How to center the text in the equation environment?

You can use an array.

\documentclass[12 pt,handout,notheorems]{beamer}
\usepackage{mathtools}
\usepackage{pgfpages}
\pgfpagesuselayout{resize to}[a4paper,landscape]
\usetheme{Boadilla}
\usefonttheme{professionalfonts} %To get the accents aligned correctly, albeit in Computer Modern Roman
\usepackage{amsmath}
\usepackage{mleftright}
\usefonttheme{structuresmallcapsserif}
\mleftright
\begin{document}
    \begin{frame}
        In conclusion,
        \begin{itemize}
            \item For minimum-time problems:
            \begin{equation*}
            \begin{array}{@{}c@{}}
                    \text{System is controllable}\\
                    \text{with respect to }u_i
            \end{array}
                \iff \text{Singular arcs do not exist.}
            \end{equation*}
            \item For minimum-time-fuel problems (also applies for minimum-fuel problems):
            \begin{equation*}
                \begin{array}{@{}c@{}}
                    \text{System is not controllable}\\
                    \text{with respect to }u_i\\
                    \text{or}\\
                    \det\left(A\right) = 0
                \end{array}
                \implies \text{Singular arcs exist.}
            \end{equation*}
        \end{itemize}
    \end{frame}
    \begin{frame}
        In conclusion,
        \begin{itemize}
            \item For minimum-time problems:
            \begin{equation*}
             \left.\begin{array}{@{}c@{}}
                    \text{System is controllable}\\
                    \text{with respect to }u_i
                \end{array}\right\}
                \iff \text{Singular arcs do not exist.}
            \end{equation*}
            \item For minimum-time-fuel problems (also applies for minimum-fuel problems):
            \begin{equation*}
              \left.\begin{array}{@{}c@{}}
                    \text{System is not controllable}\\
                    \text{with respect to }u_i\\
                    \text{or}\\
                    \det\left(A\right) = 0
               \end{array}\right\}
                \implies \text{Singular arcs exist.}
            \end{equation*}
        \end{itemize}
    \end{frame} 
\end{document}

You could replace the aligned environments with gathered environments -- and remove the & alignment points. The main difference to an array-based solution is that the contents are typeset in display-math mode by default, which implies a slight increase in line spacing.

\documentclass[12pt,handout,notheorems]{beamer}
\usepackage{pgfpages}
\pgfpagesuselayout{resize to}[a4paper,landscape]
\usetheme{Boadilla}
\usefonttheme{professionalfonts}
\usefonttheme{structuresmallcapsserif}
\usepackage{amsmath} % for 'gathered' environment
\usepackage{mleftright}\mleftright

\begin{document}
    \begin{frame}
        In conclusion,
        \begin{itemize}
            \item For minimum-time problems:
            \begin{equation*}
                \begin{gathered}
                    \text{System is controllable}\\
                    \text{with respect to $u_i$}
                \end{gathered}
                \iff \text{Singular arcs do not exist.}
            \end{equation*}
            \item For minimum-time-fuel problems (also applies for minimum-fuel problems):
            \begin{equation*}
                \begin{gathered}
                    \text{System is not controllable}\\
                    \text{with respect to $u_i$, or}\\
                    \det\left(A\right) = 0
                \end{gathered}
                \implies \text{Singular arcs exist.}
            \end{equation*}
        \end{itemize}
    \end{frame}
\end{document}

You could use a \parbox, although you will need to specify a width. I used \widthof from the calc package, but you don't have to be that precise. Note that \parbox takes you out of math mode.

\documentclass[12 pt,handout,notheorems]{beamer}
\usepackage{pgfpages}
\pgfpagesuselayout{resize to}[a4paper,landscape]
\usetheme{Boadilla}
\usefonttheme{professionalfonts} %To get the accents aligned correctly, albeit in Computer Modern Roman
\usepackage{amsmath}
\usepackage{mleftright}
\usefonttheme{structuresmallcapsserif}
\mleftright
\usepackage{calc}
\begin{document}
    \begin{frame}
        In conclusion,
        \begin{itemize}
            \item For minimum-time problems:
            \begin{equation*}
                \parbox{\widthof{System is controllable}}{\centering
                    System is controllable with respect to $u_i$}
                \iff \text{Singular arcs do not exist.}
            \end{equation*}
            \item For minimum-time-fuel problems (also applies for minimum-fuel problems):
            \begin{equation*}
                \parbox{\widthof{System is not controllable}}{\centering
                    System is not controllable with respect to $u_i$ or
                    $\det\left(A\right) = 0$}
                \implies \text{Singular arcs exist.}
            \end{equation*}
        \end{itemize}
    \end{frame}
\end{document}