Going beyond Zeno
through a pointwise asymptotically stable set
in a hybrid system
Rafal Goebel
Mathematics and Statistics, Loyola University Chicago
supported by Simons 315326
Trento, January 2017
Introduction and outline
Going past Zeno:
Hybrid inclusions, examples, and well-posedness
Pointwise asymptotic stability
Small ordinary time property
Good behavior of limits of Zeno solutions
Well-posedness past Zeno
Optimal control for pointwise asymptotic stability:
Finite length Lyapunov functions
Optimal and robust feedback stabilization in discrete time
Based on:
Set-valued Lyapunov functions for difference inclusions, G., Automatica 2011
Robustness of stability through necessary and sufficient Lyapunov-like conditions ..., G., SCL 2014
Results on optimal stabilization of a continuum of equilibria, G., CDC 2016
and joint work with R. Sanfelice:
Notions and sufficient conditions for pointwise asymptotic stability in hybrid systems, G. and Sanfelice, NOLCOS 2016
How well-posedness of hybrid systems can extend beyond Zeno times G. and Sanfelice, CDC 2016
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Beyond Zeno through a PAS set
Hybrid Inclusions
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Beyond Zeno through a PAS set
Hybrid Inclusions
A hybrid inclusion combines a differential inclusion, a difference inclusion, and constraints on
motions resulting from the inclusions.
x ∈C
ẋ ∈ F (x)
x ∈D
x + ∈ G (x) .
Above, ẋ is velocity, x + is value after a jump.
Single-valued case:
x ∈C
ẋ = f (x)
x ∈D
x + = g (x) .
Solutions:
parameterized by t and j, with (t, j) evolving in hybrid time domains;
satisfy φ(t, j) ∈ C , φ̇(t, j) ∈ F (φ(t, j)) when flowing;
satisfy φ(t, j) ∈ D, φ(t, j + 1) ∈ G (φ(t, j)) when jumping.
Hybrid Dynamical Systems: Modeling, Stability, and Robustness G., Sanfelice, Teel, Princeton University Press, 2012
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Beyond Zeno through a PAS set
Pointwise asymptotic stability in a hybrid system — an example
Agents z1 , z2 , . . . , zI ∈ Rk
agree on a target w in the convex hull of zi ’s;
converge exponentially to w ;
every T amount of time communicate and agree on a new w .
Hybrid inclusion modeling this:
state x = (z1 , z2 , . . . , zI , w , τ );
if τ ≥ 0,
żi = w − zi , ẇ = 0, τ̇ = −1;
if τ = 0,
zi+ = zi , w + ∈ con{z1 , z2 , . . . , zI }, τ + = T .
Natural to expect convergence of z to and stability of the consensus set
{z | z1 = z2 = · · · = zI }
In fact, the following set is partially pointwise asymptotically stable:
A = {(z, w ) | z1 = z2 = · · · = zI = w } × [0, T ]
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Beyond Zeno through a PAS set
Pointwise asymptotic stability in a hybrid system — a twisted example
Two agents z1 , z2 ∈ Rk
agree on a target w =
z1 + z2
;
2
w − zi
converge to w according to żi = ci p
, where ci > 0.
|w − zi |
update w when one agent reduces its distance from w by a factor of 4.
Hybrid inclusion with x = (z1 , z2 , w , τ ) ∈ R3k+1 and
C = Rk × Rk × Rk × [0, ∞),
F (x) =
!
w − z2
w − z1
c1 p
, c2 p
, 0, −1 ,
|w − z1 |
|w − z2 |
D = Rk × Rk × Rk × {0},
“
G (x) = z1 , z2 ,
z1 +z2
, min {t1 , t2 }
2
”
√
where ti :=
|a−zi |
.
ci
Then A = {z | z1 = z2 = w , τ = 0} is pointwise small ordinary time asymptotically stable.
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Beyond Zeno through a PAS set
(Nominal) well-posedness
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Beyond Zeno through a PAS set
(Nominal) well-posedness
For a differential equation ẋ = f (x) or inclusion ẋ ∈ F (x), if f or F is sufficiently regular, for
every bounded sequence of solutions
there exists a locally uniformly convergent subsequence (Arzela-Ascoli);
the limit of the subsequence is a solution;
and if solutions are unique, this reduces to continuous dependence of solutions, over bounded
time intervals, on initial conditions.
