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Lyapunov-like functions and Lie brackets Franco Rampazzo Monica Motta 11th Meeting on Nonlinear Hyperbolic PDEs and Applications On the occasion of the 60th birthday of Alberto Bressan Trieste, June 2016 Lyapunov-like functions and Lie

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Lyapunov-like functions and Lie brackets

Franco Rampazzo Monica Motta 11th Meeting on Nonlinear Hyperbolic PDEs and Applications On the occasion of the 60th birthday of Alberto Bressan Trieste, June 2016

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A control system and a target C    ˙ x(t) = f (x(t), u(t)) u(t) ∈ U x(0) = z limt→tu x(t) ∈ C

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A control system and a target C    ˙ x(t) = f (x(t), u(t)) u(t) ∈ U x(0) = z limt→tu x(t) ∈ C

TARGET C

f(x,u) u ϵ U

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Globally Asymptoticly Controllable (GAC) systems

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Globally Asymptoticly Controllable (GAC) systems The system ˙ x = f (x(t), u(t)) is (GAC) to the target C if, from any initial point x(0) = z ∃ a control u(·) s.t d(xz,u(t), C) → 0, in finite or infinite time.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Globally Asymptoticly Controllable (GAC) systems The system ˙ x = f (x(t), u(t)) is (GAC) to the target C if, from any initial point x(0) = z ∃ a control u(·) s.t d(xz,u(t), C) → 0, in finite or infinite time.

  • More precisely , a uniform estimate must hold true:

d(xz,u(t), C) ≤ β(d(z, C), t) where lim

t→Tu β(r, t) = 0 and r → β(r, t) is increasing.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Globally Asymptoticly Controllable systems

z TARGET z

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Globally Asymptoticly Controllable systems

z TARGET z

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Globally Asymptoticly Controllable systems

z TARGET z

Exit time! (Possibly = +∞)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Goal: find sufficient conditions for (GAC)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Goal: find sufficient conditions for (GAC)

A simple idea originally due to A. Lyapunov, in the case of dynamical systems (with no control):

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Goal: find sufficient conditions for (GAC)

A simple idea originally due to A. Lyapunov, in the case of dynamical systems (with no control):

Look for a function V : Rn → [0, +∞], equal to zero on the target and > 0 outside, such that ”V decreases along trajectories”

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Goal: find sufficient conditions for (GAC)

A simple idea originally due to A. Lyapunov, in the case of dynamical systems (with no control):

Look for a function V : Rn → [0, +∞], equal to zero on the target and > 0 outside, such that ”V decreases along trajectories”

In the case of control systems, where one has a trajectory for each control, one needs to say: Look for a function V : Rn → [0, +∞], equal to zero on the target and > 0 outside, such that

”V decreases along suitable trajectories”

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Goal: find sufficient conditions for (GAC)

A simple idea originally due to A. Lyapunov, in the case of dynamical systems (with no control):

Look for a function V : Rn → [0, +∞], equal to zero on the target and > 0 outside, such that ”V decreases along trajectories”

In the case of control systems, where one has a trajectory for each control, one needs to say: Look for a function V : Rn → [0, +∞], equal to zero on the target and > 0 outside, such that

”V decreases along suitable trajectories”

Such a V is called a Control Lyapunov Function

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Two Lyapunovs

Lyapunov Lyapunov

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Two Lyapunovs

Liapunov,Serjei, Russian MUSICIAN Liapunov,Aleksandr, MATHEMATICIAN, Serjei's brother

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A third Lyapunov

Figure: Aleksey Lyapunov (range of vector measures)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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TARGET

Level sets of a Lyapunov function V

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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TARGET

Level sets of a Lyapunov function V

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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TARGET

Level sets of a Lyapunov function V

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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TARGET

Level sets of a Lyapunov function V

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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TARGET

Level sets of a Lyapunov function V

DV DV DV DV DV

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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(smooth) Control Lyapunov Functions:

A C 1 map V : Rn → R+ is a Control Lyapunov Function (CLF), if V is proper; V (x) > 0 if x / ∈ C and V (x) = 0 if x ∈ C; it verifies H(x, DV(x))<0 where H(x, p) := min

u∈Up , f (x, u)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Why control Lyapunov functions are important?

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Why control Lyapunov functions are important? Lyapunov-like Theorem: If there exists a Control Lyapunov Function then the system is Globally Asymptotically Controllable

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Smooth Lyapunov control functions?

