The Computation and Theory of Optimal Control (Enhanced Edition‪)‬

The theory of optimization is a subject that has received a great deal of
attention from the days of the early Greek mathematicians to the present
time. In different ages different facets of the subject have been emphasized,
depending upon the influences of prevailing technology. Today, optimization
is highly influenced by certain engineering applications, for example,
the design of optimal control systems and the computation of optimal
trajectories for space vehicles. One distinguishing feature of modern
optimization theory is the emphasis on closed-loop, as opposed to openloop,
solutions. The differences between the two types of solution are best
described by means of a simple example. Consider the problem of
choosing some parameter a to maximize a functionl(x, a), where x is some
additional parameter that is specified and will be referred to as the state
variable. Clearly the optimal choice of 01, a*, will depend upon the variable
x, i.e., a* = a*(.). A particular solution 0 1 ~= a*(xl) is defined by a
specific value of x, x1 . The first form of the solution, 01*(x), is referred to
as the closed-loop solution, and it specifies the optimal solution for all
values of the state variable x. The second form of the solution, al, is
the open-loop solution, and it specifies the solution for a distinct value
of the state variable.