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 nal state. For notational convenience we denote the state by q instead of x. The problem is as follows: Determine a function q(t), de ned on the interval [0; T ], such that the integral J(q()) = S(q(T )) + Z T 0 F (t; q(t); _ q(t))dt (1) is maximized (or minimized), and where q(t) in addition satis es the boundary condition q(0) = q 0 (2) for given q 0 . The following result is similar to Theorem 2.2.3. The di erence is that the condition x(t) = x T 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 C 2 function q  (t) maximizes (1) and satis es (2) is that q  (t) is a solution of the di erential equation Collections: Engineering