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Smooth Lyapunov control functions?

ACTUALLY: it may be difficult or even impossible to find a smooth Lyapunov function.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Smooth Lyapunov control functions?

ACTUALLY: it may be difficult or even impossible to find a smooth Lyapunov function. This might be somehow unexpected, for the differential inequality H(x, DV(x)) < 0 seems far less demanding than the corresponding equation H(x, DV(x)) = 0....

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Smooth Lyapunov control functions?

ACTUALLY: it may be difficult or even impossible to find a smooth Lyapunov function. This might be somehow unexpected, for the differential inequality H(x, DV(x)) < 0 seems far less demanding than the corresponding equation H(x, DV(x)) = 0.... Instead , even in some geometrically promising cases, it may happen that H(x, DV(x))(= inf

u DV, f(z, u))≤ 0

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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H(x, DV (x)) = inf

u DV , f (z, u)≤ 0

(instead of inf

u DV , f (z, u)< 0).

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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H(x, DV (x)) = inf

u DV , f (z, u)≤ 0

(instead of inf

u DV , f (z, u)< 0).

TARGET Level sets of a Liapunov function V

DV DV DV DV DV f(x,u) f(x,u) f(x,u)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Zooming, we get:

TARGET

DV DV DV DV

)

f(x,u) f(x,u)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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An example in R3: the Nonholonomic Integrator.

Brockett’s nonholonomic integrator.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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An example in R3: the Nonholonomic Integrator.

Brockett’s nonholonomic integrator. f1 =   1 −x2   f2 =   1 x1   ˙ x = f (x, u) := u1f1 + u2f2 |u| = 1 Target: C = {0}.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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An example in R3: the Nonholonomic Integrator.

Brockett’s nonholonomic integrator. f1 =   1 −x2   f2 =   1 x1   ˙ x = f (x, u) := u1f1 + u2f2 |u| = 1 Target: C = {0}. (By Chow-Rashewsky Th., this system is locally controllable )

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator:

TRY the (smooth!)distance function V (x) = d(x, 0) as Lyapunov function:

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator:

TRY the (smooth!)distance function V (x) = d(x, 0) as Lyapunov function:

f1 f2 f1 f2 Level sets of V(x)=d(x,0) (=Spheres)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator:

TRY the (smooth!)distance function V (x) = d(x, 0) as Lyapunov function:

f1 f2 f1 f2 Level sets of V(x)=d(x,0) (=Spheres)

Does the distance V (x) = d(x, 0) verify H(x, DV (x)) = inf

u

  • DV (x), u1f1(x) + u2f2(x)
  • <0 ?

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator:

TRY the (smooth!)distance function V (x) = d(x, 0) as Lyapunov function:

f1 f2 f1 f2 Level sets of V(x)=d(x,0) (=Spheres)

Does the distance V (x) = d(x, 0) verify H(x, DV (x)) = inf

u

  • DV (x), u1f1(x) + u2f2(x)
  • <0 ?

No, it doesn’t!

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator:

TRY the (smooth!)distance function V (x) = d(x, 0) as Lyapunov function:

f1 f2 f1 f2 Level sets of V(x)=d(x,0) (=Spheres)

Does the distance V (x) = d(x, 0) verify H(x, DV (x)) = inf

u

  • DV (x), u1f1(x) + u2f2(x)
  • <0 ?

No, it doesn’t! In fact, on the vertical axis one has H(x, DV (x)) = inf

u

  • DV (x), u1f1(x) + u2f2(x)
  • =0

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator: So the distance V (x) = d(x, 0) is not a Lyapunov function

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator: So the distance V (x) = d(x, 0) is not a Lyapunov function because the system dynamics is in the kernel of DV (x).

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator: So the distance V (x) = d(x, 0) is not a Lyapunov function because the system dynamics is in the kernel of DV (x). Maybe another smooth function would work?

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator: So the distance V (x) = d(x, 0) is not a Lyapunov function because the system dynamics is in the kernel of DV (x). Maybe another smooth function would work? NO!

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Brockett’s nonholonomic integrator: So the distance V (x) = d(x, 0) is not a Lyapunov function because the system dynamics is in the kernel of DV (x). Maybe another smooth function would work? NO! Actually, by algebraic topological arguments (essentially the hairy ball theorem) one can prove that No (smooth) Lyapunov functions exist

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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”What to do?”

