1. 1 of 9 ON ALMOST LYAPUNOV FUNCTIONS Daniel Liberzon University of Illinois, Urbana-Champaign, U.S.A. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AA A A A Joint work with Charles Ying and Vadim Zharnitsky CDC, LA, Dec 2014

2. 2 of 9 MOTIVATING REMARKS To verify we typically look for a Lyapunov function s.t. Computing pointwise – easy, checking – hard For polynomial and, can use semidefinite programming (SOS) Randomized approach: check the inequality at sufficiently many randomly generated points. Then, with some confidence, we know that it holds outside a set of small volume ( Chernoff bound; see, e.g., [ Tempo et al.,’12 ]). Taking this property as a starting point, what can we say about convergence of trajectories?

3. 3 of 9 SET - UP Assume: for some and subset, For, let be the radius of the ball in with volume

4. 4 of 9 MAIN RESULT for some radius of the ball with volume Theorem: & a cont., incr. fcn with s.t. if then with we have:

5. 5 of 9 CLARIFYING REMARKS cannot contain a ball of radius If then we can take and recover the classical asymptotic stability theorem If another equilibrium then In fact, this weaker condition is enough for the theorem to hold, and it can be checked by sampling enough points (on a lattice). Gives a meaningful result for small enough contains a nbhd of and cannot be arb. small – nothing else known about

6. 6 of 9 IDEA of PROOF For any consider - ball around it In this ball s.t. Corresp. solution satisfies, for some time, Distance between the two trajectories grows as ( Lip const of ) If is large compared to then decay of initially dominates and “pulls” towards 0

7. 7 of 9 LAST STEP in MORE DETAIL (using MVT) (for ) (from and MVT again) If where is large enough compared to then crossing time exists and is smaller than We can then repeat the procedure with in place of and iterate until upper bound on

8. 8 of 9 OPEN QUESTION upper bound on possible behavior of itself Does our result actually allow this to happen? but holds on a larger set Can this set still be a strict subset of ?

9. 9 of 9 CONCLUSIONS Less conservative results will need to be tailored to specific system structure Developed a stability result that calls for to hold only outside a set of small volume Remains to test our ability to handle situations where at some points in the region of interest Proof compares convergence rate of nearby stable trajectories with expansion rate of distance between trajectories (entropy)