can any one explain
1. what is singular control ?
2.what is the difference between normal control and singular control in stochastic optimal control theory.
3. what is the role of singular control in stochastic optimal control theory?
Singular control is the control strategy when, in an optimal deterministic control problem, the Hamiltonian is independent of u(t) for an interval [t1,t2]. Thus, the intantaneous optimal control input u(t) on the interval cannot be determined from the first order necessary condition for optimality. The control strategy on this interval is called singular control. It may be determined based on higher order necessary conditions for example. I am not sure how the Hamiltonian method can be applied to stochastic optimal control (there are papers on it, but I don't understand them). Therefore, it is not clear how singular control can be defined for stochastic optimal control.
Dear Professor,
I suggest to see a basic papers by Haussmann, Cadenilas and the book of Oksendal et al. (see the attached files); for singular control, impulsive control and the regular controls with their applications in finances.
(1) A singular stochastic control η(⋅) is of bounded variation, non-decreasing continuous on the left with right limits and η(0₋)=0, and E[sup_{t∈[0,T]}|u(t)|²+|η(T)|²]
Hallo. In the deterministic case singular control is such control that first order optimality condition degenerate, not give information about extremum. In this case it must be get second order necessary opt conditions
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