Shota Rustaveli National Science Foundation of GeorgiaStart Date: 2017-11-29 End Date: 2019-12-01
The first part of the project deals with proving of the variation formulas of solutions for the nonlinear controlled functional differential equations with several constant delays in the phase coordinates and the continuous initial condition. Linear representation of a solution variation with respect to perturbations of the initial data is called the variation formula of solution and the discontinuous initial condition means that the values of the initial function and trajectory do not coincide at the initial moment. The variation formula of solution plays the basic role in proving the necessary conditions of optimality and under sensitivity analyses of differential models. In the project , an order of estimation of the increment of a solution is established with respect to small parameter and the value of the increment is calculated at the initial moment when variation of the initial moment take place from the right or from the left. In an neighborhood of the final point of the main interval the variation formulas of solutions are proved for the three cases when variation of the initial moment take place from the right or from the left and from both sides. In the formulas are detected the effects of variations of the initial moment and delays and the discontinuous initial condition, the form of equation in “variations” established. For Marchuk’s immune response model the variation formulas are derived and the sensitivity analysis of the model is performed.
The second part of the project deals the nonlinear optimization problems with several constant delays in the phase coordinates and controls with the general boundary conditions and functional. Using the variation formulas are proved the following necessary optimality conditions: for the initial and final moments in the form of inequalities and equalities; for delays in the form of inequalities and equalities; for the initial vector in the form of equality, and for the initial function and control function in the form of the linearized integral and pointwise maximum principle. The general optimality conditions are concretized : for the optimization problem with the fixed ends and integral functional; for the optimization problem with the free right-hand side and integral functional; for time optimization problem; for the linear optimization problem. For the optimization problem with the free right-hand side corresponding to Marchuk’s model the necessary optimality conditions are obtained and structures of the optimal control and initial function are established and computed simulation of test optimization problems are performed.
For the nonlinear functional differential equation with the discontinuous (continuous) initial condition it is proved the variation formulas of solution for a wide class of variations, where the signs of the delay parameters and the initial moment variations are independent of each other. There is considered three cases where the initial moment's variation occurs from left, right, and both side. For the nonlinear optimization problem, with the general boundary condition and the functional containing the delays in both the phase coordinates and the controls, the necessary optimization conditions are proved. The results obtained in the project represent new knowledge in the theory of functional differential equations and optimal control. The results obtained within the project and dissertation were reported in scientific seminars: TSU I. Vekua Institute of Applied Mathematics (VIAM); University of Nantes Jean Lerey Math Laboratory (France); International conferences: Tbilisi, Cyprus, Baku; Shanghai and Beijing at two conferences in China; Israel. During the reporting period of the project, 7 papers were performed, including 2 papers published in 2018, 2 papers are published in impact journals in 2019, and 3 papers are published in the materials of the international conferences. With the financial support of the Foundation, I was in France, in the Mathematics Laboratory of the University of Nantes, with Professor A. Nachaoui to carry out some scientific works within the project , who later became the co-supervisor of my dissertation. 1 scientific article has been prepared together with Nachaoui, which was presented in journal with Impact factor. With the financial support of Shota Rustaveli Georgian National Science Foundation, it became possible to submit a dissertation in a timely manner. One important step in this direction has already been taken, namely, the dissertation was reviewed on October 24, 2019, at the meeting of the Dissertation Standing Sectoral Commission of the TSU Mathematics Department.