Singular control explained

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) = \phi(x, λ ,t)u + … 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 \phi(x, λ ,t) , specifically:u(t) = \begin{cases} b, & \phi(x, λ ,t)< 0 \ ?, & \phi(x, λ ,t) =0 \ a, & \phi(x, λ ,t)> 0 .\end{cases}

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 _{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 is to repeatedly differentiate \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 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:^[1] ( -1 ) ^{k} (\partial)/(\partial u) \left[ {\left( (d)/(dt) \right)} ^{2k} H _{u} \right] \ge 0 , k =0 , 1 , …

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.

External links

• Book: Arthur E. Jr. . Bryson . Yu-Chi . Ho . Applied Optimal Control . Waltham . Blaisdell . 1969 . Singular Solutions of Optimization and Control Problems . 246–270 . 9780891162285 . https://books.google.com/books?id=P4TKxn7qW5kC&pg=PA246 .

Notes and References

1. M. I. . Zelikin . Mikhail Zelikin . V. F. . Borisov . Singular Optimal Regimes in Problems of Mathematical Economics . Journal of Mathematical Sciences . 2005 . 130 . 1 . 4409–4570 [Theorem 11.1] . 10.1007/s10958-005-0350-5 . 122382003 .