Innovative approach of mathematical modeling of learning processes as an optimal control object

Pic. 4. Graph of optimal program control in time.

Pic. 5. Graph of the optimal trajectory for program control in
of time.

curve 2 – , On the picture 5, curve 1 – curve 2 –
The training load of the student with this program management is determined by the area of ​​the integrand curve in Pig. 4 and is equal to 120 hours (at a given teacher at 30 hours), therefore, for a real learning process, feedback control is also necessary.
We are talking about deterministic systems with “continuous time”, i.e. about systems whose evolution is described by differential equations. But of no less importance are systems with “discrete time”. Their role is determined not only by the fact that when constructing computational procedures we always carry out discretization of the variables of the problem posed - we replace differential equations with finite-difference ones. These are primarily multi-step decision making tasks. Almost always the development of socio-economic systems is described by finite-difference equations. The discretization step is determined by the cycle of the educational process (month, quarter, semester, academic year).
Conclusion
The proposed mathematical model of learning based on the control theory can be useful, first of all, by the manager of the teaching process and teachers. On the basis of the coefficients of learning and forgetting, determined with the help of special tests, it is possible to predict in some approximation the level of current knowledge of both the individual student and the group of students (or the flow of students). Thus, the learning process can be controlled more accurately compared to the traditional approach.
The use of these solutions in practice will improve the quality of education and preserve the knowledge of graduates of educational institutions in the long-term plan with a minimum load of teaching staff.
Based on the proposed mathematical models of optimal control, you can create an automated control system that allows you to optimally plan the pedagogical process.
The presented model of optimal control can be used by managers of different levels of education management to solve applied problems of analysis, management and regulation of the educational process.
The theory of mathematical modeling can be used in advanced training courses for engineering and technical workers of industrial enterprises by industry.