“The harm to Apple was deliberate, not accidental,” Apple attorneys said in court papers filed 21 September in the US District Court in San Jose, California.
Samsung “wilfully diluted its trade dress, taking billions in sales in the fast-growing US smartphone market at a key moment in the transition between feature phones and smartphones,” attorneys said.
Cupertino, California-based Apple sued Samsung in April 2011, and Samsung countersued as part of a battle being waged on four continents over a smartphone market valued by Bloomberg Industries at $219.1bn (£1.3bn).
The companies have also sued each other in the UK, Australia and South Korea.
In August, Apple won a jury verdict of more than $1.05bn against Samsung after a finding that Samsung infringed six of seven patents for the mobile devices.
Apple is requesting an enhancement of $135m under the Patent Act and $400m under the Lanham Act, for a combined total of $535m, compared with $268m from the verdict under both statutes, court papers show.
The court may increase the damages up to three times the amount assessed, according to the court papers.