For a hybrid inclusion (C , F , D, G ), under
Basic Assumptions: C , D closed; F , G closed graph; F (x) nonempty, convex for all x ∈ C ;
G (x) nonempty for all x ∈ D.
one has that, for every bounded sequence of solutions
there exists a graphically convergent subsequence;
the graphical limit of the subsequence is a solution;
and more...
In short: (C , F , D, G ) is well-posed.
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Beyond Zeno through a PAS set
(Nominal) well-posedness
Consequences of nominal well-posedness include:
(a) Solutions, over bounded hybrid time domains, depend on initial conditions in an
outer-semicontinuous way and this can be characterized in terms of distances between
graphs of solutions.
(b) Solutions, over bounded hybrid time domains, depend on initial conditions continuously
when uniqueness of solutions can be ensured.
(c) The Krasovskii-LaSalle invariance principle, and other arguments relying on invariance,
apply.
(d) For a compact asymptotically stable set the basin of attraction is open and from it, the
convergence to the set is uniform and it admits a KL bound.
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Beyond Zeno through a PAS set
(Nominal) well-posedness
Consequences of nominal well-posedness include:
(a) Solutions, over bounded hybrid time domains, depend on initial conditions in an
outer-semicontinuous way and this can be characterized in terms of distances between
graphs of solutions.
(b) Solutions, over bounded hybrid time domains, depend on initial conditions continuously
when uniqueness of solutions can be ensured.
(c) The Krasovskii-LaSalle invariance principle, and other arguments relying on invariance,
apply.
(d) For a compact asymptotically stable set the basin of attraction is open and from it, the
convergence to the set is uniform and it admits a KL bound.
However:
Even if solutions have limits, the limits need not depend regularly on initial conditions.
Infinite-horizon reachable sets need not depend regularly on initial conditions.
Even if solutions are Zeno, the Zeno times need not depend regularly on initial conditions.
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Beyond Zeno through a PAS set
Pointwise asymptotic stability
a.k.a. semistability
Goebel
Beyond Zeno through a PAS set
Pointwise asymptotic stability
a.k.a. semistability
Definition
The closed set A is pointwise asymptotically stable (PAS) if
every point a ∈ A is Lyapunov stable, that is, for every a ∈ A, ε > 0 there exists δ > 0
such that every solution from a + δB remains in a + B;
every solution is convergent and its limit is in A.
PAS is AS if A={a}, PAS =⇒ AS if A compact, PAS ?? AS if A closed
PAS is not AS, even if every a ∈ A is an equilibrium.
Standard Lyapunov conditions are not sufficient for PAS.
Examples:
Steepest descent / negative gradient flow: ẋ ∈ −∂f (x) with f convex, A = arg min f
Saddle-point dynamics: ẋ ∈ −∂x h(x, y ), ẏ ∈ ∂y h(x, y ) with h convex-concave, A the set
of saddle points
Convex optimization algorithms (proximal-point, and more) with Fejer property:
kx + − ak ≤ kx − ak for every a ∈ A = arg min f
Numerous consensus algorithms, with A = {x | x1 = x2 = · · · = xn }
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Beyond Zeno through a PAS set
Pointwise asymptotic stability
a.k.a. semistability
Definition
The closed set A is pointwise asymptotically stable (PAS) if
every point a ∈ A is Lyapunov stable, that is, for every a ∈ A, ε > 0 there exists δ > 0
such that every solution from a + δB remains in a + B;
every solution is convergent and its limit is in A.
Singular perturbation of autonomous linear systems, Campbell and Rose 1979
A continuous algorithm for finding the saddle points of convex-concave functions, Venets 84
Nontangency-based Lyapunov tests ..., Bhat, Bernstein 03
several articles by Haddad et al. 08, 09, 10,. . .
