# The problem is to solve optimal control problem by economy optimality for the process

### Содержание работы

3. The problem is to solve optimal control problem by economy optimality for the process for minimum of expences when the initial conditions are: , .

Solution

1)  First of all we must write phase coordinates: So, this process can be described by the system of differential equations: where - control parameter.

2)  Write Hamilton-Pontryagin function for our case: ,

where - auxiliary functions.

For finding of control parameter let’s use the maximal principal, which states that in optimal process Hamilton-Pontryagin function gets maximum value, for existing of which the requirement is following: So, we have for our case:  For finding of auxiliary functions let’s use Hamilton-Pontryagin equation in general form are the following:  Let’s write it for our case:     Substituting the got values of auxiliary functions in formula for finding of control parameter we obtain: ,

Let’s denote our constant in more compact view where, , .

3)  Let’s return to initial coordinates and synthesize the phase trajectory . , so we have     4)  Let’s use the initial conditions: and Because of , we will have , that’s why we eliminate one constant, it will be equal to . , thus we eliminate the second constant .  From the system of two last equations we found the rest of constants  5)  So, we have the control parameter and the equation of phase trajectory (fig. 3.1):  Fig. 3.1 Phase trajectory

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