From derivation of Hamilton-Jacobi (HJ) equation one can see that it is only applicable for conserved systems, but from some books and Wikipedia one reads the HJ equation as
$$\frac{\partial{S}}{\partial{t}}+H(q,\frac{\partial{S}}{\partial{q}},t)~=~0$$
there is an extra variable $t$ in Hamilton's function. When Hamilton's function explicitly depends on time , this means that energy is not conserved. So where is truth?
Two points:
The Hamilton-Jacobi equation is a partial differential equation for a function $S(q,t)$. Given a Hamiltonian $H(q, p ,t)$. Even if $H$ explicitly depends on time, there is nothing preventing one from writing down and attempting to solve the Hamilton Jacobi equation given such a Hamiltonian. This leads into the second point.
By "derivation" of the Hamilton Jacobi equation, I'm guessing you're referring to the fact that one can show that Hamilton's principal function (defined in terms of the Lagrangian of a classical system that can depend explicitly on time) satisfies the Hamilton Jacobi equation for the corresponding Hamiltonian. I just looked back at my notes regarding the demonstration of this fact, and I can't see anywhere where it is necessary to assume that the Hamiltonian is not explicitly time-dependent.
