About distributed control in model of testosterone regulation

Abstract

Abstract

In the paper, stability of integro-differential equation is studied.
The model of testosterone regulation is considered. The model describes an interaction of: the concentration of hormone (GnRH)
which will be denoted as $x_{1}$, with  the concentration of the hormone (LH)-$x_{2}$ and  the concentration of testosterone (Te)-$x_{3}$
and can be written in the form
\begin{equation*}
\begin{array}{l}
            \displaystyle\left\{
            \begin{array}{l}
                \displaystyle x_{1}^{\prime }(t)+b_{1}x_{1}(t)=0, \\[1.0ex]
                \displaystyle x_{2}^{\prime }(t)+b_{2}x_{2}(t)-g_{1}x_{1}(t)=0,~ \\
                \displaystyle x_{3}^{\prime
                }(t)+b_{3}x_{3}(t)-c_{1}\!\int\limits_{0}^{t}\!e^{-\alpha
                    _{1}(t-s)}x_{2}(s)\,ds=0, ~~t\geq 0~.%
            \end{array}
            ~~~\right.
        \end{array}
%        $
\end{equation*}
The values $b_{i}$, $i=1,2,3$ correspond to the respective half-life times of
\abbr{GnRH}, \abbr{LH} and \abbr{Te}.
The aim of the work is to propose a concept to hold the concentration of testosterone above a corresponding level.
In order to achieve this, distributed input control in the form of integral term is used.