In the files of Chancery Court, the case is listed as "Charles M. Oberly III, Attorney General of the State of Delaware vs. Howard Hughes Medical Institute, a Delaware Corporation."

But what it really embodies is a struggle between two wealthy men who came by very different routes into the shadowy world of the late industrialist Howard R. Hughes. The potential prize is control of a rich piece of Hughes' legacy, the Hughes Aircraft Co., a giant defense contractor on the leading edge of technology in missiles, satellites and electronics.

The mountain of legal papers and depositions filed in the case is a gold mine of information for a potential biographer of Hughes, recounting his sad decline from dynamic tycoon into quirky recluse, shut up in hotel rooms and communicating only in writing with his operating executives. His most loyal lieutenants and lawyers admit in their testimony that they didn't actually see him or talk to him for years on end.

But the core question is a simple one: did Hughes approve a 1971 change in the bylaws of a medical foundation he created in 1953? If the court says no, control of Hughes Aircraft could be up for grabs.

One of the protagonists is Frank William Gay, 62, who has spent almost his entire working life in the Hughes network of companies, rising from chauffeur and errand-boy to senior executive of Hughes' personal holding company, the Summa Corp. Gay is now the senior member of the executive committee that controls the Howard Hughes Medical Institute, which is the sole stockholder of Hughes Aircraft.

The other is William R. Lummis, 53, a prominent Houston lawyer. Lummis was Hughes' first cousin but met the reclusive tycoon only once, when Lummis was 9 years old, according to court documents. After Hughes, reduced to a drug-dependent, 90-pound shell of a man, died in 1976, a Nevada probate court installed Lummis in Gay's place as head of Summa.

Lummis is now chairman and chief executive officer of the Las Vegas-based Hughes Corp., a newly created holding company that owns Summa, operator of Hughes' Nevada gaming and real estate interests, and the Hughes Helicopters Corp., a maker of helicopters for private business and for the Army.

As administrator of Hughes' estate and director of the tangled web of business operations Hughes left behind, Lummis has been trying to break Gay's hold on the medical institute--with such doggedness, in fact, that Gay's lawyers complained here about what they called his "spectacularly litigious attitude."

The Delaware case, which was submitted for a decision last week to Chancellor judge Grover Brown, has potentially far-reaching consequences because of its implications for Hughes Aircraft. If Brown rules against the medical institute and Gay, Hughes Aircraft could be sold or merged with the Hughes Corp., or the company's stock could be sold to the public or distributed to its employes.

Despite its name, the company does not make or operate any airplanes. The name is a throwback to Howard Hughes' days as an airline and aircraft pioneer, the man who built Trans World Airlines and created the "Spruce Goose," the wooden airborne white elephant that flew only once.

What Hughes Aircraft does make is sophisticated weapons, missiles, satellites, state-of-the-art defense electronics, radar, guidance systems, torpedoes, and military communications systems. Hughes makes communications satellites, and all or part of such advanced weapons as the Trident, Maverick and Phoenix missiles. With 64,300 workers, it is the largest industrial employer in California. Its 1982 sales of $4.4 billion made Hughes one of the biggest defense contractors and one of the biggest privately held corporations in the nation.

All Hughes Aircraft stock has been held since 1953 by the Howard Hughes Medical Institute, a nonprofit Delaware corporation that uses the income from the company to fund advanced medical research in association with prominent medical schools.

In 1982, according to papers submitted to the court, the company returned $42 million to its owner, the medical institute. This is less than 1 percent of revenues, which means, according to its critics, that the institute is receiving an inadequate return on its sole investment and therefore should either diversify its holdings or shake up Hughes Aircraft.

But officials of Hughes Aircraft, according to company sources, say the modest return reflects very high expenditures on the research and development that make Hughes Aircraft a leader in the field. They reportedly fear research would be curtailed if Hughes Aircraft were responsible for delivering profits to public stockholders.

In his lifetime, Hughes had sole legal power over the institute because its articles of incorporation provided that it be "managed and controlled by a Trustee" and he was the trustee. The corporate bylaws contained no provision for filling a vacancy in the trusteeship.

Hughes delegated operational control of the institute to an "executive committee." In February 1971, he appointed Gay, who was then running Hughes Air West, and Chester Davis, his longtime attorney, to the three-member panel. Two months later, the corporate bylaws were amended to provide that "in the absence of a Trustee, the Executive Committee shall also have and may exercise the powers of the Trustee."

Hughes left no will. After his death, Gay, Davis and the third member of the committee, a doctor who was the institute's research director, invoked that bylaw to abolish the office of trustee and establish themselves in control of the institute and, by extension, of Hughes Aircraft. Davis died in May of this year, and the research director has been replaced, so it is actually Gay who is exercising control as successor to Hughes.

Lummis, as administrator of the estate, sued, claiming that Hughes never authorized the change in the bylaws and asking that he be named trustee in Gay's place. The State of Delaware, where the institute and Hughes Aircraft are incorporated, agreed that the transfer of control to the executive committee was invalid and joined Lummis as co-plaintiff. The State of Florida, where the medical institute is based, also charged that the bylaw amendment was "not authorized," and sought to intervene in the case on Lummis' side.

Florida asked for a court-ordered accounting of the institute's affairs, a request that state officials said privately was stirred by consulting contracts given by both the institute and Hughes Aircraft to a firm headed by Gay's son, and by Gay and Davis' decision after Hughes' death to start paying themselves $75,000-a-year salaries. Before Hughes died, they served on the institute's executive committee without pay, Gay testified.

The court, however, ruled that Lummis had no standing to sue, and denied Florida's petitition to intervene. That left the state of Delaware as sole plaintiff, though Lummis, through his attorneys, has continued to work closely with the attorney general's office.

Delaware wants the court to name an outside trustee--not necessarily Lummis--to assume control of the institute. According to Deputy Attorney General Bartholomew J. Dalton, such a person could "do any of the things Hughes could have done. He could take the assets of the institute and diversify, or sell, or something short of that. He would do what's reasonable and prudent."

Most of the testimony taken in depositions has centered on the question of who had access to Hughes in his Bahamas retreat and could verify his approval of the bylaw change. A key witness was Nadine Henley, Hughes' personal secretary and administrative assistant for more than 30 years.

Henley, who described herself as "Madam Nit-Pick" because she is so meticulous, said she added the revised language to the bylaws with the clear understanding that Hughes had approved it, but she could not remember who told her Hughes had approved it. Even she, the virtual nerve center of Hughes' operations, did not have any direct communication with him.

Lawyers for the medical institute and Gay said in a legal brief that "the mere fact that she could not remember the name of the person who so advised her eight years after the event is hardly startling, and is no basis for undoing the bylaws of a Delaware corporation."

Even if he wins the case, Gay is already committed to expanding the executive committee that would control the institute and Hughes Aircraft. Before Davis' death, the incumbent members voted to add four new members: Irving Shapiro, former chairman of E. I. duPont de Nemours & Co; Donald Frederickson, former director of the National Institutes of Health; Lawrence Fouraker, former dean of the Harvard Business School; and William B. Graham, chairman of Baxter Travenol Corp.