<abstract xmlns="http://www.w3.org/1999/xhtml">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 <italic>x</italic>1, with the concentration of the hormone (LH)-<italic>x</italic>2 and the concentration of testosterone (Te)-<italic>x</italic>3 and can be written in the form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2018-0013_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><msub><mrow><msup><mi>x</mi><mo>′</mo></msup></mrow><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>b</mi></mrow><mn>1</mn></msub><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><msup><mi>x</mi><mo>′</mo></msup></mrow><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>b</mi></mrow><mn>2</mn></msub><msub><mrow><mi>x</mi></mrow><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>g</mi></mrow><mn>1</mn></msub><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mrow><msup><mi>x</mi><mo>′</mo></msup></mrow><mn>3</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>b</mi></mrow><mn>3</mn></msub><msub><mrow><mi>x</mi></mrow><mn>3</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>c</mi></mrow><mn>1</mn></msub><mrow><munderover><mo>∫</mo><mn>0</mn><mi>t</mi></munderover><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo><msub><mrow><mi>α</mi></mrow><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>−</mo><mi>s</mi><mo>)</mo></mrow></msup><msub><mrow><mi>x</mi></mrow><mn>2</mn></msub><mo>(</mo><mi>s</mi><mo>)</mo><mi>d</mi><mi>s</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi> </mi><mi>t</mi><mo>≥</mo><mn>0</mn><mo>.</mo></mrow></mrow></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>$$\left\{ {\matrix{ {{{x'}_1}(t) + {b_1}{x_1}(t) = 0,} \hfill \cr {{{x'}_2}(t) + {b_2}{x_2}(t) - {g_1}{x_1}(t) = 0,} \hfill \cr {{{x'}_3}(t) + {b_3}{x_3}(t) - {c_1}\int\limits_0^t {{e^{ - {\alpha _1}(t - s)}}{x_2}(s)ds = 0,\;t \ge 0.} } \hfill \cr } } \right.$$</tex-math></alternatives></disp-formula>The values <italic>bi</italic>, <italic>i</italic> = 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.</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 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 bi, 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.

About Distributed Control in Model of Testosterone Regulation

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"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:...

About Distributed Control in Model of Testosterone Regulation

Published Online: Jan 25, 2019

Page range: 149 - 154

Received: Dec 12, 2017

DOI: https://doi.org/10.2478/tmmp-2018-0013

Keywords
integro-differential equations, stability, reduction method, testosterone regulation

© 2018 Olga Pinhasov, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

About Distributed Control in Model of Testosterone Regulation

Published Online: Jan 25, 2019

Page range: 149 - 154

Received: Dec 12, 2017

DOI: https://doi.org/10.2478/tmmp-2018-0013

Keywordsintegro-differential equations, stability, reduction method, testosterone regulation

© 2018 Olga Pinhasov, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
integro-differential equations, stability, reduction method, testosterone regulation