Arc-length-based Lyapunov tests ..., Bhat, Bernstein 10
links to consensus:
Stability of multiagent systems with time-dependent communication links, Moreau 05
Stability of leaderless discrete-time multi-agent systems, Angeli, Bliman 06
set-valued Lyapunov functions in discrete time
Set-valued Lyapunov functions for difference inclusions, G. 2011
Robustness of stability through necessary and sufficient Lyapunov-like conditions ..., G. 2014
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Beyond Zeno through a PAS set
Strict set-valued Lyapunov function for a closed set A
Definition
A set-valued mapping W : Rn ⇒ Rn is a strict set-valued Lyapunov function if it has right
growth and regularity properties and there exist continuous and positive definite with respect to
A functions c, d : Rn → R so that
for every solution φ : [0, T ] → Rn to ẋ ∈ F (x) such that φ(t) ∈ C for every t ∈ (0, T ),
t
Z
c(φ(s)) dsB ⊂ W (φ(0))
W (φ(t)) +
∀t ∈ [0, T ].
0
W (G (x)) + d(x)B ⊂ W (x)
∀x ∈ D.
Compare to V (G (x)) + d(x) ≤ V (x)
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Beyond Zeno through a PAS set
Strict set-valued Lyapunov function for a closed set A
Definition
A set-valued mapping W : Rn ⇒ Rn is a strict set-valued Lyapunov function if it has right
growth and regularity properties and there exist continuous and positive definite with respect to
A functions c, d : Rn → R so that
for every solution φ : [0, T ] → Rn to ẋ ∈ F (x) such that φ(t) ∈ C for every t ∈ (0, T ),
t
Z
c(φ(s)) dsB ⊂ W (φ(0))
W (φ(t)) +
∀t ∈ [0, T ].
0
W (G (x)) + d(x)B ⊂ W (x)
∀x ∈ D.
Compare to V (G (x)) + d(x) ≤ V (x)
Theorem
If there exists a strict set-valued Lyapunov function then A is PAS.
Converse results: exist for discrete time, expected for hybrid.
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Beyond Zeno through a PAS set
Weak set-valued Lyapunov function for a closed set A
Definition
A set-valued mapping W : Rn ⇒ Rn is a weak set-valued Lyapunov function if it has right
growth and regularity properties and
for every solution φ : [0, T ] → Rn to ẋ ∈ F (x) such that φ(t) ∈ C for every t ∈ (0, T ),
W (φ(t)) ⊂ W (φ(0))
W (G (x)) ⊂ W (x)
Goebel
∀t ∈ [0, T ].
∀x ∈ D.
Beyond Zeno through a PAS set
Weak set-valued Lyapunov function for a closed set A
Definition
A set-valued mapping W : Rn ⇒ Rn is a weak set-valued Lyapunov function if it has right
growth and regularity properties and
for every solution φ : [0, T ] → Rn to ẋ ∈ F (x) such that φ(t) ∈ C for every t ∈ (0, T ),
W (φ(t)) ⊂ W (φ(0))
W (G (x)) ⊂ W (x)
∀t ∈ [0, T ].
∀x ∈ D.
Theorem
If
there exists a continuous weak set-valued Lyapunov function W ;
every weakly invariant set on which W is constant is contained in A;
and C , F , D, G satisfies the Hybrid Basic Assumptions;
then A is PAS.
Proof: invariance principle.
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Beyond Zeno through a PAS set
Consequences of PAS
Theorem
If C , F , D, G satisfies Basic Assumptions and the closed set A ⊂ Rn is PAS, then
(a) the set-valued mapping L defined by
L(x) =
[
ff
lim
t+j→∞
φ(t, j) | φ(0, 0) = x
is outer semicontinuous and locally bounded;
(b) the set-valued mapping R∞ defined by
R∞ (x) = R∞ (x)
is outer semicontinuous and locally bounded and R∞ (x) = R∞ (x) ∪ L(x) for every x,
where the set-valued mapping R∞ is defined by
R∞ (x) =
[
{φ(t, j) | φ(0, 0) = x, (t, j) ∈ domφ} ;
In single-valued case, outer semicontinuity and local boundedness is continuity.
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Beyond Zeno through a PAS set
Small ordinary time property
Definition
A set A ⊂ Rn is pointwise small ordinary time asymptotically stable (PSOTAS) if it is pointwise
asymptotically stable and every a ∈ A is SOT stable:
for every ε > 0 there exists δ > 0 such that lengtht dom φ < ε if φ(0, 0) ∈ a + δB.
