Search

In this paper, we study generic properties of the optimal, in a certain sense, highly active antiretroviral therapy (or HAART). To address this problem, we consider a control model based on the 3-dimensional Nowak-May within-host HIV dynamics model. Taking into consideration that precise forms of functional responses are usually unknown, we introduce into this model a nonlinear incidence rate of a rather general form given by an unspecified function of the susceptible cells and free virus particles. We also add a term responsible to the loss of free virions due to infection of the target cells. To mirror the idea of highly active anti-HIV therapy, in this model we assume six controls that can act simultaneously. These six controls affecting different stage of virus life cycle comprise all controls possible for this model and account for all feasible actions of the existing anti-HIV drugs. With this control model, we consider an optimal control problem of minimizing the infection level at the end of a given time interval. Using an analytical mathematical technique, we prove that the optimal controls are bang-bang, find accurate estimates for the maximal possible number of switchings of these controls and establish qualitative types of the optimal controls as well as mutual relationships between them. Having the estimate for the number of switchings found, we can reduce the two-point boundary value problem for Pontryagin Maximum Principle to a considerably simpler problem of the finite-dimensional optimization, which can be solved numerically. Despite this possibility, the obtained theoretical results are illustrated by numerical calculations using BOCOP-2.0.5 software package, and the corresponding conclusions are made. © Copyright @ 2020 Ellina V. Grigorieva et al.

Biological treatment of waste water involves bacteria, which remove waste products from the treated water and kill pathogens. Thus, this problem is closely linked to both the population dynamics and chemical kinetics. Aerobic process is very efficient but rather costly, as it requires aeration of the treated water. An aim of this paper is to explore the possibility to apply the ideas and methods of the optimal control theory to the problem of optimizing the costs of this process. In this paper, we consider a three-dimensional nonlinear model of water treatment with includes a bounded control. In order to foretell possible outcomes of the control, we analytically investigate the corresponding attainable set, constructing a parametric description of this set by moments of switching of piecewise constant controls. This allows us to visualize the attainable set using MATLAB. These results lead to a number of practically relevant conclusions.