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

Pic.3. Periodic structure of calculating experiment.

Consider a way out “teacher-student” systems as her response to managed and unmanaged out The model can be expressed as a combination of two processes: and
The state of the system as a certain (internal) characteristic of the system, the value of which currently determines the current value of the output value (knowledge, teachings). The state can be considered as a kind of information repository (knowledge) necessary to predict the influence of the present on the future. This means the existence of such a mapping. what
An explicit dependence on introduced to take into account the possibility of a change in the dependence of the output on the state over time. This mapping is called the output mapping.
To build a model, we describe the relationship between the input and the state, i.e. introduce parametric mapping family set for all parameter values and . This means that the state at any time uniquely determined by the state in the moment and segment implementation of the entrance from till . This mapping is called transitive mapping.
The study of the learning process as a cybernetic research system (theoretically) solves the problems of developing a mathematical model of the control object, identifying the special values ​​of feedback in the system, emphasizing optimal control and system synthesis, transmitting and processing information and its quantitative description, developing a computational experiment .
Optimum software control of the learning process using the Lagrange-Pontryagin method for continuous learning processes is given in [5-7].
Requires optimal way to get from the point to the point , where .In the first approximation, we take the straight line connecting the starting and ending points
(1)
To calculate the optimal programmed control and optimal trajectory, the following formulas were proposed in [7–9]:
, (2)
(3)
where is the additional variable
(4)
In the same way, optimal software control is calculated for self-learning. and optimal trajectory
, (5)
(6)
where we take as an additional variable
(7)
As specific process parameters: , with a total load given by the teacher - 30 hours. Then the graphs of optimal programmed control and optimal trajectory for different initial conditions are presented in Pic. 4, curve 1 –