The limits within such a controller can recover define a target basket. If you can land within the target basket, you don't fall down. Unstable moves should end with a landing inside that target basket, leading to a stable state.
Next, you want to go from a standing start to a fast run in one stride. You lower the torso (probably by bending the knees, but it could be a piston-type motion), fall forward, and at some angle, launch. While in the air, you make the appropriate maneuvers to land on one foot within the target basket of the basic controller. Doing that is a two-point boundary value problem. That's planning one move ahead. One wildly unstable step and a controlled landing within the target basket of the basic controller. That's Boston Dynamics's flip.
Now if you could do two unstable footfalls in succession, you'd have a much more powerful system. You could do two unstable steps and then land in the target basket and let the basic controller take over for a safe stop. But you now have the option of taking one unstable step and then planning two more if possible for fast changes in speed and direction. There's always the possibility of sticking with the original plan that lands you in the target basket for a safe stop. But, while in flight for the first unstable step, you can try planning a new two-step plan. If that problem can be solved, you can continue with agile motion. If not, you fall back to the original plan and end up stopped but safe.
Two point boundary problems can be solved using known techniques. See "shooting method" and "relaxation method", both of which are optimization schemes. Solving two two-point boundary problems in sequence is harder and requires looking through a big solution space, but everybody is doing gradient descent in bumpy spaces now and it's less mysterious than it used to be.
Good PhD thesis problem for someone. Startup potential, not so much.