Theorem
Suppose that a closed set A ⊂ Rn is PAS and there exists a continuously differentiable
V : Rn → R such that
(a) V is positive definite with respect to A;
(b) there exists c > 0 and ρ ∈ [0, 1) such that
∇V (x) · f ≤ −c (V (x))ρ
(c) V (g ) ≤ V (x)
∀x ∈ C , f ∈ F (x);
∀x ∈ D, g ∈ G (x);
and there exist no nontrivial flowing solutions φ satisfying φ(t, j) ⊂ A for all (t, j) ∈ dom φ.
Then A is PSOTAS.
Goebel
Beyond Zeno through a PAS set
The twisted example, revisited
Two agents z1 , z2 ∈ Rk
agree on a target w =
z1 + z2
;
2
w − zi
, where ci > 0.
converge to w according to żi = ci p
|w − zi |
update w when one agent reduces its distance from w by a factor of 4.
Consider the set A = {(z1 , z2 , w ) | z1 = z2 = w } × {0}. Then
the set-valued mapping
W (η) = co{z1 , z2 , w } × [0, max{τ, min {t1 , t2 }}]
is a weak set-valued Lyapunov function;
`
´
the functon V (x) = 21 kz1 − w k2 + kz2 − w k2 satisfies
V̇ (x(t)) ≤ −23/4 min{c1 , c2 } (V (x(t)))3/4 .
PSOTAS of A follows!
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Beyond Zeno through a PAS set
Consequences of PSOTAS
Theorem
If C , F , D, G satisfies Hybrid Basic Assumptions and the closed set A ⊂ Rn is PSOTAS, then
(a) every complete solution φ is Zeno: its Zeno time tφ is finite
tφ := sup{t | ∃j (t, j) ∈ dom φ} < ∞;
(b) for every graphically convergent φi to φ,
lim tφi = tφ ;
i→∞
(c) the function
T (x) = sup{tφ | φ(0, 0) = x}
is locally bounded, upper semicontinuous, and the supremum defining it is attained.
If solutions are unique, T is continuous.
Goebel
Beyond Zeno through a PAS set
Summary so far
Suppose that:
C , F , D, G satisfies Hybrid Basic Assumptions;
the closed set A is PSOTAS.
Then:
limits of complete solutions depend reasonably on initial conditions;
complete solutions are Zeno and their Zeno times depend reasonably on initial conditions.
Idea:
solutions can be extended past their Zeno times, from their Zeno limits, via another
hybrid system (not a new idea)
so that their extensions depend reasonably on the pre-Zeno initial conditions!
Goebel
Beyond Zeno through a PAS set
Going past Zeno
Hybrid system 1:
C1 , F1 , D1 , G1 satisfies Basic Assumptions;
the closed set A1 is PSOTAS.
Re-initialization map:
Ψ is locally bounded and has closed graph.
Hybrid system 2:
C2 , F2 , D2 , G2 satisfies Basic Assumptions;
Solution (φ, ψ), where:
φ solves system 1;
ψ solves system 2;
!
ψ(tφ , 0) ∈ Ψ
lim
t→tφ ,j→∞
φ(t, j) , where tφ is the Zeno time of φ.
Goebel
Beyond Zeno through a PAS set
Going past Zeno
Hybrid system 1:
C1 , F1 , D1 , G1 satisfies Basic Assumptions;
the closed set A1 is PSOTAS.
Re-initialization map:
Ψ is locally bounded and has closed graph.
Hybrid system 2:
C2 , F2 , D2 , G2 satisfies Basic Assumptions;
Solution (φ, ψ), where:
φ solves system 1;
ψ solves system 2;
!
ψ(tφ , 0) ∈ Ψ
lim
t→tφ ,j→∞
φ(t, j) , where tφ is the Zeno time of φ.
ψ over bounded hybrid time domains depends outer-semicontinuously on φ(0, 0).
Consequence: if system 2 has a compact GAS set A2 , then
convergence to A2 is uniform from compact sets of φ(0, 0), etc.
Goebel
Beyond Zeno through a PAS set
The twisted example, revisited
Hybrid system 1: two agents z1 , z2 ∈ Rk
agree on a target w = (z1 + z2 )/2;
converge to w according to żi = ci (w − zi )/
p
|w − zi |;
update w when one agent reduces its distance from w by a factor of 4.
