or a difference equation 
x =f 
nt+1 re mn? a) 
They might even take the form of an integro-differential-difference 
equation or a time delay equation, but they cannot take on a form such 
that, the solution at some time ty is dependent on the solution in the 
future, t > t,. The dynamic equations must reflect this principle of 
al 
nonanticipation. One does not violate this principle by choosing a 
control in anticipation of the future and thus influencing the future 
state of the system based on estimated future information; in fact, the 
choice of such a control is actually based on the history of the state 
of the system available at the time of the choice. 
If no further specification of system performance is given, every 
control function which yielded a physical realizable state of the system 
for t > to would be a solution to the control problem. One can have a 
meaningful control problem only if there is a desired objective, a goal 
to be achieved by the process. Moreover, it is not sufficient merely to 
have a goal; there must be a control by which this goal can be achieved. 
This (controll Could be the case of no controll, £(t, x, 1) = (es =)5 
however, it must exist. Since it is not the purpose of these notes to 
delve into all the mathematical problems, it will be assumed that there 
exists at least one control by which the objective can be achieved. It 
will further be assumed that any control function used in the sequel 
yields a unique state function x (t) with x (to) = X35 the state func- 
tion is obtained by solving the dynamic equations. 
In general, there are a number of controls which could yield the 
desired system state. From among this set of possible controls, one 
would like to choose the "best" control with respect to some performance 
criterion. For example, one would like to choose the control so that 
the process is carried out with a minimum cost in fuel, or time, or 
money. In the sequel, it is assumed that the performance criterion can 
be expressed in terms of a cost function; furthermore, it is assumed 
that the cost function is additive with respect to the contribution from 
each time interval. An example of such a cost function is 
