If at the expiry of the contractual term of a lease which is protected by the Landlord and Tenant Act 1954 ("the Act"), a tenant remains in occupation for the purposes of its business then there is created what is known as a s.24 continuation tenancy – a common law tenancy subject to the same terms as the previous lease save for a variation to the methods of termination, and with the right to apply to Court for a renewal tenancy.
If you as landlord find out that the tenant has since ceased to occupy the premises for the purposes of its business (so that it no longer attracts the protection of the Act), then that does not, of itself, bring the s.24 continuation tenancy to an end.
In those circumstances, if the tenant has not yet made an application to the Court for a renewal tenancy, and has not returned to occupy the premises, then you are able to serve a notice pursuant to s.24(3)(a) giving between three and six months' notice to terminate the s.24 continuation tenancy and remove the tenant's right to renew. This would remove the right even if the tenant subsequently re-occupies the premises for its business.
This is particularly useful if you suspect that the tenant has left but is planning on coming back, as it prevents the tenant continuing to assert its renewal rights. However, if you are going to serve notice then make sure you have evidence of the lack of occupation at the time the notice is served, in case the tenant later denies that was the case.