We will keep di¤erentiating the switching functions to obtain higher order derivatives until
we reach a derivative that will explicitly determine the control. The order of derivative where
thecontrolu appearsforthe…rsttimewillindicatetheorderofthesingularcontrol. Itfollows
from some facts in Lie algebra that the order of this derivative has to be even. Therefore,
the order of singular control will be introduced as 12 of that derivative. For example, if the
order of the derivative is 2n, then the order of the singular control is n. However, …nding
an explicit form of singular control u isn’t su¢ cient in order to conclude the optimality of
the singular control. The Legendre-Clebsch theorem gives an additional necessary condition
stated as follow:
Theorem 8 [1] Considering a singular control u(t) of order k, if u(t) is minimizing the
objective then it satis…es the following property called Legendre Clebsch condition
d2k @H(1)k @  0: (4.32)
@udt2k @u
Remark 9 In the case of optimal control problem where we maximize the objective, the
condition takes the following form
@H(1)k @ d2k  0: (4.33)
@udt2k @u
23
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