Vector notation

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

$u(t)=\{\begin{array}{ll}b,& \varphi (x,\lambda ,t)<0\\ ?,& \varphi (x,\lambda ,t)=0\\ a,& \varphi (x,\lambda ,t)>0.\end{array}$

If $\varphi$ 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 $\varphi$ switches from negative to positive.

The case when $\varphi$ remains at zero for a finite length of time $t_1\le t\le t_2$ 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 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. 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:

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

. 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.

References

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