Seven years after the death of mystery billionaire Howard Hughes, the state of Delaware asked a judge today to strip his former chauffeur of control of the medical research institute Hughes founded and of the giant Hughes Aircraft Co., a major defense contractor.

This latest chapter in the seemingly endless struggle to sort out the affairs of the reclusive tycoon unfolded in the unlikely setting of a tiny hearing room of Delaware Chancery Court. The stakes, however, are as high as the arena is modest: control of a foundation that handed out more than $39 million in research grants last year and of an industrial empire on the leading edge of defense technology.

Delaware claims that F. William Gay, Hughes' former chauffeur, and Chester Davis, one of his lawyers, improperly altered the bylaws of the Howard Hughes Medical Institute in 1971 to give them control after Hughes' death in 1976.

Davis died in May, so the case now pits the state against Gay, whom it asked Chancery Court Judge Grover C. Brown to replace with an independent trustee.

If appointed, an independent trustee probably would have to preside over the sale of 75 percent of the aircraft company's stock by 1989 as required by tax laws, and likely would bring in a new research director at the medical institute.

Brown scheduled a full-scale trial of the complex case, which Delaware first filed in 1978. But at the last minute, attorneys for the state and the medical institute decided not to call any witnesses. They submitted their case to Brown on the basis of the mountains of depositions and legal briefs already in the record. It is expected to be several months before Brown rules.

The Howard Hughes Medical Institute is the sole stockholder of Hughes Aircraft, a major manufacturer of missiles, satellites and defense electronic equipment. It has more than 64,000 employes and had sales of $4.4 billion in 1982, making it one of the largest privately held industrial corporations in the nation.

Both the medical institute, which is based in Florida, and Hughes Aircraft, which has its headquarters in California, are incorporated in Delaware. That gives the state the legal standing to challenge what it views as an improper seizure of control of the institute. Delaware was brought into the case by William H. Lummis, administrator of the Hughes assets in Delaware.

"We maintain that the members of the institute's executive committee, namely Gay and Davis, usurped power," said Assistant Attorney General Bartholemew J. Dalton.

The state's position is that Hughes, the sole trustee of the institute, appointed no successor, and, under state law, Chancery Court must name an independent trustee.

Attorneys for the medical institute argued that Hughes approved a change in the bylaws in 1971 permitting the executive committee--Davis and Gay--to take on the powers of the trustee and assume control if no other trustee were named.

A team of defense lawyers headed by Sherwin Markman of the Washington firm of Hogan & Hartson argued that, even if the 1971 change were invalid, the executive committee still had the right to assume the powers of the trustee because Hughes authorized it in an earlier resolution, adopted by the institute in 1968.

"This court need not involve itself with the institute's internal affairs," their brief said. They argued that Hughes, who did not want to leave control in the hands of the state of Delaware, consented to the bylaw change that expanded the powers of the executive committee and that the committee is the authorized successor to Hughes as overall head of the institute.

In fact, Hughes exercised no personal control for many years over the institute or over Hughes Aircraft. The executive committee ran his affairs. The issue is who has legal authority now that Hughes is dead.

As is always the case when Hughes' affairs are at issue, there is no agreement on who had access to him, or what papers he saw, or what he agreed to, or even where he was at the time the bylaws were changed.

According to Dalton, Davis claimed before his death that "he had a conversation with Hughes and somehow some off-the-cuff memo got into the bylaws. In fact, Hughes was in the Bahamas and the bylaws were in Los Angeles and there's no evidence that Hughes ever knew about it."