Gen. Martin Dempsey, bottom left, chairman of the Joint Chiefs of Staff, and military officials arrive to brief members of the U.S. Senate on the White House strategy to combat the Islamic State September 11, 2014 at the U.S. Capitol in Washington. (Win Mcnamee/Getty Images)

In declaring that the United States would degrade and “ultimately destroy” an al-Qaeda offshoot in Iraq and Syria, President Obama articulated an objective that the United States has yet to achieve against any of the Islamist adversaries it has faced since the Sept. 11, 2001, attacks.

Through two wars, thousands of drone strikes and hundreds of covert operations around the world, the United States has substantially weakened al-Qaeda and its affiliates, eroding their capabilities in ways that have reduced the threat they pose to the United States.

The scope of that conflict is poised to expand again as U.S. military officials said Thursday that they were given new authority to begin targeting leaders of an al-Qaeda rival known as the Islamic State.

But even as Obama warned that “it will take time to eradicate a cancer” like the Islamic State, the timing of his remarks — coming 13 years after the attacks on the World Trade Center and the Pentagon — underscored how elusive the finish line has been for the United States in a series of conflicts that have come to resemble a permanent war.

Although the conventional wars in Iraq and Afghanistan have essentially concluded, the United States is still battling al-Qaeda affiliates in countries including Pakistan, Somalia and Yemen. None of those groups has been eradicated, or even degraded to a degree that would allow U.S. counterterrorism operations to end.

View Graphic U.S. airstrikes in Iraq targeting Islamic State

The only apparent exception to this pattern had been al-Qaeda in Iraq, a group that was seen as virtually dismantled until its reincarnation as the Islamic State. After conquering parts of Syria and Iraq in the span of six months, and beheading two U.S. journalists, the group is again in U.S. crosshairs.

“We’re not going to see an end to this in our lifetime,” said Charles F. Wald, a retired Air Force general who oversaw the start of the air war in Afghanistan in 2001. Airstrikes and ground operations by allies can degrade the Islamic State and force it to surrender its territorial gains, Wald said. But “there isn’t going to be any time where we all of a sudden can declare victory. This is what the world is going to be like for us for a long time.”

A day after Obama’s speech, U.S. intelligence officials said the CIA has completed a new assessment of the Islamic State’s strengths, showing that it has more than doubled in size in recent months.

“CIA assesses the Islamic State of Iraq and the Levant (ISIL) can muster between 20,000 and 31,500 fighters across Iraq and Syria, based on a new review of all-source intelligence reports from May to August, an increase from our previous assessment of at least 10,000 fighters,” according to a CIA spokesman, using another name for the group, which he discussed on the condition of anonymity. “This new total reflects an increase in members because of stronger recruitment since June following battlefield successes and the declaration of a caliphate, greater battlefield activity, and additional intelligence.”

At the same time, details of how the Pentagon will pursue the new offensive began to emerge. U.S. military officials said they have new authority to carry out strikes against the group’s leaders, including Abu Bakr al-

Baghdadi, who earlier this year declared himself the head of a restored caliphate.

Such targets had been off-

limits under the more narrow terms of an air campaign that Obama had described as a humanitarian effort to protect members of religious minorities and also shield American diplomats from Islamic State fighters in Iraq.

Pentagon officials described their altered mission as a shift to offense from defense. “We’re going to intensify our efforts inside Iraq, there’s no question about that,” said Adm. John Kirby, the Pentagon press secretary.

In remarks to reporters Thursday, Kirby declined to answer directly when asked whether the Pentagon now had a green light to hunt down individuals, but he said: “One of the ways you get at, and you destroy the capabilities of an enemy like ISIL is to be pretty aggressive against them. And that does include disrupting their ability to command and control, and to lead their own forces.”

Kirby said the 475 additional U.S. troops that Obama ordered to Iraq will arrive over “the next week or so.” Among them are a detachment of about 125 personnel who will operate armed U.S. surveillance aircraft for the first time from Irbil, the capital of Iraq’s semiautonomous Kurdish region. Kirby said that officials are still considering which types of aircraft to send to Irbil but that the fleet will not include drones.

