That depends on the system. Such an equilibrium must exist in that system and be reachable in that system and if you want something to "come to rest there" the equilibrium must also be stable.

For any system you need to characterize the equilibria and trajectories and see what's there. Sometimes you might not like any of the options. Sometimes you might want to go for some trajectory rather than an equilibrium.

Systems are all different. You need to characterize the one you're interested in and see.