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# 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|ECE1659H Some applications to lower closure and denseness

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## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|Some applications to lower closure and denseness

We illustrate the power of the above apparatus by some applications to a variety of problems; we refer to $[18,22]$ for more extensive expositions.

As our first application, we derive an extension of the so-called fundamental theorem for Young measures in [25]. Here $L$ is a locally compact space that is countable at infinity; its usual Alexandrov compactification is denoted by $\hat{L}:=L \cup{\infty}$. The space $\hat{L}$ is metrizable, and its metric is denoted by $\hat{d}$. On $L$ we use the natural restriction of $\hat{d}$, and denote it by $d$. Let $\mathcal{C}_0(L)$ be the usual space of continuous functions on $L$ that converge to zero at infinity. Although it could be avoided by the additional introduction of transition subprobabilities (see the comments below), the Alexandrov compactification $\hat{L}$ of $L$ figures explicitly in the result. Also, below $\nu$ denotes a $\sigma$-finite measure on $(\Omega, \mathcal{A})$.

Corollary $5.1$ (i) Let $\left(f_n\right)$ in $\mathcal{L}^0(\Omega ; L)$ and the closed set $C \subset L$ be such that $\lim n$ $\nu\left(f_n^{-1}(L \backslash G)\right)=0$ for every open $G, C \subset G \subset L$. Then there exist a subsequence $\left(f{n^{\prime}}\right)$ of $\left(f_n\right)$ and $\delta_$ in $\mathcal{R}(\Omega ; \hat{L})$ such that $\delta_(\omega)(L \backslash C)=0$ for a.e. $\omega$ in $\Omega$ and
$$\lim n \int{\Omega} \phi(\omega) c\left(f_{n^{\prime}}(\omega)\right) \nu(d \omega)=\int_{\Omega}\left[\int_L \phi(\omega) c(x) \delta_(\omega)(d x)\right] \nu(d \omega)$$ for every $\phi \in \mathcal{L}^1(\Omega ; \mathbf{R})$ and every $c \in \mathcal{C}0(L)$. (ii) Moreover, if for that subsequence there exists a sequence $\left(K_r\right)$ of compact sets in $L$ such that $\lim {r \rightarrow \infty} \sup {n^{\prime}} \nu\left(\left{\omega \in \Omega: f{n^{\prime}}(\omega) \notin K_r\right}=0\right.$ then $\delta_(\omega)({\infty})=0$ for a.e. $\omega$ in $\Omega$ and
$$\lim n \int_A \phi(\omega) c\left(f{n^{\prime}}(\omega)\right) \nu(d \omega)=\int_A\left[\int_L \phi(\omega) c(x) \delta_*(\omega)(d x)\right] \nu(d \omega)$$
for every $A \in \mathcal{A}, \phi \in \mathcal{L}^1(A ; \mathbf{R})$ and $c \in \mathcal{C}(L)$ for which $\left(1_A c\left(f_{n^{\prime}}\right)\right)$ is relatively weakly compact in $\mathcal{L}^1(A ; \mathbf{R})$.

## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|A condition of Legendre type for optimal control problems, linear in control

For the control system, linear in the control, and nonlinear in the state variable,
$$\dot{x}=f(x, t)+F(x, t) u,$$
on a fixed time interval $[0, T]$, where both the control and state variables are multidimensional, we consider the following optimal control problem: to minimize a terminal functional $J=\varphi_0(p)$ subject to terminal constraints $\varphi_i(p) \leq 0, i=1, \ldots \nu, g(p)=$ 0 , and to a pointwise control constraint $u(t) \in U(t)$.

Here $p=(x(0), x(T)), \quad \operatorname{dim} g=q, \quad$ and $U(t)$ is a convex solid set, Hausdorff continuous in $t$. All the data functions are assumed to be at least $C^2-$ smooth in $x, p$, and $C^1-$ smooth in $t$. This is a fairly general statement, which includes (after simple reformulations) problems with integral functionals and integral constraints, time-optimal problems, and others.

Denote by $W=A C^m \times L_{\infty}^r[0, T]$ the space of all pairs of functions $w=(x, u)$, where $x(t)$ is absolutely continuous $m-$ dimensional, and $u(t)$ is measurable essentially bounded $r$ – dimensional, and let $w^0=\left(x^0, u^0\right)$ be an examined extremal.

## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代 考|Some applications to lower closure and denseness

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$$\lim n \int_A \phi(\omega) c\left(f n^{\prime}(\omega)\right) \nu(d \omega)=\int_A\left[\int_L \phi(\omega) c(x) \delta_*(\omega)(d x)\right] \nu(d \omega)$$

## 数学代写|凸分析和最优控制代写Convex Analysis and Optimal Control代考|A condition of Legendre type for optimal control problems, linear in control

$$\dot{x}=f(x, t)+F(x, t) u,$$

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