Is it legal to accept additional rent over the HUD payment standard for a Section 8 tenant?

I have an upscale condo for rent in a very nice area and have an interested applicant on the Section 8 program. The prospective tenant is retired, has good references, and seems like a good prospect, but the area housing authority's payment standard for a condo of that size is $300 below what I'm asking for rent. The prospective tenant and his adult children (not living with him) have offered to pay the difference each month out of pocket (and presumably "under the table"). I have no doubt they can afford to do so, and they said they've been doing this for many years at his current residence. Sounds to me like an abuse of the system, and I question the legality of accepting under the table payments on top of the set rent agreed to by the area housing authority. As a landlord, I think this seems risky. I'm waiting to hear back from the local housing authority for the official word. Anyone have advice on this?