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

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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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