guidance, flight mechanics and trajectory optimization

2.4.8 The Problem of Bolza
The preceding sections have delt with the Dynamic Programming formulation
of the problem of Lagrange. In this section the Bolza Problem will be
considered, since optimal trajectory and control problems are usually cast
in this form. The Bellman equation for this case will be-developed and
some solutions presented. Also, some comparisons and parallels will be
drawn between the Dynamic Programming approach and the Pontryagin Maximum
Principle (Ref. 2.4.3).
The problem of Bolza is usually stated in the following form: given
the dynamical system
ii = J(%,Lo ; L’=/,R (2.4.81A)
or in the vector notation
where the state x is a n-dimensional vector,
and the control u is a r-dimensional vector,
i= f(X) u) (2.4.81~)
which is required to lie in some closed set u in the r-dimensional control
space; determine the control history u(t) for which the functional
J =s” $
4
(x,u)dt +$(x~,$) = minimum
is minimized subject to the terminal constraints
82
(2.4.82)
(2.4.83)
(2.4.84)
(2.4.85)