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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)
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