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 ${\displaystyle u}$, i.e., is of the form: ${\displaystyle H(u)=\phi (x,\lambda ,t)u+\cdots }$ and the control is restricted to being between an upper and a lower bound: ${\displaystyle a\leq u(t)\leq b}$. To minimize ${\displaystyle H(u)}$, we need to make ${\displaystyle u}$ as big or as small as possible, depending on the sign of ${\displaystyle \phi (x,\lambda ,t)}$, specifically:

${\displaystyle u(t)={\begin{cases}b,&\phi (x,\lambda ,t)<0\\?,&\phi (x,\lambda ,t)=0\\a,&\phi (x,\lambda ,t)>0.\end{cases}}}$

If ${\displaystyle \phi }$ 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 ${\displaystyle b}$ to ${\displaystyle a}$ at times when ${\displaystyle \phi }$ switches from negative to positive.

The case when ${\displaystyle \phi }$ remains at zero for a finite length of time ${\displaystyle t_{1}\leq t\leq t_{2}}$ is called the singular control case. Between ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ the maximization of the Hamiltonian with respect to ${\displaystyle u}$ gives us no useful information and the solution in that time interval is going to have to be found from other considerations. One approach is to repeatedly differentiate ${\displaystyle \partial H/\partial u}$ with respect to time until the control u again explicitly appears, though this is not guaranteed to happen eventually. One can then set that expression to zero and solve for u. This amounts to saying that between ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ the control ${\displaystyle u}$ 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

${\displaystyle (-1)^{k}{\frac {\partial }{\partial u}}\left[{\left({\frac {d}{dt}}\right)}^{2k}H_{u}\right]\geq 0,\,k=0,1,\cdots }$

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.