In this paper the model of infection diseases by Marchuk is considered. Mathematical questions which are important in its study are discussed. Among them there are stability of stationary points, construction of the Cauchy matrices of linearized models, estimates of solutions. The novelty we propose is in a distributed feedback control which affects the antibody concentration. We use this control in the form of an integral term and come to the analysis of nonlinear integro-differential systems. New methods for the study of stability of linearized integro–differential systems describing the model of infection diseases are proposed. Explicit conditions of the exponential stability of the stationary points characterizing the state of the healthy body are obtained. The method of the paper is based on the symmetry properties of the Cauchy matrices which allow us their construction.

The Marchuk model of infectious diseases is considered. Distributed control to make convergence to stationary point faster is proposed. Medically, this means that treatment time can be essentially reduced. Decreasing the concentration of antigen, this control facilitates the patient’s condition and gives a certain new idea of treating the disease. Our approach involves the analysis of integro-differential equations. The idea of reducing the system of integro-differential equations to a system of ordinary differential equations is used. The final results are given in the form of simple inequalities on the parameters. The results of numerical calculations of simulation models and data comparison in the case of using distributive control and in its absence are given.

In this paper, a new approach to stability of integro-differential equations x (t) + β1 t t−τ 1 (t) e −α 1 (t−s) x (s) ds + β2 t t−τ 2 (t) e −α 2 (t−s) x (s) ds = 0 and x (t) + β1 t−τ 1 (t) 0 e −α 1 (t−s) x (s) ds + β2 t−τ 2 (t) 0 e −α 2 (t−s) x (s) ds = 0 is proposed. Under corresponding conditions on the coefficients α1, α2, β1 and β2 the first equation is exponentially stable if the delays τ1 (t) and τ2 (t) are large enough and the second equation is exponentially stable if these delays are small enough. On the basis of these results, assertions on stabilization by distributed input control are proven. It should be stressed that stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper demonstrate that this is not the case.

The impulsive delay differential equation is considered (Lx)(t) = x′(t) + ∑i=1 m p i(t)x(t − τ i(t)) = f(t),  t ∈ [a, b], x(t j) = β j x(t j − 0),  j = 1,…, k,  a = t 0 < t 1 < t 2 < ⋯

In this paper, the methods of the stability theory of differential equations with time delays are used in the study of an actual engineering problem of a drone (UAV) autonomous flight. We describe correct operation of autopilot for supply stability of desirable drone flight. There exists a noticeable delay in getting information about position and orientation of a drone to autopilot in the presence of vision-based navigation (visual navigation). In spite of this fact, we demonstrate that it is possible to provide stable flight at a constant height in a vertical plane. We describe how to form relevant controlling signal for autopilot in the case of the navigation information delay and provide control parameters for particular case of flight.