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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”What to do?” Nonsmooth Answer:

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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”What to do?” Nonsmooth Answer: ”Avoid bad points where H(x, DV ) = 0 by allowing ... Lyapunov functions which are nonsmooth at those bad points”

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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”What to do?” Nonsmooth Answer: ”Avoid bad points where H(x, DV ) = 0 by allowing ... Lyapunov functions which are nonsmooth at those bad points” (this rules out the distance function in the example)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Formal definition of nonsmooth Lyapunov Function: (replace DV with D∗V )

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Formal definition of nonsmooth Lyapunov Function: (replace DV with D∗V )

A map V : Rn → R+ is a Control Lyapunov Function (CLF), if V is continuous, locally semiconcave, proper; V (x) > 0 if x / ∈ C and V (x) = 0 if x ∈ C; It verifies the partial differential inequality :

H(x, D∗V (x)) = min

u∈UD∗V (x) , f (x, u) < 0

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Formal definition of nonsmooth Lyapunov Function: (replace DV with D∗V )

A map V : Rn → R+ is a Control Lyapunov Function (CLF), if V is continuous, locally semiconcave, proper; V (x) > 0 if x / ∈ C and V (x) = 0 if x ∈ C; It verifies the partial differential inequality :

H(x, D∗V (x)) = min

u∈UD∗V (x) , f (x, u) < 0

Here D∗V (x) denotes the set of limiting gradients of V at x: D∗V (x) . =

  • w : w = lim

k DV (xk),

lim

k zk = z

  • .

Remark: In general D∗V (x) is not convex.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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H(x, D∗V (x)) = min

u∈UD∗V (x) , f (x, u) < 0

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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H(x, D∗V (x)) = min

u∈UD∗V (x) , f (x, u) < 0

a non-homogeneous special case: Hℓ(x, D∗V (x)) = min

u∈U (D∗V (x) , f (x, u) + p0ℓ(x, u)) < 0

for some p0 ≥ 0, where ℓ(x, u) ≥ 0 is a current cost. V is called p0-Minimum Restraint Function: if p0 > 0 its existence guarantees (Motta-Rampazzo 2013): Global Asymptotic Controllability, A bound on the value function W = inf Tx ℓ(x(t), u(t))dx W ≤ V /p0 .

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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H(x, D∗V (x)) = min

u∈UD∗V (x) , f (x, u) < 0

a non-homogeneous special case: Hℓ(x, D∗V (x)) = min

u∈U (D∗V (x) , f (x, u) + p0ℓ(x, u)) < 0

for some p0 ≥ 0, where ℓ(x, u) ≥ 0 is a current cost. V is called p0-Minimum Restraint Function: if p0 > 0 its existence guarantees (Motta-Rampazzo 2013): Global Asymptotic Controllability, A bound on the value function W = inf Tx ℓ(x(t), u(t))dx W ≤ V /p0 . I am NOT speaking of this special case today

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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In the case of Brockett’s nonholonomic integrator

  • ne can try V = max
  • x2

1 + x2 2, |x3| −

  • x2

1 + x2 2

  • ,

which has singularities on the vertical axis:

f1 f2

V(x)= cost.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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In the case of Brockett’s nonholonomic integrator

  • ne can try: V = max
  • x2

1 + x2 2, |x3| −

  • x2

1 + x2 2

  • ,

which has singularities on the vertical axis H = 0 avoided!

f1 f2

V(x)= cost. D*V

Notice: NO VERTICAL GRADIENTS

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Why nonsmooth Lyapunov control functions are important?

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Why nonsmooth Lyapunov control functions are important?

Because they are useful, namely we can extend the smooth Lyapunov-like theorem:

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Why nonsmooth Lyapunov control functions are important?

Because they are useful, namely we can extend the smooth Lyapunov-like theorem:

Nonsmooth Lyapunov-like Theorem: If there exists a Control Lyapunov Function then the system is Globally Asymptotically Controllable

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Why nonsmooth Lyapunov control functions are important?

Because they are useful, namely we can extend the smooth Lyapunov-like theorem:

Nonsmooth Lyapunov-like Theorem: If there exists a Control Lyapunov Function then the system is Globally Asymptotically Controllable

Many important results since the 80’s, with various notions of nonsmooth gradients and/or generalized notions of ODE solutions. Quite incomplete list of authors includes : Sontag, Artstein,Bacciotti,Clarke, Subbotin,Malisoff, Rifford

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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What if we insist with smooth functions ?

