Is the viscosity solution of Hamilton-Jacobi equation of practical use in optimal control?

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.