Hamilton-Jacobi-Bellman PDEs minimize trajectory time under a range of time dependent external “control” inputs u(t). This is a feed-forward, open lop control. Read Bardi, Capuzzo and Dolcetta 1997.
“arg min” means that the `value function' is minimized under a range of variations of the “argument” u(t).
For Hodgkin-Huxley they minimize the power of electrical control input. That is done by Euler-Lagrange equations, with Lagrange multipliers of form λ_1(t)(eqs of motion), λ_2(t)(power constraint). They solve these by the shooting method.