Optimal control of glycemia regulation dynamics in patients with type I diabetes mellitus

The work is devoted to the problem of mathematical modeling and search for optimal control of the dynamics of the insulin-glucose balance in human blood, represented by a nonsmooth system of differential equations with a constant delay. Aim. This study is aimed at developing and numerically solving the problem of optimal control of the glycemic profile in patients with type 1 diabetes mellitus by insulin therapy, based on the optimality conditions for nonsmooth systems with a constant delay in the phase variable. Methods. The general research technique of the problem under study is based on the mathematical theory of optimal control, the theory of numerical methods, the theory of differential equations with a nonsmooth right-hand side and with a lagging argument. When implementing the software package, the methods of object-oriented design are used. Results. In this study, on the basis of the initial dynamic model proposed by N.A. Shirokova, the problem of optimal control of the dynamics of glycemic regulation in patients with type 1 diabetes mellitus with a nonsmooth right side and a constant delay in the phase variable is constructed. On the basis of the necessary optimality condition obtained for the constructed optimal control problem, algorithmic and software tools have been developed, with the help of which optimal programs are obtained, and their meaningful interpretation is presented. Conclusion. The results obtained on the basis of the software implementation of numerical algorithms of the developed nonsmooth problem of optimal control of the insulin-glucose balance with a constant lag in the phase variables, make it possible to obtain the data that are necessary for monitoring the situation regarding the change in the glycemic profile, for predicting diabetes mellitus and choosing an effective treatment.