The campaign is aimed at a terrorist group whose rapid expansion and brutal tactics have alarmed Western security officials, although there is significant confusion and debate over how significant a threat it poses to the United States. The Islamic State has seized cities in Syria and northern Iraq and amassed cash and weapons at a rate eclipsing any al-Qaeda rival. But so far it has not been tied to a transnational terror plot.

Obama conceded that point during his speech, but he warned of a growing danger if the Islamic State was left unchecked.

The group is a descendant of al-Qaeda in Iraq, a violent terrorist force founded by Abu Musab al-Zarqawi that sought to ignite a sectarian conflagration in that country before it was subdued by Sunni tribal leaders who were dismayed by its tactics and backed by U.S. cash and commando teams. The group’s collapse was so complete that U.S. intelligence agencies estimated it had lost 95 percent of its membership and strength by the time U.S. forces left Iraq in 2011.

But the group’s remnants relocated to Syria and took advantage of the chaos created by civil war there — as well as Sunni discontent with the Shiite-dominated government in Iraq — to regroup. This year, it severed ties with al-Qaeda and rebranded itself the Islamic State.

“Defeating a group doesn’t necessarily mean you will have been successful at eradicating every single person who was ever aligned with the group,” Matt Olsen, the director of the National Counterterrorism Center, said during a recent briefing for reporters on the Islamic State. “Even if you’ve been successful at eliminating the threat that they pose, it doesn’t mean every single person ever affiliated with the group will have adopted a new worldview.”

Supporters of Obama said his terms were aimed to a large extent at preparing the nation for another extended conflict. “Rallying the public to a war effort is hard to do if you’re only talking about degrading an enemy,” said Rep. Adam B. Schiff (D-Calif.), a member of the House Intelligence Committee.

“Within a matter of years we can get to a point where ISIL no longer poses a significant threat to the homeland or region,” Schiff said. “In terms of extinguishing it as a group that bears any resemblance to the Islamic State? That could take much longer.”

Obama cited U.S. counterterrorism operations against al-Qaeda affiliates in Yemen and Somalia as models for the strategy against the Islamic State. But the comparison is problematic because key factors in those countries don’t exist in Syria, including effective allies on the ground and free rein for U.S. aircraft including armed drones.

In Somalia, the United States has spent much of the past decade organizing and largely paying for an international proxy force of about 18,000 African Union troops to counter the al-Qaeda linked group al-Shabab.

The African force has gradually wrested territory away from the Islamist group, while the U.S. military has intermittently targeted leaders with raids and drone strikes, including a Sept. 1 air attack that killed its co-founder.

In Yemen, the United States has virtually unchecked authority to patrol the country’s skies with armed drones. Yemeni counterterrorism forces are backed by U.S. funding, training and intelligence.

Experts said the United States can expect similar cooperation from Iraq, but not Syria. President Bashar al-Assad’s military has lost territory but almost certainly retains control of the country’s airspace, said James O. Poss, a retired U.S. Air Force major general. Even if U.S. drones were able to evade Syria’s ground-based missile batteries, the Syrian air force could easily scramble fighter jets to chase and shoot down unmanned aircraft, which fly at relatively slow speeds.

To send drones into Syria, “we’d have to do so with the tacit approval of the Syrian air force,” Poss said. “It depends on how strident the Syrians would want to be in defending their airspace.”

Even with drones and ground forces, the campaigns that Obama cited in Yemen and Somalia are far from over. Al-

Shabab killed 67 people last year in a three-day siege of a crowded Nairobi mall. Al-Qaeda in the Arabian Peninsula, as the Yemen-based affiliate is known, has gone several years without launching a major attack but is still seen as the most direct danger to the United States, surpassing that posed by the Islamic State.

“We’ve not seen any sign that their interest or capability has abated,” Olsen said.