Transversality conditions for infinite horizon optimality: Higher order differential problems Original Research Article

Michel [P. Michel, Some clarifications on the transversality condition, Econometrica 58 (1990) 705–723] and Ekeland and Scheinkman [I. Ekeland, J.A., Scheinkman, Transversality conditions for some infinite horizon discrete time optimization problems, Mathematics of Operations Research 11 (1986) 216–229] presented the transversality condition for the first-order differential problems View the MathML sourcemaxx∫0∞v(x(t),ẋ(t),t)dt, which may have unbounded objective functions. Kamihigashi [T. Kamihigashi, Necessity of transversality conditions for infinite horizon problems, Econometrica 69 (2001) 995–1012] showed a generalization of their transversality condition that does not assume concavity. Using the variational approach, this paper deals with higher order differential problems: View the MathML sourcemaxx∫0∞v(x(t),ẋ(t),ẍ(t),…,x(n)(t),t)dt. We derive two conditions: Euler’s condition and the transversality condition, for such problems in a simple manner. They are imperative for solving variational problems. Furthermore, two assumptions are necessary to induce the two conditions. We construct a counterexample in which the transversality condition is not satisfied without the two assumptions.