Conjugate Intervals for the Linear Fixed-Endpoint Control Problem

For certain optimal control problems with piecewise continuous controls, recently Loewen and Zheng (Ref. 1) and Zeidan (Ref. 2) defined two sets of generalized conjugate points for which, under normality assumptions, the second-order conditions in terms of the accessory problem imply their emptiness. However, simple examples show that checking the existence of such points may be more difficult than directly finding variations that make the second variation negative. In this paper, for the linear fixed-endpoint control problem, we introduce a new set whose emptiness is equivalent to the nonnegativity of the second variation along admissible variations. Moreover, we achieve by means of this set the main objective of introducing a characterization of this condition, namely, to obtain a simpler way of verifying it.