \theta)=\liminf {h \rightarrow 0^{+}, \theta^{\prime} \rightarrow \theta} \frac{u\left(x+h \theta^{\prime}\right)-u(x)}{h},$$
respectively.
It is readily seen that, for any $x \in A$ and $\theta \in \mathbb{R}^{n}$
$$\partial^{-}(-u)(x, \theta)=-\partial^{+} u(x, \theta) .$$
Remark 3.1.4 Whenever $u$ is Lipschitz continuous in a neighborhood of $x$, the lower Dini derivative reduces to
$$\partial^{-} u(x, \theta)=\liminf _{h \rightarrow 0+} \frac{u(x+h \theta)-u(x)}{h}$$
for any $\theta \in \mathbb{R}^{n}$. Indeed, if $L>0$ is the Lipschitz constant of $u$ we have
$$\left|\frac{u\left(x+h \theta^{\prime}\right)-u(x)}{h}-\frac{u(x+h \theta)-u(x)}{h}\right| \leq L\left|\theta^{\prime}-\theta\right|,$$
and (3.5) easily follows. A similar property holds for the upper Dini derivative.

## 统计代写|最优控制作业代写optimal control代考|Directional derivatives

We begin our exposition of the differential properties of semiconcave functions showing that they possess (one-sided) directional derivatives
$$\partial u(x, \theta):=\lim {h \rightarrow 0^{+}} \frac{u(x+h \theta)-u(x)}{h}$$ at any point $x$ and in any direction $\theta$. Theorem 3.2.1 Let $u: A \rightarrow \mathbb{R}$ be semiconcave. Then, for any $x \in A$ and $\theta \in \mathbb{R}^{n}$, $$\partial u(x, \theta)=\partial^{-} u(x, \theta)=\partial^{+} u(x, \theta)=u{-}^{0}(x, \theta) .$$
Proof – Let $\delta>0$ be fixed so that $B_{\delta|\theta|}(x) \subset A$. Then, for any pair of numbers $h_{1}, h_{2}$ satisfying $0<h_{1} \leq h_{2}<\delta$, estimate (2.1) yields
$$\left(1-\frac{h_{1}}{h_{2}}\right) u(x)+\frac{h_{1}}{h_{2}} u\left(x+h_{2} \theta\right)-u\left(x+h_{1} \theta\right) \leq h_{1}\left(1-\frac{h_{1}}{h_{2}}\right)|\theta| \omega\left(h_{2}|\theta|\right) .$$
Hence,
\begin{aligned} &\frac{u\left(x+h_{1} \theta\right)-u(x)}{h_{1}} \ &\geq \frac{u\left(x+h_{2} \theta\right)-u(x)}{h_{2}}-\left(1-\frac{h_{1}}{h_{2}}\right)|\theta| \omega\left(h_{2}|\theta|\right) . \end{aligned}
Taking the liminf as $h_{1} \rightarrow 0^{+}$in both sides of the above inequality, we obtain

$$\partial^{-} u(x, \theta) \geq \frac{u\left(x+h_{2} \theta\right)-u(x)}{h_{2}}-|\theta| \omega\left(h_{2}|\theta|\right)$$
Now, taking the limsup as $h_{2} \rightarrow 0^{+}$, we conclude that
$$\partial^{-} u(x, \theta) \geq \partial^{+} u(x, \theta) .$$
So, $\partial u(x, \theta)$ exists and coincides with the lower and upper Dini derivatives.
To complete the proof of $(3.15)$ it suffices to show that
$$\partial^{+} u(x, \theta) \leq u_{-}^{0}(x, \theta),$$
since the reverse inequality holds by definition and by Remark 3.1.4. For this purpose, let $\varepsilon>0, \lambda \in] 0, \delta[$ be fixed. Since $u$ is continuous, we can find $\alpha \in$ ] $0,(\delta-\lambda) \theta$ [ such that
$$\frac{u(x+\lambda \theta)-u(x)}{\lambda} \leq \frac{u(y+\lambda \theta)-u(y)}{\lambda}+\varepsilon, \quad \forall y \in B_{\alpha}(x) .$$
Using inequality (3.16) with $x$ replaced by $y$, we obtain
$$\left.\frac{u(y+\lambda \theta)-u(y)}{\lambda} \leq \frac{u(y+h \theta)-u(y)}{h}+|\theta| \omega(\lambda|\theta|), \quad \forall h \in\right] 0, \lambda[.$$
Therefore,
$$\frac{u(x+\lambda \theta)-u(x)}{\lambda} \leq \inf {y \in B{u}(x), h \in|0, \lambda|} \frac{u(y+h \theta)-u(y)}{h}+|\theta| \omega(\lambda|\theta|)+\varepsilon .$$
This implies, by definition of $u_{-}^{0}(x, \theta)$, that
$$\frac{u(x+\lambda \theta)-u(x)}{\lambda} \leq u_{-}^{0}(x, \theta)+|\theta| \omega(\lambda|\theta|)+\varepsilon .$$
Hence, taking the limit as $\varepsilon, \lambda \rightarrow 0$, we obtain inequality (3.17).

## 统计代写|最优控制作业代写optimal control代考|Generalized differentials

\begin{对齐} D^{-} u(x) &=\left{p \in \mathbb{R}^{n}: \liminf {y \rightarrow x} \frac{u(y)-u( x)-\langle p, yx\rangle}{|yx|} \geq 0\right} \ D^{+} u(x) &=\left{p \in \mathbb{R}^{n}： \limsup {y \rightarrow x} \frac{u(y)-u(x)-\langle p, yx\rangle}{|yx|} \leq 0\right} \end{aligned}\begin{对齐} D^{-} u(x) &=\left{p \in \mathbb{R}^{n}: \liminf {y \rightarrow x} \frac{u(y)-u( x)-\langle p, yx\rangle}{|yx|} \geq 0\right} \ D^{+} u(x) &=\left{p \in \mathbb{R}^{n}： \limsup {y \rightarrow x} \frac{u(y)-u(x)-\langle p, yx\rangle}{|yx|} \leq 0\right} \end{aligned}

D−(−在)(X)=−D+在(X).

∂+在(X,θ)=林H→0+,θ′→θ在(X+Hθ′)−在(X)H和∂−在(X,θ)=林恩夫H→0+,θ′→θ在(X+Hθ′)−在(X)H,

∂−(−在)(X,θ)=−∂+在(X,θ).

∂−在(X,θ)=林恩夫H→0+在(X+Hθ)−在(X)H

|在(X+Hθ′)−在(X)H−在(X+Hθ)−在(X)H|≤大号|θ′−θ|,

## 统计代写|最优控制作业代写optimal control代考|Directional derivatives

∂在(X,θ):=林H→0+在(X+Hθ)−在(X)H在任何时候X并且在任何方向θ. 定理 3.2.1 令在:一种→R是半凹的。那么，对于任何X∈一种和θ∈Rn,∂在(X,θ)=∂−在(X,θ)=∂+在(X,θ)=在−0(X,θ).

(1−H1H2)在(X)+H1H2在(X+H2θ)−在(X+H1θ)≤H1(1−H1H2)|θ|ω(H2|θ|).

∂−在(X,θ)≥∂+在(X,θ).

∂+在(X,θ)≤在−0(X,θ),

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