Abstract
In this paper we propose a method for stability studies of functional differential systems. The idea of our method is to reduce the analysis of an n-dimensional system to one for an (n+m)-dimensional system, where m is a natural number, to obtain stability and then to come back and make conclusions on the stability of the given n-dimensional system. As an example, a model describing testosterone by distributed inputs feedback control is considered. The aim of the regulation is to hold testosterone concentration above an appropriate level. The feedback control with integral term is proposed. We have to increase the testosterone level to the normal one. The control we proposed could destroy the stability of the model. That is why we have to choose the parameters of our distributed control, namely a dosage or intensity of assimilation of a medicine in a human body in such a form that the stability of our system is preserved. Thus the problem of regulation of testosterone level leads us to the stability analysis of the functional differential system describing a connection between the concentrations of hormones (GnRH), (LH), and testosterone (Te). Constructing the system, we discard the connections which seem nonessential. To estimate the effect of these connections is an important problem. We construct the Cauchy matrix of integro-differential system to estimate this influence.
Highlights
Functional differential equation of the formX (t) + B(t)X(t) + (KX)(t) = f (t), t ∈ [0, ∞), (1.1)where B(t) is an n × n matrix with essentially bounded coefficients and K : Cn → Ln∞ is a linear bounded operator acting from the space of continuous functions Cn to the space of essentially bounded functions Ln∞, f ∈ Ln∞ → Rn), appears as a mathematical model describing processes in medicine, biology, and technology [18].Domoshnitsky et al Boundary Value ProblemsThe operator K can be, for example, of the integral form t (KX)(t) = k(t, s)X(s) ds. (1.2) the control with distributed input control frequently appears as a challenging problem, only a few papers are devoted to it
Where B(t) is an n × n matrix with essentially bounded coefficients and K : Cn → Ln∞ is a linear bounded operator acting from the space of continuous functions Cn to the space of essentially bounded functions Ln∞, f ∈ Ln∞ → Rn), appears as a mathematical model describing processes in medicine, biology, and technology [18]
Noise in the feedback delay control is the main obstacle appearing in mathematical models because of the fact that it is impossible to base our control on the value of process X(t) at a moment tj only, and we have to use an average value of the process X(t) = col{x1(t), . . . , xn(t)} at a corresponding neighborhood of the point tj
Summary
Where B(t) is an n × n matrix with essentially bounded coefficients and K : Cn → Ln∞ is a linear bounded operator acting from the space of continuous functions Cn to the space of essentially bounded functions Ln∞, f ∈ Ln∞ (all functions we understand as x : [0, ∞) → Rn), appears as a mathematical model describing processes in medicine, biology, and technology [18]
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