Porsche plans to lift its stake in Volkswagen to 31 percent from 27.3 percent. Under German law, it is required to make an offer for the entire company once it owns more than 30 percent of the shares.

The offer of 100.92 euros ($134.40) a share is the minimum amount Porsche can legally offer other shareholders and is well below Volkswagen"s most recent closing price of 117.70 euros. A Porsche spokesman said it was not seeking to acquire a majority stake in Volkswagen any time soon.

"We expect very few, if any, shareholders to sell us their shares at this price," the spokesman, Michael Baumann, said. "This gives us the freedom to move further without taking other special steps."

The move confirms, however, that Porsche is determined to tighten its grip on Volkswagen, with which it has deep historical links. The two companies cooperate in manufacturing sport utility vehicles, and two of Porsche"s most senior executives serve on Volkswagen"s board.

In a statement, Volkswagen"s chief executive, Martin Winterkorn, said he welcomed Porsche"s increased stake.

Analysts said Porsche was forced to cross this threshold sooner than it might have wished because of the recent rally in Volkswagen"s share price. Many analysts had expected it to wait until a European court issued a ruling on a German law that protects Volkswagen from hostile takeover bids.

The court is expected to strike down the law, which places a cap on the voting rights of shareholders at 20 percent, regardless of how many shares they own. If it does not, Porsche could find itself holding a large stake in Volkswagen without commensurate voting power.

"It"s clearly a high-risk approach, because Porsche doesn"t know exactly what is going to happen at the court," said Ferdinand Dudenhöffer, the director of the Center for Automotive Research in Gelsenkirchen.

Shares of Volkswagen have risen 37 percent this year, largely on speculation about Porsche"s intentions.

Porsche first acquired shares in Volkswagen in September 2005, saying it wanted to protect the independence of its partner. It has steadily increased its stake, occasionally ruffling Volkswagen"s other major shareholder, the state of Lower Saxony, which owns 20.5 percent.

The links between the two companies are longstanding, complex and personal. Ferdinand Porsche, the patriarch of the Porsche family, designed the Volkswagen Beetle at the behest of Hitler.

The chairman of Volkswagen"s supervisory board, Ferdinand K. Piëch, is a member of the Porsche family, which still controls the sports-car company. He was once chief executive of Volkswagen, and continues to exert influence over its day-to-day affairs.

Analysts regard Porsche"s growing stake in Volkswagen as a way for Mr. Piëch, an automotive engineer and billionaire industrialist, to reassemble the pieces of his grandfather"s empire.

If all of Volkswagen"s shareholders were to accept Porsche"s offer, Mr. Baumann said, the deal would be valued at about 35 billion euros ($46.6 billion). Porsche has lined up financing for that, though he made it clear that the carmaker does not intend to raise the offer beyond the current discounted level.

Porsche"s investment in Volkswagen has already shaken up the larger company. Last fall, Volkswagen dismissed its chief executive, Bernd Pischetsrieder, and replaced him with Mr. Winterkorn, a protégé of Mr. Piëch, who ran the company"s luxury car division, Audi.

Soon afterward, Volkswagen"s No. 2 executive, Wolfgang Bernhard, resigned. Mr. Bernhard, a former chief operating officer at Chrysler, had pushed a stringent cost-cutting program.

Since then, analysts say, Volkswagen has shifted its emphasis to high-quality engineering and design. These are hallmarks of Porsche, and of Mr. Piëch, when he was chief executive of VW.

"In the long-term, we"ll have to see how competitive Volkswagen will be in the market," Mr. Dudenhöffer said. "The risk is Piëch wants to do everything. He wants to buy control, and he"s reversing Volkswagen"s strategy."