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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