Missing } inserted error

Question

My equation is giving me an error

Missing } inserted. \end{split} error.

I had imported the code from Mathtype.

\begin{equation}
\begin{split}
$$ - {\delta _{max}} \le \delta (k) \le {\delta _{max}}$$ \\
$$ - {{\dot \delta }_{\max }} \le \delta (k + 1) - \delta (k) \le {{\dot \delta }_{\max }}$$ \\
\end{split}
\end{equation}

Answer 1

Note that this answer addresses the original question asked and not the quite different question which replaced it after this answer was posted.

equation already puts you in maths mode, so you don't need to switch into it. Moreover, you don't want to try to fit an unnumbered display environment into split inside equation. Furthermore, $$ ... $$ ought not be used in LaTeX anyway. Use \[ ... \] instead. But not here, of course:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\begin{split}
 - {\delta _{max}} \le \delta (k) \le {\delta _{max}} \\
 - {{\dot \delta }_{\max }} \le \delta (k + 1) - \delta (k) \le {{\dot \delta }_{\max }} \\
\end{split}
\end{equation}
\end{document}

But did you really want the sequence of variables m, a and x? Or did you mean

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation}
\begin{split}
 - {\delta _{\max}} \le \delta (k) \le {\delta _{\max}} \\
 - {{\dot \delta }_{\max }} \le \delta (k + 1) - \delta (k) \le {{\dot \delta }_{\max }} \\
\end{split}
\end{equation}
\end{document}

Answer 2

I suppose that, for the new version of your equation, you want one of these:

\documentclass{article}
\usepackage{mathtools}
\DeclareMathOperator\obstacle{Obstacle}

\begin{document}

\begin{equation}
  \obstacle(k) = \begin{cases}
  0 & \text{if}\quad  x(k),y(k)  \in   \bigcup _{q = 1}^Q  A(q),    \\[1ex]
  \infty & \text{if}\quad x(k),y(k)  \notin  \bigcup _{q = 1}^Q  A(q) .
  \end{cases}
\end{equation}
\bigskip
\begin{equation}
  \obstacle(k) = \begin{dcases}
  0 & \text{if}\quad  x(k),y(k)  \in \bigcup _{q = 1}^{\smash{Q}}  A(q),   \\[-2ex]
  \infty & \text{otherwise. }
  \end{dcases}
\end{equation}

\end{document} 

Answer 3

Note that this answer addresses the form of the posting after some major interim edits were rolled back. I believe the current form of the posting is fairly close to its original form.

You obviously have to remove the utterly inappropriate $$ directives: TeX is already in display math mode after \begin{equation}; there is no sense in having $$ inside an equation environment.

In addition, you should rethink the equations' layout so as to maximize the chances that you readers will immediately "get" what this pair of grouped inequalities is about. In particular, you may want to pursue one of the following two layouts. The first uses a simple array environment and doesn't require loading any special LaTeX packages. The second, which uses a split environment (provided by the amsmath package) is a bit more terse but should be able to convey its content even more efficiently.

\documentclass{article}
\usepackage{amsmath}

\begin{document}
\begin{equation}
\renewcommand\arraystretch{1.3}
\setlength\arraycolsep{0pt}
\begin{array}{r @{{}\le{}} c @{{}\le{}} l}
-\delta_{\max}       & \delta(k)             & \delta_{\max} \\
-\dot{\delta}_{\max} & \delta(k+1)-\delta(k) & \dot{\delta}_{\max}
\end{array}
\end{equation}

\begin{equation}\begin{split}
|\delta(k)| &\le \delta_{\max} \\
|\delta(k+1)-\delta(k)| &\le \dot{\delta}_{\max}
\end{split}\end{equation}
\end{document}