Re-initialization map:
Ψ is a continuous function.
Hybrid system 2:
ẋ = f2 (x), where f2 is a Lipschitz continuous function.
Solution (φ, ψ), where:
φ solves system 1;
ψ solves system 2;
!
ψ(tφ , 0) = Ψ
lim
t→tφ ,j→∞
φ(t, j) , where tφ is the Zeno time of φ.
ψ(t) depends continuously on φ(0, 0), for t > tφ .
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Beyond Zeno through a PAS set
PAS from optimal control
Goebel
Beyond Zeno through a PAS set
PAS from optimal control — the big picture
If A is pointwise asymptotically controllable with locally exponential convergence rate for the
hybrid inclusion with input
(x, u) ∈ C
ẋ ∈ F (x, u)
(x, u) ∈ D
x + ∈ G (x, u) ,
then minimization of cost like
J Z
X
j=0
tj+1
dA (φ(t, j)) + ku(t, j)k dt +
tj
J−1
X
`
´
dA φ(tj+1 , j) + ku(tj+1 , j)k
j=0
over infinite horizon yields optimal solutions that result in PAS.
Goebel
Beyond Zeno through a PAS set
Length-based sufficient condition for PAS in discrete time
Motivation:
Arc-length-based Lyapunov tests ..., Bhat, Bernstein 10
∞
Z
For ẋ = f (x), inequality V̇ ≤ −kf k implies
kẋ(t)k dt < ∞... finite length!
0
Goebel
Beyond Zeno through a PAS set
Length-based sufficient condition for PAS in discrete time
Motivation:
Arc-length-based Lyapunov tests ..., Bhat, Bernstein 10
∞
Z
For ẋ = f (x), inequality V̇ ≤ −kf k implies
kẋ(t)k dt < ∞... finite length!
0
Theorem
Suppose that
A ⊂ Rn is a closed set,
V : Rn → [0, ∞) is positive definite with respect to A and continuous at every a ∈ A,
α : Rn → [0, ∞) is continuous and positive definite with respect to A.
If
V (g (x)) + α(x) + kg (x) − xk ≤ V (x)
∀x ∈ Rn ,
then A is pointwise asymptotically stable for x + = g (x).
Proof idea:
•
•
V (g (x)) + α(x) ≤ V (x) ensures asymptotic stability of A (not pointwise!)
V (g (x)) + kg (x) − xk ≤ V (x) ensures finite length, so convergence, of each solution
•
V (a) = 0 at a ∈ A and continuity implies small length of solutions from near a,
and so Lyapunov stability of a
Goebel
Beyond Zeno through a PAS set
Pointwise asymptotic controllability in discrete time
Let G : Rn × U → Rn be a function, U ⊂ Rk be a set. Consider
x + = G (x, u), u ∈ U.
The control system is pointwise asymptotically controllable to a set A ⊂ Rn with locally
exponential convergence rate:
(a) for every ξ ∈ Rn , there exists an open-loop control uξ : {0, 1, 2, . . . } → U such that the
resulting solution φξ from ξ converges, and limj→∞ φξ (j) ∈ A;
(b) for every a ∈ A, for every ε > 0 there exists δ > 0 such that, for every ξ with kξ − ak < δ,
the solution φξ from (a) is such that kφξ (j) − ak < ε for every j;
(c) there exists an open set O containing A and constants M, γ > 0 such that, for every
ξ ∈ O, the solution φξ from (a) satisfies
kφξ (j) − aξ k ≤ Me −γj kξ − aξ k
∀j,
where aξ = limj→∞ φξ (j).
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Beyond Zeno through a PAS set
Optimal feedback stabilization in discrete time
Theorem
If G is continuous, U compact, A closed and PAC Assumption holds for
x + = G (x, u), u ∈ U,
then there exists a feedback u : Rn → U such that A is robustly PAS for
x + = G (x, u(x)).
Goebel
Beyond Zeno through a PAS set
Optimal feedback stabilization in discrete time
Theorem
If G is continuous, U compact, A closed and PAC Assumption holds for
x + = G (x, u), u ∈ U,
then there exists a feedback u : Rn → U such that A is robustly PAS for
x + = G (x, u(x)).