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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What if we insist with smooth functions ? For instance, some function V such that it is useful as a Lyapunov function (i.e., a Lyapunov-likeTheorem holds true) it has more chances to be smooth ???

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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IDEA: USE NON-COMMUTATIVITY

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A movie on non-commutativity, in R3: the ”Nonholonomic integrator”

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A movie on non-commutativity, in R3: the ”Nonholonomic integrator”

f1 =   1 −x2   f2 =   1 x   ˙ x = u1f1 + u2f2

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A movie of non-commutativity, in R3: the ”Nonholonomic integrator”

f1 f2 Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A movie of non-commutativity, in R3: the ”Nonholonomic integrator”

f1 f2

Φt

f1(x)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A movie of non-commutativity, in R3: the ”Nonholonomic integrator”

f1 f2

Φt

f2 ◦ Φt f1(x)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A movie of non-commutativity, in R3: the ”Nonholonomic integrator”

f1

  • f1

f2

Φ−t

f1 ◦ Φt f2 ◦ Φt f1(x)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A movie of non-commutativity, in R3: the ”Nonholonomic integrator”

f1

  • f1
  • f2

f2

Φ−t

f2 ◦ Φ−t f1 ◦ Φt f2 ◦ Φt f1(x) = x

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Lie brackets Definition Lie bracket of C 1 vector fields f, g: [f, g] := Dg · f − Df · g

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Lie brackets Definition Lie bracket of C 1 vector fields f, g: [f, g] := Dg · f − Df · g

Basic properties:

1

[f , g] is a vector field (i.e. it is an intrinsic object)

2

[f , g] = −[g, f ] (antisymmetry) ( = ⇒ [f , f ] = 0)

3

[f , [g, h]] + [g, [h, f ]] + [h, [f , g]] = 0 (Jacobi identy)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Remind:The asymptotics of a Lie bracket:

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Remind:The asymptotics of a Lie bracket:

Set Ψ[f1,f2](t)(x) := Φ−t

f2 ◦ Φ−t f1 ◦ Φt f2 ◦ Φt f1(x)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Remind:The asymptotics of a Lie bracket:

Set Ψ[f1,f2](t)(x) := Φ−t

f2 ◦ Φ−t f1 ◦ Φt f2 ◦ Φt f1(x)

Asymptotic formula: Ψ[f1,f2](t)(x) − x = t2[f1, f2](x) + o(t2)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Remind:The asymptotics of a Lie bracket:

Set Ψ[f1,f2](t)(x) := Φ−t

f2 ◦ Φ−t f1 ◦ Φt f2 ◦ Φt f1(x)

Asymptotic formula: Ψ[f1,f2](t)(x) − x = t2[f1, f2](x) + o(t2)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Sophus Lie

Figure: Continuous (Lie!) groups, geometry, ODEs

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Observe: Lie brackets show up in higher order necessary conditions for minima controllability boundary conditions of HJ equations, to guarantee continuity of time optimal functions BUT

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Observe: Lie brackets show up in higher order necessary conditions for minima controllability boundary conditions of HJ equations, to guarantee continuity of time optimal functions BUT they are not included in HJ equations or HJ inequalities

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A new PDI (defining a Lyapunov’like function)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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A new PDI (defining a Lyapunov’like function)

For simplicity we limit our attention to control-linear systems: ˙ x =

m

  • i=1

uigi(x) u = ±e1, . . . , ±em

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 83

A new PDI (defining a Lyapunov’like function)

For simplicity we limit our attention to control-linear systems: ˙ x =

m

  • i=1

uigi(x) u = ±e1, . . . , ±em Set F(1) :=

  • ±f1, . . . , ±fm
  • Franco RampazzoMonica Motta

Lyapunov-like functions and Lie brackets

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SLIDE 84

A new PDI (defining a Lyapunov’like function)

For simplicity we limit our attention to control-linear systems: ˙ x =

m

  • i=1

uigi(x) u = ±e1, . . . , ±em Set F(1) :=

  • ±f1, . . . , ±fm
  • F(2) := F(1)∪
  • [fi, fj]

i, j = 1, . . . , m

  • . . .

