When you produce a tone with your instrument, you are not only using your lips but also all the muscles involved in breathing. You need to make sure that you coordinate all these muscles with your lips properly to get the best possible result.
One comment we hear about a singer sometimes is that "His tone really comes from deep". That's because they need to use their entire torso to move the air through their vocal cords to produce a nice tone. We can learn from that and learn to support our chops with a good air stream.
It's not very complicated after all. Here is something you can do to practice your air support:
1) Stand up or sit with your back straight.
2) Fill your lungs with air completely and let the air come out without making any effort to push it out. When you breath, extend your entire rib cage as much as possible.
3) After a few breaths, place your hand in front of your mouth and blow in it. Try to maintain the same air pressure for as long as possible. You will have to apply gentle pressure from your rib cage and diaphragm to keep the air pressure. Don't blow to hard, It's not necessary.
Do this a few time and be aware the muscles you are using to keep the air running.
4) Repeat the exercise using your mouthpiece this time. Hold long notes. You can try various pitches. You'll notice that it gets harder to stay relaxed as you get higher. Try to keep your torso muscles from getting tense as you go higher.
5) Now put your mouthpiece on your instrument and do the same thing with various notes. Once you feel comfortable, try scales and arpeggios.
Try to be aware that you're are using your entire body to produce your tone. Not only your chops. Feel which muscles are working.
You will have more power in the high register and more endurance if your lips are well supported by a good air stream that comes from the bottom.
I am associate principal horn of the Singapore Symphony Orchestra and the author of the progressive methods. I'm happy to share my experience as a horn player and teacher with you.