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 {\displaystyle u} , i.e., is of the form: H ( u ) = ϕ ( x , λ , t ) u + ⋯ {\displaystyle H(u)=\phi (x,\lambda ,t)u+\cdots } and the control is restricted to being between an upper and a lower bound: a ≤ u ( t ) ≤ b {\displaystyle a\leq u(t)\leq b} . To minimize H ( u ) {\displaystyle H(u)} , we need to make u {\displaystyle u} as big or as small as possible, depending on the sign of ϕ ( x , λ , t ) {\displaystyle \phi (x,\lambda ,t)} , specifically:
u ( t ) = { b , ϕ ( x , λ , t ) < 0 ? , ϕ ( x , λ , t ) = 0 a , ϕ ( x , λ , t ) > 0. {\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 ϕ {\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 b {\displaystyle b} to a {\displaystyle a} at times when ϕ {\displaystyle \phi } switches from negative to positive. The case when ϕ {\displaystyle \phi } remains at zero for a finite length of time t 1 ≤ t ≤ t 2 {\displaystyle t_{1}\leq t\leq t_{2}} is called the singular control case. Between t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} the maximization of the Hamiltonian with respect to u {\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 ∂ H / ∂ u {\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 t 1 {\displaystyle t_{1}} and t 2 {\displaystyle t_{2}} the control u {\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 ) k ∂ ∂ u [ ( d d t ) 2 k H u ] ≥ 0 , k = 0 , 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.
References
… excerpt ends here. Continue reading the full article.