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 85

A new PDI (defining a Lyapunov’like function)

For simplicity we limit our attention to control-linear systems: ˙ x =

m

  • i=1

uigi(x) u = ±e1, . . . , ±em Set F(1) :=

  • ±f1, . . . , ±fm
  • F(2) := F(1)∪
  • [fi, fj]

i, j = 1, . . . , m

  • . . .

F(j) := F(j−1) ∪ {Lie brackets of degree j}

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 86

A new PDI (defining a Lyapunov’like function)

For simplicity we limit our attention to control-linear systems: ˙ x =

m

  • i=1

uigi(x) u = ±e1, . . . , ±em Set F(1) :=

  • ±f1, . . . , ±fm
  • F(2) := F(1)∪
  • [fi, fj]

i, j = 1, . . . , m

  • . . .

F(j) := F(j−1) ∪ {Lie brackets of degree j}

H(j)(x, p) := inf

v∈F(j)(x) p, v

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SLIDE 87

A new PDI (defining a Lyapunov’like function)

For simplicity we limit our attention to control-linear systems: ˙ x =

m

  • i=1

uigi(x) u = ±e1, . . . , ±em Set F(1) :=

  • ±f1, . . . , ±fm
  • F(2) := F(1)∪
  • [fi, fj]

i, j = 1, . . . , m

  • . . .

F(j) := F(j−1) ∪ {Lie brackets of degree j}

H(j)(x, p) := inf

v∈F(j)(x) p, v

Notice that H(1) = H and

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SLIDE 88

A new PDI (defining a Lyapunov’like function)

For simplicity we limit our attention to control-linear systems: ˙ x =

m

  • i=1

uigi(x) u = ±e1, . . . , ±em Set F(1) :=

  • ±f1, . . . , ±fm
  • F(2) := F(1)∪
  • [fi, fj]

i, j = 1, . . . , m

  • . . .

F(j) := F(j−1) ∪ {Lie brackets of degree j}

H(j)(x, p) := inf

v∈F(j)(x) p, v

Notice that H(1) = H and H = H(1) ≥ H(2) ≥ . . . H(k−1) ≥ H(k)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 89

A new PDI (defining a Lyapunov’like function)

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SLIDE 90

A new PDI (defining a Lyapunov’like function)

Definition

Let U : Rn \ C → R be continuous function, locally semiconcave, positive definite, and proper. If H(h)(x, D∗U(x)) < 0 we say that U is a degree-h Control Lyapunov Function

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SLIDE 91

A new PDI (defining a Lyapunov’like function)

Definition

Let U : Rn \ C → R be continuous function, locally semiconcave, positive definite, and proper. If H(h)(x, D∗U(x)) < 0 we say that U is a degree-h Control Lyapunov Function Remark: if h1 ≤ h2, U is a degree-h1 CLF = ⇒ U is a degree-h2 CLF

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SLIDE 92

A new PDI (defining a Lyapunov’like function)

Definition

Let U : Rn \ C → R be continuous function, locally semiconcave, positive definite, and proper. If H(h)(x, D∗U(x)) < 0 we say that U is a degree-h Control Lyapunov Function Remark: if h1 ≤ h2, U is a degree-h1 CLF = ⇒ U is a degree-h2 CLF Indeed: H = H(1) ≥ H(2) ≥ . . . H(k−1) ≥ H(k)

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SLIDE 93

Why degree-h control Lyapunov functions are useful? (h ≥ 1)

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SLIDE 94

Why degree-h control Lyapunov functions are useful? (h ≥ 1) Because a ”Lyapunov-like” theorem holds true!

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SLIDE 95

Why degree-h control Lyapunov functions are useful? (h ≥ 1) Because a ”Lyapunov-like” theorem holds true! degree-h Lyapunov-like Theorem: If there exists a degree-h Control Lyapunov Function then the system is Globally Asymptotically Controllable

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SLIDE 96

Why degree-h control Lyapunov functions are useful? (h ≥ 1) Because a ”Lyapunov-like” theorem holds true! degree-h Lyapunov-like Theorem: If there exists a degree-h Control Lyapunov Function then the system is Globally Asymptotically Controllable Moreover, degree-h Control Lyapunov Functions are likely smoother than stan- dard CLFs

