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. The most well known is probably Merton's portfolio problem in financial economics. 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: $H(u)=phi(x,lambda,t)u+cdots$ and the control is restricted to being between an upper and a lower bound: $ale\; u(t)le\; b$. To minimize $H(u)$, we need to make $u$ as big or as small as possible, depending on the sign of $phi(x,lambda,t)$, specifically:

:

If $phi$ 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 $b$ to $a$ at times when $phi$ switches from negative to positive.

The case when $phi$ remains at zero for a finite length of time $t\_1le\; tle\; t\_2$ 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 $partial\; H/partial\; u$ 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).

The term bang-singular control refers to a control that has a bang-bang portion as well as a singular portion.