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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 {\displaystyle u} , i.e., is of the form: H = ϕ u + ⋯ {\displaystyle H=\phi u+\cdots } and the control is restricted to being between an upper and a lower bound: a ≤ u ≤ b {\displaystyle a\leq u\leq b}. To minimize H {\displaystyle H} , we need to make u {\displaystyle u} as big or as small as possible, depending on the sign of ϕ {\displaystyle \phi } , specifically:
If ϕ {\displaystyle \phi } 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 {\displaystyle b} to a {\displaystyle a} at times when ϕ {\displaystyle \phi } switches from negative to positive.
The case when ϕ {\displaystyle \phi } remains at zero for a finite length of time t 1 ≤ t ≤ t 2 {\displaystyle t_{1}\leq t\leq t_{2}} is called the singular control case. Between t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} the maximization of the Hamiltonian with respect to u {\displaystyle 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 {\displaystyle \partial H/\partial 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 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} the control u {\displaystyle 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: