Hi Lawrence:
This type of thing happens a lot. We suggest notifying your tenants as soon as you find out that they are violating their lease by allowing others to move in and stay beyond your guest policy without approval. If you accept rent after having knowledge and not having sent a notice, you might be deemed by a court to have accepted the situation. This could jeopardize an eviction for violation of the lease.
The first "notice" doesn't have to be a formal "notice to quit" to initiate an eviction--a simple letter using a friendly tone might be in order, something like "...you may have forgotten that your lease limits the time guest can stay in your unit. Please refer to your lease clause ___ which stipulates that any additional persons living in the unit must be approved..." "...please contact us so we can make arrangements..., etc." Some landlords will tell you to send a legal notice to quit along with the letter to start the process--just in case. Not a bad idea if you don't think they'll take you seriously. The attached letter could say something like "...we've included a copy of a notice to quit which will be initiated if the guest has not moved or started our application process..." On the other hand, if this has been going on for a while and you've already given them notice, you might be forced to start the formal eviction process without the niceties.
If you have to pursue eviction, it can be tough to prove...unless of course they admit it.
If you don't care for your current tenants and believe they will be a problem and the lease is nearing the end, you could just ask for possession of your property at the end of the lease. (Just be sure not to mention any other reason other than to obtain possession.)
On a side note, fair housing folks hate it when landlords unnecessarily deny additional tenants unless there is a particular issue with local zoning or property maintenance codes. Giving the existing tenants a chance to resolve their problem and come into compliance is something fair housing advocates like to see.
Note: I'm not an attorney nor giving legal advice--just passing on information. And, this is for Michigan only.
Good luck!