% m211oh8581.tex (rev.: *oh81.*) - Equilibrium Points
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\begin{titlepage}
\vspace*{-20pt}
\begin{center}\tcr{\rmH Equilibrium Points} \end{center} %\vspace{0pt}
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\newcommand{\ueq}{\vec{u}_{\scr eq}}
\newcommand{\vu}[1]{\vec{u}_{\scr #1}}
\newcommand{\vv}[1]{\vec{v}_{\scr #1}}

\begin{description}  

\item[\tcr{Definition:}] A vector \mc{\ueq} is called an \tcr{equilibrium point} of the \tcg{discrete linear system}\vspace{-5pt}
\tcb{\beq
\vu{n+1} = A\vu{n} + \vec{b}\vspace{-5pt} \label{1}
\eeq}
if it satisfies the equation \vspace{-3pt}
\tcb{\beq
\ueq = A\ueq + \vec{b},\vspace{-3pt}
\eeq}
or, equivalently, if we have \vspace{-5pt}
\tcb{$$
(I - A)\ueq = \vec{b}.\vspace{-5pt}
$$}
The equilibrium point \mc{\ueq} is called \tcr{stable} if\vspace{-20pt} we have
\tcb{\beq
\vu{n} \ra \ueq, \quad \mbox{as}\; n \ra \infty.\vspace{-30pt}
\eeq}

\item[\tcr{Theorem 16:}] Suppose that the d.l.s. \sref{1} has an equilibrium point \mc{\ueq}. Then, letting\vspace{-3pt}
\tcb{\beq
\vv{n} = \vu{n} - \ueq\vspace{-3pt}
\eeq}
denote the \tcg{deviation from the equilibrium point}, we have\vspace{-3pt}
\tcb{\beq
\vv{n+1} = A\vv{n}\vspace{-3pt}
\eeq}
and hence \vspace{-3pt} 
\tcb{\beq
\vu{n} = A^n(\vu{0} - \ueq) + \ueq.
\eeq}
In particular, if \mc{A^n \ra 0}, then \mc{\ueq} is stable.  
\end{description} 
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