“From the start, President Obama has stated that the role of the U.S. military would be limited in time and scope,” Secretary of State Hillary Rodham Clinton said Thursday evening in announcing the plan.

But even that agreement — brokered by Mrs. Clinton and the foreign ministers of Britain, France and Turkey — frayed almost immediately over how far the military campaign should go in trying to erode the remaining pillars of Colonel Qaddafi’s power by striking his forces on the ground and those devoted to protecting him. It was salvaged, one diplomat said, only by papering over the differences concerning the crucial question of who actually controls military strikes on Libya’s ground forces.

“There were differences in the scope of what NATO would do and what would remain with the national militaries,” a senior administration official said, expressing hope that the agreement on NATO command would be a step toward resolving them.

The questions swirling around the operation’s command mirrored the larger strategic divisions over how exactly the coalition will bring it to an end — or even what the end might look like, and whether it might even conceivably include a Libya with Colonel Qaddafi remaining in some capacity. While few countries have openly sided with the Libyan leader, officials said on Thursday that most of the allies expected that the use of military force would lead to talks between the government and the rebels.

“I don’t think anyone is ruling out some kind of negotiated settlement,” the official said. Colonel Qaddafi has responded defiantly, making the likelihood of his negotiated departure seem exceedingly remote.