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| 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.
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| The most common difficulty in applying Pontryagin's principle arises when the Hamiltonian depends linearly on the control <math>u</math>, i.e., is of the form: <math>H(u)=\phi(x,\lambda,t)u+\cdots</math> and the control is restricted to being between an upper and a lower bound: <math>a\le u(t)\le b</math>. To minimize <math>H(u)</math>, we need to make <math>u</math> as big or as small as possible, depending on the sign of <math>\phi(x,\lambda,t)</math>, specifically:
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| : <math>u(t) = \begin{cases} b, & \phi(x,\lambda,t)<0 \\ ?, & \phi(x,\lambda,t)=0 \\ a, & \phi(x,\lambda,t)>0.\end{cases}</math>
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| If <math>\phi</math> 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 <math>b</math> to <math>a</math> at times when <math>\phi</math> switches from negative to positive.
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| The case when <math>\phi</math> remains at zero for a finite length of time <math>t_1\le t\le t_2</math> is called the '''singular control''' case. Between <math>t_1</math> and <math>t_2</math> 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 <math>\partial H/\partial u</math> 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 <math>t_1</math> and <math>t_2</math> the control <math>u</math> 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''':
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| :<math>(-1)^k \frac{\partial}{\partial u} \left[ {\left( \frac{d}{dt} \right)}^{2k} H_u \right] \ge 0 ,\, k=0,1,\cdots</math>
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| .<ref>Bryson, Ho: Applied Optimal Control, Page 246</ref> This condition is also called the generalized [[Legendre-Clebsch condition]]). | |
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| 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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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Singular Control}}
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| [[Category:Control theory]]
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