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. 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 aleq 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: 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 tleq 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 would be to repeatedly differentiate ∂ H / ∂ u {displaystyle 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 {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 will be optimal if it satisfies the Kelley condition: 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.