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My understanding is, given an optimal control problem, one can show that the optimal cost satisfies a Hamilton-Jacobi PDE and use dynamic programming to figure out the optimal control. However, sometimes this PDE has no strong solution, and the deep theory of viscosity solution was invented to make sense of a "solution" in this situation.

Does such "optimal" cost have any practical meaning? Phrased slightly differently, can one still figure out what the control should be, and implement it?

I feel like it has to be a function, not a distribution, in order to be implementable.

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  • $\begingroup$ Sorry, my original question has a mistake: "one can use dynamic programming to show that the optimal cost satisfies a Hamilton-Jacobi PDE" should be "one can show that the optimal cost satisfies a Hamilton-Jacobi PDE and use dynamic programming to figure out the optimal control". $\endgroup$ – Isley Apr 29 '17 at 22:11
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    $\begingroup$ You can always feel free to edit your question (and please do so instead of commenting. Just use the "edit" button beneath the question and its tags. (I've done the edit for you this time) $\endgroup$ – Wasabi Apr 29 '17 at 23:58

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