Summary: Simplest problem of the calculus of variations with end cost
We consider a variation on the problem stated in Section 2.2. The value of the state at time T is
not prescribed, rather there is a penalty on the final state. For notational convenience we denote
the state by q instead of x.
The problem is as follows:
Determine a function q(t), defined on the interval [0, T], such that the integral
J(q(·)) = S(q(T)) +
F(t, q(t), q(t))dt (1)
is maximized (or minimized), and where q(t) in addition satisfies the boundary condition
q(0) = q0 (2)
for given q0.
The following result is similar to Theorem 2.2.3. The difference is that the condition x(t) = xT is
now replaced by a penalty S(x(T)).
Consider the simplest problem in the calculus of variations and suppose Assumptions 2.2.1 and
2.2.2 are met. Then a necessary condition that a C2
(t) maximizes (1) and satisfies