About Distributed Control in Model of Testosterone Regulation

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 x₁, with the concentration of the hormone (LH)-x₂ and the concentration of testosterone (Te)-x₃ and can be written in the form

{x′1(t)+b1x1(t)=0,x′2(t)+b2x2(t)−g1x1(t)=0,x′3(t)+b3x3(t)−c1∫0te−α1(t−s)x2(s)ds=0, t≥0.$$\left\{ {\matrix{ {{{x'}_1}(t) + {b_1}{x_1}(t) = 0,} \hfill \cr {{{x'}_2}(t) + {b_2}{x_2}(t) - {g_1}{x_1}(t) = 0,} \hfill \cr {{{x'}_3}(t) + {b_3}{x_3}(t) - {c_1}\int\limits_0^t {{e^{ - {\alpha _1}(t - s)}}{x_2}(s)ds = 0,\;t \ge 0.} } \hfill \cr } } \right.$$

The values b_i, i = 1, 2, 3 correspond to the respective half-life times of GnRH, LH and 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.