Proof idea: let dA (x) be the distance from A: dA (x) = mina∈A kx − ak, let
V (ξ)
=
inf
∞
X
u:{0,1,2,... }→U
dA (φ(j)) + kφ(j + 1) − φ(j)k
j=0
and consider optimal (or suboptimal) feedback.
Goebel
Beyond Zeno through a PAS set
Optimal feedback stabilization in discrete time
Theorem
If G is continuous, U compact, A closed and PAC Assumption holds for
x + = G (x, u), u ∈ U,
then there exists a feedback u : Rn → U such that A is robustly PAS for
x + = G (x, u(x)).
Proof idea: let dA (x) be the distance from A: dA (x) = mina∈A kx − ak, let
V (ξ)
=
∞
X
inf
u:{0,1,2,... }→U
dA (φ(j)) + kφ(j + 1) − φ(j)k
j=0
and consider optimal (or suboptimal) feedback.
Under further assumptions on g , for example kG (x, u) − xk ≤ ν(dA (x) + kuk), can consider
V (ξ)
=
inf
u:{0,1,2,... }→U
Goebel
∞
X
dA (φ(j)) + ku(j)k
j=0
Beyond Zeno through a PAS set
Robustness of optimal feedback in discrete time
Definition
A is robustly PAS for
x + = G (x, u(x)) =: g (x)
if there exists ρ : Rn → [0, ∞), continuous and positive definite with respect to A and such that
A is PAS for
x + ∈ gρ (x),
[
where gρ : Rn ⇒ Rn is a set-valued mapping defined by gρ (x) =
y + ρ(y )B.
y ∈g (x+ρ(x)B)
This covers measurement error, actuator error, and external disturbance.
Given the optimal value function V , define a set-valued mapping W :
W (x) = x + V (x)B.
This W is a continuous set-valued Lyapunov function, and a result of
Robustness of stability through necessary and sufficient Lyapunov-like conditions ..., G. 2014
implies robustness.
Goebel
Beyond Zeno through a PAS set
The End
Summary:
In the right circumstances (PSOTAS), Zeno solutions to hybrid systems can be extended
past their Zeno times and well-posedness is preserved.
PAS can be achieved via optimal control.
Further questions for hybrid systems:
Converse set-valued Lyapunov results and relation to robustness of PAS?
Infinitesimal characterization of set-valued Lyapunov functions (in continuous and then
hybrid time)?
Thank you for your attention.
Goebel
Beyond Zeno through a PAS set
Bonus: Finite-length Lyapunov function in hybrid setting
Definition
A continuously differentiable function V : Rn → [0, ∞) is a finite-length Lyapunov function if V
is positive definite with respect to A, and there exist continuous c, d : Rn → [0, ∞), positive
definite with respect to A, such that the following hold:
for every x ∈ C , f ∈ F (x),
∇V (x) · f ≤ −c(x) − kf k,
for every x ∈ D, g ∈ G (x),
V (g ) − V (x) ≤ −d(x) − kg − xk.
Inspired by Arc-length-based Lyapunov tests ..., Bhat, Bernstein 10
which used ∇V (x) · f (x) ≤ −kf (x)k for ẋ = f (x)
Goebel
Beyond Zeno through a PAS set
Bonus: Finite-length Lyapunov function in hybrid setting
Definition
A continuously differentiable function V : Rn → [0, ∞) is a finite-length Lyapunov function if V
is positive definite with respect to A, and there exist continuous c, d : Rn → [0, ∞), positive
definite with respect to A, such that the following hold:
for every x ∈ C , f ∈ F (x),
∇V (x) · f ≤ −c(x) − kf k,
for every x ∈ D, g ∈ G (x),
V (g ) − V (x) ≤ −d(x) − kg − xk.
Inspired by Arc-length-based Lyapunov tests ..., Bhat, Bernstein 10
which used ∇V (x) · f (x) ≤ −kf (x)k for ẋ = f (x)
Theorem
If there exists a finite-time Lyapunov function then A is PAS.
Key to proof / justification of name:
J Z tj+1
J
X
X
kφ̇(t, j))k ds +
kφ(tj , j + 1) − φ(tj , j)k ≤ V (φ(0, 0)) − V (φ(T , J))
j=0
tj
j=1
Goebel
Beyond Zeno through a PAS set