Singular control
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. A more technical explanation follows. The most common difficulty in applying Pontryagin's principle arises when the Hamiltonian depends linearly on the control formula_1, i.e., is of the form: formula_2 and the control is restricted to being between an upper and a lower bound: formula_3. To minimize  formula_4, we need to make formula_1 as big or as small as possible, depending on the sign of formula_6, specifically: If formula_8 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 formula_9 to formula_10 at times when formula_8 switches from negative to positive. The case when formula_8 remains at zero for a finite length of time formula_13 is called the singular control case. Between formula_14 and formula_15 the maximization of the Hamiltonian with respect to formula_1 gives us no useful information and  the solution in that time interval is going to have to be found from other considerations. (One approach would be to repeatedly differentiate formula_17 with respect to time until the control u again explicitly appears, which is guaranteed to happen eventually. One can then set that expression to zero and solve for u. This amounts to saying that between formula_14 and formula_15 the control formula_1 is determined by the requirement that the singularity condition continues to hold. The resulting so-called singular arc will be optimal if it satisfies the Kelley condition: 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.