Covariance Matrix

To simplify calculations we have the following matrix Or alternatively a more compact form

$$\begin{array}{rl}{\mathcal{C}}_x(t)& =E[(x-m_x)(x-m_x{)}^T]\\ {\dot{\mathcal{C}}}_x(t)& =\cdots =A{\mathcal{C}}_x+{\mathcal{C}}_x{A}^T+BE[(u-m_u)(x-m_x{)}^T]+E[(x-m_x)(u-m_u{)}^T]B^T\end{array}$$

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The autocovariance of white noise

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And we can rewrite ${\dot{\mathcal{C}}}_x$ as

$${\dot{\mathcal{C}}}_x=A{\mathcal{C}}_x+{\mathcal{C}}_x{A}^T+B{Q}_u(t)B^T$$