Did you grow the annual Sweet Alyssum? Your expired plants will not return as a perennial, they have lived their lifespan and should be removed.
Here is a link with more information.
https://www.gardeningknowhow.com/ornamental/flowers/alyssum/growing-sweet-alyssum.htm
If you planted the perennial Basket-Of-Gold Alyssum, which is a perennial, you can trim the dead portions away and the plants will began to grow when the weather and soil warms.
https://www.gardeningknowhow.com/ornamental/flowers/basket-of-gold/growing-basket-of-gold.htm