For each segment of a piecewise Lyapunov function that exhibits asymptotic stability, we can utilize the LaSalle-Yoshizawa theorem and solve it using a differential inclusion. This allows us to merge all the piecewise Lyapunov functions and demonstrate that the entire system is asymptotically stable.
Now, if each segment of my piecewise Lyapunov function achieves finite-time convergence, specifically in the form of square root, how can I utilize the differential inclusion? Can you explain whether the overall system is finite-time stable? Upon searching through some resources, I found that these studies generally assume each segment to be asymptotically stable and then prove that when the function obtained through the differential inclusion exhibits square root convergence, the overall system is finite-time stable. However, if my current equations do not exhibit asymptotic stability for each segment but instead demonstrate finite-time stability, how can I prove it?