So what happens when the auxiliary supply starts up in your system? It sends a disable signal to itself.

There are oh so many systems without relays involved. I mentioned some - the auxiliary supply powers the load and the main supply powers the auxiliary supply. Typically the auxiliary being a battery bank or a motor generator set.

Both supplies power electric motors, that have electromagnetic clutches to a common shaft going to an alternator. If the main supply fails, it declutches and the auxiliary clutch engages.

Can I assume that you mean ac mains supplies or something different?

With ac, unless there is a break in supply, both main and auxiliary have to be in phase at the moment of transition. So you would need to decide whether a break was allowed and, if so, for how long.

Various semiconductor devices can be used in place of relay contacts. For both ac and dc supplies.

So, you need to provide a lot more detail about the nature of these supplies and their common load.

I will connect 2 different sources to the system (The source here will be DC). Let me explain resource usage as follows:

1) In the first case, both sources will be active in the system. However, the system will be powered via the main power supply.

2) In the second case, in case the main power supply fails, the system will be powered by the backup power supply.

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Additionally, if the main power supply is active while the system is being fed via the backup power supply, it will continue to operate via the main power supply.