拉普拉斯变换#
这一节介绍的是拉普拉斯变换的基本概念
什么是拉普拉斯变换?#
下面测试一下公式好不好用
测试一下行内公式:\(\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}\).再测试一下行间公式:
下面测试一下表格好不好用
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下面是插图的测试,下面是一张星空的图片:
我们测试一下各种漂亮的代码块
Note
这是一个笔记.
Warning
这是一个警告!
然后我们测试一下定理和命题,引理,证明
定理
Theorem 1
Given \(y \in \mathbb R^n\) and linear subspace \(S \subset \mathbb R^n\), there exists a unique solution to the minimization problem
The minimizer \(\hat y\) is the unique vector in \(\mathbb R^n\) that satisfies
\(\hat y \in S\)
\(y - \hat y \perp S\) The vector \(\hat y\) is called the orthogonal projection of \(y\) onto \(S\).
证明
Proof. We’ll omit the full proof. But we will prove sufficiency of the asserted conditions. To this end, let \(y \in \mathbb R^n\) and let \(S\) be a linear subspace of \(\mathbb R^n\). Let \(\hat y\) be a vector in \(\mathbb R^n\) such that \(\hat y \in S\) and \(y - \hat y \perp S\). Let \(z\) be any other point in \(S\) and use the fact that \(S\) is a linear subspace to deduce
Hence \(\| y - z \| \geq \| y - \hat y \|\), which completes the proof.
公理
Axiom 1
Every Cauchy sequence on the real line is convergent.
引理
Lemma 2
If \(\hat P\) is the fixed point of the map \(\mathcal B \circ \mathcal D\) and \(\hat F\) is the robust policy as given in (7), then
定义
Definition 1
The economical expansion problem (EEP) for \((A,B)\) is to find a semi-positive \(n\)-vector \(p>0\) and a number \(\beta\in\mathbb{R}\), such that $\( &\min_{\beta} \hspace{2mm} \beta \\ &\text{s.t. }\hspace{2mm}Bp \leq \beta Ap \)$
第二个引理
Lemma 2
If \(\hat P\) is the fixed point of the map \(\mathcal B \circ \mathcal D\) and \(\hat F\) is the robust policy as given in (7), then