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ON ALMOST LYAPUNOV FUNCTIONS

Daniel Liberzon

University of Illinois, Urbana-Champaign, U.S.A.

Joint work with Charles Ying and Vadim Zharnitsky

CDC, LA, Dec 2014

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MOTIVATING REMARKS

_x = f (x)

• To verify we typically look for a Lyapunov function s.t.

x(t) ! 0 V_V (x) := @V

@x (x) ¢f (x) < 0 8x 6= 0

• Computing pointwise – easy, checking – hard _V (x) _V (x) < 0 8x

• For polynomial and , can use semidefinite programming (SOS) V f

• 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?

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SET - UP

0

V = c(x)

D

_x = f (x); x 2 Rn

V : Rn ! R; V (0) = 0; V (x) >0 8x6= 0

D := V ¡ 1([0;c]) ½ Rn; c > 0

M := maxx2D

jVx(x)j

jf (x) ¡ f (y)j · L jx ¡ yj 8x;y 2 D

Assume: for some and subset , a > 0 ­ ½ D

_V (x) · ¡ aV (x) 8x 2 D n­

For , let be the radius of the ball in with volume " > 0 ½(") Rn "

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MAIN RESULT V = c(x)

• V (x(t)) · V (x0) + 2M ½(") () x(t) 2 D) 8t ¸ 0

• for someV (x(T )) · R(") T ¸ 0

• V (x(T )) · R(") + 2M ½(") 8t ¸ T

x0_V (x) · ¡ aV (x) 8x 2 D n­

radius of the ball with volume ½(") = "

M = maxx2D jVx(x)j

0Theorem: & a cont., incr. fcn

with

9 ¹" > 0R : [0; ¹"] ! [0;1 ) R(0) = 0

vol(­ ) < " · ¹" 8x0 2 DV (x0) < c ¡ 2M ½(")

s.t. if then

with

2M ½(")

V(x) = R(")

we have:

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CLARIFYING REMARKS

• cannot contain a ball of radius ½(")vol(­ ) < " ) ­

V = c(x)

0

V(x) = R(")

• If then we can take and recover the classical asymptotic stability theorem

_V (x) · ¡ aV (x) 8x 2 D " ! 0

• If another equilibrium thenz V (z) · R(")9

z

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"

_V (x) · ¡ aV (x) 8x 2 D n­

_V (z) = 0 contains a nbhd of and

cannot be arb. small

z"V (z) > 0

) ­

vol(­ ) < " – nothing else known about ­•

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IDEA of PROOF V = c(x)

0

x0

¹x0• For any consider - ball around itx0 2 D ½(")

• In this ball s.t.9 ¹x0 _V (¹x0) · ¡ aV (¹x0)

• Corresp. solution satisfies, for some time, _V (¹x(t)) · ¡ a

2V (¹x(t)) ) V (¹x(t)) · e¡ a2tV (¹x0)

¹x(t)

½

• Distance between the two trajectories grows as

jx(t) ¡ ¹x(t)j · jx0 ¡ ¹x0jeL t · ½eL t ( Lip const of ) L = f

• If is large compared to then decay of initially dominates and “pulls” towards 0

V (¹x0) ½ V (¹x(t))¹x(t) x(t)

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LAST STEP in MORE DETAIL

V (x(t)) · V (¹x(t)) + M jx(t) ¡ ¹x(t)j (using MVT)

· e¡ a2tV (¹x0) + M jx0 ¡ ¹x0jeL t

(for )0· t · T +

· e¡ a2t(V (x0) + M ½) + M ½eL t (from

and MVT again)

jx0¡ ¹x0j · ½

V (x0)

V (x0) + 2M ½

°V (x0)0<° <1

T½;°

If where is large enough compared to

then crossing time exists and is smaller than

V (x0) ¸ R R ½T½;° T +

We can then repeat the procedure with in place

of and iterate until

x(T½;°)x0 V (x(t)) = R

V (x(t))upper bound on

t

V

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OPEN QUESTION

V (x0)

V (x0) + 2M ½

°V (x0)

t

0<° <1

T½;°

V (x(t))upper bound on

V

V (x(t))possible behavior of itself

Does our result actually allow this to happen?

_V (x) · ¡ aV (x) 8x 2 D n­

but holds on a larger set _V (x) · 0

Can this set still be a strict subset of ? D

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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

_V (x) < 0

• Remains to test our ability to handle situations where

at some points in the region of interest_V (x) > 0

• Proof compares convergence rate of nearby stable trajectories with expansion rate of distance between trajectories (entropy)