COLUMBUS, Ohio — The White House the prize, President Barack Obama and Mitt Romney raced through a final full day of campaigning on Monday through Ohio and other battleground states holding the keys to victory in a tight race. Both promised brighter days ahead for a nation still struggling with a sluggish economy and high joblessness.

“Our work is not done yet,” Obama told a cheering crowd of nearly 20,000 in chilly Madison, Wis., imploring his audience to give him
another four years.

Romney projected optimism as he neared the end of his six-year quest for the presidency. “If you believe we can do better. If you believe America should beon a better course.

If you’re tired of being tired … then I ask you to vote for real change,” he said in a Virginia suburb of the nation’s capital. With many of the late polls in key states tilting slightly against him, he decided to campaign on Election Day in Ohio and Pennsylvania, where he and Republicans made a big, late push.

The presidency aside, there are 33 Senate seats on the ballot Tuesday, and according to one Republican official, a growing sense of resignation among his party’s rank and file that Democrats will hold their majority.

The situation was reversed in the House, where Democrats made no claims they were on the verge of victory in pursuit of the 25 seats they need to gain control.

National opinion polls in the presidential race made the popular vote a virtual tie.

In state-by-state surveys, it appeared Obama held small advantages in Nevada, Ohio, Iowa and Wisconsin — enough to deliver a second term if they endured, but not so significant that they could withstand an Election Day surge by Romney supporters. Both men appealed to an ever smaller universe of undecided voters.

More than 30 million absentee or early ballots have been cast, including in excess of 3 million in Florida. The state also had a legal controversy, in the form of a Democratic lawsuit seeking an extension of time for pre-Election Day voting.

There were other concerns, logistical rather than legal.

Officials in one part of New Jersey delivered voting equipment to emergency shelters so voters displaced by Superstorm Sandy last week could cast ballots. New York City made arrangements for shuttle buses to provide transportation for some in hard-hit areas unable to reach their polling places.

In his longest campaign day, Romney raced from Florida to a pair of speeches in Virginia to Ohio and then an election eve rally in New Hampshire.

Obama selected Wisconsin, Ohio and Iowa for his final campaign day, an itinerary that reflected his campaign’s decision to try to erect a Midwestern firewall against Romney’s challenge. Vice President Joe Biden and Republican running mate Paul Ryan of Wisconsin went through their final campaign paces, as well.