- Singular control
In
optimal control , problems of singular control are problems that are difficult to solve because a straightforward application ofPontryagin's minimum principle fails to yield a complete solution. Only a few such problems have been solved. The most well known is probablyMerton's portfolio problem infinancial economics . 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: ale 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) = egin{cases} b, & phi(x,lambda,t)<0 \ ?, & phi(x,lambda,t)=0 \ a, & 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_1le tle 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 would be to repeatedly differentiate 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).
The term bang-singular control refers to a control that has a bang-bang portion as well as a singular portion.
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