Samsung Electronics Co. rested its case in a patent-infringement trial with Apple Inc. (Nasdaq: AAPL) after putting on witnesses who said the iPhone maker would owe the South Korean company as much as $421.8 million in royalties.

Vincent O’Brien of OSKR LLC, a damages expert testifying on Samsung’s behalf, told the jury Thursday in federal court in San Jose, California, that he calculated royalties of $22.8 million based on claims that Apple infringed three of Samsung’s feature patents.

A second expert, University of California, Berkeley business school professor David Teece, testified that damages for Apple’s infringement of two other Samsung patents should be in the range of $290 million to $399 million, based on royalty rates of 2 percent to 2.75 percent.

Apple sued Samsung in April 2011, accusing it of copying patented designs for mobile devices, and Suwon, South Korea- based Samsung countersued. The case is the first to go before a federal jury in a battle being waged on four continents for dominance in a smartphone market valued by Bloomberg Industries at $219.1 billion.

The trial began July 30 and U.S. District Judge Lucy Koh limited each side to 25 hours to present their cases. Jurors are now hearing rebuttal witnesses and Koh said she wants jury deliberations to begin by Aug. 21

CEO Talks

The judge urged the companies to narrow the claims in their dispute and directed them to report to the court on talks between their respective chief executives that she proposed yesterday.

Koh said she remains “pathologically optimistic” the companies can settle. Separately, in the event the jury is left to decide the case, she said she wants a statement from the companies by Aug. 18 reporting whether “there has been some successful horse trading” to streamline and simplify claims.

During cross-examination of Samsung’s second damages witness, Joe Mueller, a lawyer representing Apple, showed Teece a July 25, 2011, letter from Samsung to Apple proposing Apple pay a 2.4 percent royalty rate to license any of 86 patents. Teece had said a 2.4 percent rate amounts to $349 million in damages.

‘No Idea’

Mueller got Teece to agree he had “no idea” how Samsung arrived at the rate. Samsung had never published the rate, and “you have no evidence that Samsung has ever asked any other company for such a rate,” Mueller said.

Apple said in a pretrial filing that the 2.4 percent rate is “unfair, unreasonable, and discriminatory.”

Samsung “has never sought or received” the 2.4 percent royalty “from any licensee, and indeed cannot even explain where that number came from,” Apple argued in the July 25 filing. “Samsung’s royalty demands are multiple times more than Apple has paid any other patentees for licenses to their declared-essential patent portfolios,” Apple said.

Samsung countered in its own filing that long before Apple announced the release of any of its products using Samsung’s technology, Samsung offered a “fair and reasonable” royalty rate on its patents to “virtually every major player in the mobile phone industry,” including Apple.

Royalty Rates

Samsung’s offer is “consistent with the royalty rates other companies charge for use of their standards-essential patents,” Samsung said in its filing. Apple “simply rejected Samsung’s opening offer, refused to negotiate further and to this day has not paid Samsung a dime for Apple’s use of Samsung’s standards-essential technology.”

Apple, based in Cupertino, California, claims it is owed at least $2.5 billion in damages and is also seeking to make permanent a preliminary ban it won on U.S. sales of a Samsung tablet, and extend the ban to Samsung smartphones.

The case is Apple Inc. v. Samsung Electronics Co. Ltd., 11- cv-01846, U.S. District Court, Northern District of California (San Jose).