In optimal control, problems of singular control are problems that are difficult to solve because a straightforward application of Pontryagin's minimum principle fails to yield a complete solution.
Only a few such problems have been solved, such as Merton's portfolio problem in financial economics or trajectory optimization in aeronautics.
The most common difficulty in applying Pontryagin's principle arises when the Hamiltonian depends linearly on the control
and the control is restricted to being between an upper and a lower bound:
as big or as small as possible, depending on the sign of
is positive at some times, negative at others and is only zero instantaneously, then the solution is straightforward and is a bang-bang control that switches from
switches from negative to positive.
remains at zero for a finite length of time
is called the singular control case.
the maximization of the Hamiltonian with respect to
gives us no useful information and the solution in that time interval is going to have to be found from other considerations.
with respect to time until the control u again explicitly appears, though this is not guaranteed to happen eventually.
is determined by the requirement that the singularity condition continues to hold.
The resulting so-called singular arc, if it is optimal, will satisfy the Kelley condition:[1] Others refer to this condition as the generalized Legendre–Clebsch condition.
The term bang-singular control refers to a control that has a bang-bang portion as well as a singular portion.