We recall that the results in the previous sections are valid for time scales whose backward jump operator has the form , in particular for the time scale .

(i)
The Discrete Case
If and (of special interest the case when ), then our work becomes on the discrete time scale . In this case the functional under optimization will have the form
and that , for where
The necessary condition for to possess an extremum for a given function is that satisfies the following EulerLagrange equations:
Furthermore, the equation
holds along for all admissible variations satisfying , .
In this case the optimalcontrol problem would read as follows.
Find the optimal control variable defined on the time scale , which minimizes the performance index
subject to the constraint
such that
The necessary conditions for this optimal control are
and also
Note that condition (5.9) disappears when the Lagrangian is independent of the delayed derivative of .
Example 5.1.
In order to illustrate our results we analyze an example of physical interest. Namely, let us consider the following discrete action:
subject to the condition
The corresponding EulerLagrange equations are as follows:
We observe that when the delay is removed, that is, , the classical discrete EulerLagrange equations are reobtained.

(ii)
The Quantum Case
If and , then our work becomes on the time scale . In this case the functional under optimization will have the form
where
Using the integral theory on time scales, the functional in (5.14) turns to be
The necessary condition for to possess an extremum for a given function is that satisfies the following EulerLagrange equations:
Furthermore, the equation
holds along for all admissible variations satisfying , .
In this case the optimalcontrol problem would read as follows.
Find the optimal control variable defined on the time scale, which minimizes the performance index
subject to the constraint
such that
where is a constant and and are functions with continuous first and second partial derivatives with respect to all of their arguments.
The necessary conditions for this optimal control are
and also
Note that condition (5.25) disappears when the Lagrangian is independent of the delayed derivative of .
Example 5.2.
Suppose that the problem is that of finding a control function defined on the time scale such that the corresponding solution of the controlled system
satisfying the conditions
is an extremum for the integral functional (quadratic delay cost functional):
According to (5.24) and (5.25), the solution of the problem satisfies
and of course
When the delay is absent (i.e., ), it can be shown that the above system is reduced to a secondorder difference equation. Namely, reduced to
If we solve recursively for this equation in terms of an integer power series by using the initial data, then the resulting solution will tend to the solutions of the second order linear differential equation:
Clearly the solutions for this equation are and . For details see [16].