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

:

If $\backslash phi$ 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 $\backslash phi$ switches from negative to positive.

The case when $\backslash phi$ remains at zero for a finite length of time $t\_1\backslash le\; t\backslash le\; t\_2$ 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 is to repeatedly differentiate $\backslash partial\; H/\backslash partial\; u$ 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 $t\_1$ and $t\_2$ the control $u$ is determined by the requirement that the singularity condition continues to hold. The resulting so-called singular arc, if it is optimal, will satisfy the Kelley condition:{{cite journal |first1=M. I. |last1=Zelikin |author-link=Mikhail Zelikin |first2=V. F. |last2=Borisov |title=Singular Optimal Regimes in Problems of Mathematical Economics |journal=Journal of Mathematical Sciences |year=2005 |volume=130 |issue=1 |pages=4409–4570 [Theorem 11.1] |doi=10.1007/s10958-005-0350-5 |s2cid=122382003 }}

:$(-1)^k\; \backslash frac\{\backslash partial\}\{\backslash partial\; u\}\; \backslash left[\; \{\backslash left(\; \backslash frac\{d\}\{dt\}\; \backslash right)\}^\{2k\}\; H\_u\; \backslash right]\; \backslash ge\; 0\; ,\backslash ,\; k=0,1,\backslash cdots$

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.