Singular control

Article summary

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,\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 \phi(x,\lambda,t) , specifically: : u(t) = \begin{cases} b, & \phi(x,\lambda,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.…

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