The psoas muscle, is a combination of the iliopsoas, psoas major and psoas minor muscles. It originates on the lumbar spine, travels over the front of the pelvis and inserts on the femur. It is the only muscle which directly connects the core with the legs. Most muscles go core to pelvis or pelvis to legs. In the triathlon aerodynamic position, the psoas is in a shortened state. Try this stretch on a long ride or in a race; stand up and arch your back (stick out your butt, lift your shoulders, drop your belly towards your top tube). This action takes tension off the psoas muscle. If your lower back is aching and this stretch brings you relief it is your psoas that is weak.
The Psoas muscle connects the spine to the legs.
To strengthen the psoas specifically for cycling is tricky as you have to strengthen it in a shortened state to be strong in the tri position. Here is one (of many) of the best psoas exercises specifically designed for cycling: I call it the Psoas crunch. Kneel on all fours with one end of an elastic exercise-band (swim cords work great here) tied to your right ankle and the other end to an attachment point beyond your feet. Extend your left arm and right leg out and then crunch them in bringing your right elbow towards your left knee. Round your back, exhale and pull your belly button hard up to your spine as you crunch in, hold for two seconds and repeat. To add variety to this exercise do it without the exercise-band, support your arms on an exercise-ball instead of the floor or turn over and do this exercise lying on your back.
On the bike, long intervals of one leg pedaling in the tri position is effective for conditioning the psoas. With your bike on a trainer and in a very easy gear unclip one foot and pedal solely with the other foot. I like to rest my lazy foot on my trainer or on a chair beside my bike. Start with 4 X 30 seconds on each leg. When you can comfortably do that keeping good form (no chain clunking or dead spots) do 4 X 60 seconds. Gradually build up the interval period to three minutes. 4 X 3 minutes of one leg pedaling is a tough session.
Stretch the psoas as follows. Put your front foot on a low bench, turn your back foot out and press your hips forward and toward the floor. Hold for 20 seconds. You should feel this stretch on the front side of your hip/inner thigh of your rear leg.