Maybe. the issue is one of Trademarks. Have you filed a Federal Trademark with the PTO? there are state laws in the area of trademarks, but they are not as strong as the federal right. Unlike copyright you can file a trademark at any time after commercial use of the mark. You also have a commonlaw right to the markin the same industry and the same market.
This will depend upon two facts. First, did you file a trademark with the Patent and Trademark Office? If not, then you do not have nationwide trademark protection of your business name.
However, you can have trademark protection based upon when and where you first used the trademark in commerce. If you did not file, then nationwide protection will likely not be granted. Nonetheless, your mark can be protected within specific regions in which it is used, so long as you can show that you began using the mark in commerce in that region first.
If you do not have any stores in this other state, and you do not advertise in this other state, then your trademark protection will likely not extend to this state. You may want to check into registering your trademark so as to prevent this from happening in the future.
Unless you have you business name protected, ie tradmarked, then there is not much you can do if your name is being used in another state. Even if you have a dba you are only protected as far as the county in which your dba is filed,
Attorneys Howell and McDaniel have both provided useful information.
I write only to clear up a few things. And to spot some issues that you and your own attorney will have to deal with.
First, without knowing what your restaurant's trade name is no one can tell you if that name is even protectable under trade name law. Many names are not. Paul's Pizza Palace, Seafood Sally's, The Rare Steak -- these and many other trade names are "generic" and cannot be claimed as the exclusive property of any one person.
Second, assuming your trade name is protectable, then the second query is to determine if your restaurant's name is being used AS A TRADEMARK. Again, many are not -- and so they are not, therefore, protectable as trademarks [which means they cannot be registered as trademarks by either a state or the federal trademark office]. Note, however, that even if the trade name does not serve as a trademark, the trade name is protectable [by the assumption in this scenario] under trade name law -- though not trademark law.
Third, assuming your trade name does serve as a trademark, it is NOT necessary to federally register that trademark in order to either own those trademark rights or to enforce those rights. Registration provides significant advantages but it is not required.
Fourth, even if your trade name serves as a trademark and is federally registered that registration DOES NOT provide you with the exclusive, immediate nationwide rights to use the mark in commerce.
If another person began to use the same mark, in good faith and in a location geographically remote from you, BEFORE your registration issued then that person may forever use the mark even after your registration issues -- and it is YOU who cannot enter the territory where the other person uses the mark.
If the other person began to use the same mark in a location geographically remote from you AFTER your registration issued then that person may continue to use the mark until you start to use the mark in that geographic area.
Have fun. And hire a trademark attorney registered to practice in your state. Trade name is, in the main, state law.