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 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)) + T 0 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)). 1 Theorem 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 function q (t) maximizes (1) and satisfies Collections: Engineering