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 u , i.e., is of the form: H ( u ) = ϕ ( x , λ , t ) u + ⋯ and the control is restricted to being between an upper and a lower bound: a ≤ u ( t ) ≤ b . To minimize H ( u ) , we need to make u as big or as small as possible, depending on the sign of ϕ ( x , λ , t ) , specifically:
u ( t ) = { b , ϕ ( x , λ , t ) < 0 ? , ϕ ( x , λ , t ) = 0 a , ϕ ( x , λ , t ) > 0. If ϕ 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 b to a at times when ϕ switches from negative to positive.
The case when ϕ remains at zero for a finite length of time t 1 ≤ t ≤ t 2 is called the singular control case. Between t 1 and t 2 the maximization of the Hamiltonian with respect to u 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 ∂ H / ∂ u 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 t 1 and t 2 the control u 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:
( − 1 ) k ∂ ∂ u [ ( d d t ) 2 k H u ] ≥ 0 , k = 0 , 1 , ⋯ . This condition is also called 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.