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SLIDE 97

Application to the nonholonomic integrator

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 98

Application to the nonholonomic integrator

˙ x = u1f1 + u2f2

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 99

Application to the nonholonomic integrator

˙ x = u1f1 + u2f2 f1 =   1 −x2   , f2 =   1 x1   , [f1, f2] =   2  

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 100

Application to the nonholonomic integrator

˙ x = u1f1 + u2f2 f1 =   1 −x2   , f2 =   1 x1   , [f1, f2] =   2  

Trivial calculations give: H1(x, p) = H(x, p) = −

  • (p1 − x2p3)2 + (p2 + x2p3)2

H2(x, p) = min

  • −
  • (p1 − x2p3)2 + (p2 + x2p3)2 , − |p3|

16

  • Franco RampazzoMonica Motta

Lyapunov-like functions and Lie brackets

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SLIDE 101

Application to the nonholonomic integrator

˙ x = u1f1 + u2f2 f1 =   1 −x2   , f2 =   1 x1   , [f1, f2] =   2  

Trivial calculations give: H1(x, p) = H(x, p) = −

  • (p1 − x2p3)2 + (p2 + x2p3)2

H2(x, p) = min

  • −
  • (p1 − x2p3)2 + (p2 + x2p3)2 , − |p3|

16

  • Remember the BAD GRADIENT on the x3-axis: Dd(x) = (0, 0, 1):

H

  • (0, 0, x3), Dd(x)
  • = H1

(0, 0, x3), Dd(x)

  • = 0

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 102

Application to the nonholonomic integrator

˙ x = u1f1 + u2f2 f1 =   1 −x2   , f2 =   1 x1   , [f1, f2] =   2  

Trivial calculations give: H1(x, p) = H(x, p) = −

  • (p1 − x2p3)2 + (p2 + x2p3)2

H2(x, p) = min

  • −
  • (p1 − x2p3)2 + (p2 + x2p3)2 , − |p3|

16

  • Remember the BAD GRADIENT on the x3-axis: Dd(x) = (0, 0, 1):

H

  • (0, 0, x3), Dd(x)
  • = H1

(0, 0, x3), Dd(x)

  • = 0

Istead Dd(x) = (0, 0, 1) IS NOT A BAD GRADIENT FOR H2(x, p): H2 (0, 0, x3), Dd(x)

  • = − 1

16 < 0

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SLIDE 103

So the distance function -while NOT being a standard Lyapunov function-

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 104

So the distance function -while NOT being a standard Lyapunov function- is a (C ∞!) Lyapunov function of degree-2

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 105

So the distance function -while NOT being a standard Lyapunov function- is a (C ∞!) Lyapunov function of degree-2 In other words, the new Partial Differential Inequality H2 < 0 does have a C ∞ solution (while H < 0 has no C 1 solutions)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 106

So the distance function -while NOT being a standard Lyapunov function- is a (C ∞!) Lyapunov function of degree-2 In other words, the new Partial Differential Inequality H2 < 0 does have a C ∞ solution (while H < 0 has no C 1 solutions)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 107

Two Variations

  • n the Nonholonomic Integrator

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 108

Variation 1 on the Nonholonomic Integrator:higher order

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 109

Variation 1 on the Nonholonomic Integrator:higher order

˙ x = u1f1 + u2f2

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 110

Variation 1 on the Nonholonomic Integrator:higher order

˙ x = u1f1 + u2f2 f1 =   1 x2

2

  , f2 =   1 x2

1

  , [f1, f2] =   2x1 − 2x2  

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 111

Variation 1 on the Nonholonomic Integrator:higher order

˙ x = u1f1 + u2f2 f1 =   1 x2

2

  , f2 =   1 x2

1

  , [f1, f2] =   2x1 − 2x2  

So : [f1, f2] = 0 on x1 = x2, in particular on the x3-axis.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 112

Variation 1 on the Nonholonomic Integrator:higher order

˙ x = u1f1 + u2f2 f1 =   1 x2

2

  , f2 =   1 x2

1

  , [f1, f2] =   2x1 − 2x2  

So : [f1, f2] = 0 on x1 = x2, in particular on the x3-axis. BUT [f1, [f1, f2]] =   2  

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 113

Variation 1 on the Nonholonomic Integrator:higher order

˙ x = u1f1 + u2f2 f1 =   1 x2

2

  , f2 =   1 x2

1

  , [f1, f2] =   2x1 − 2x2  

So : [f1, f2] = 0 on x1 = x2, in particular on the x3-axis. BUT [f1, [f1, f2]] =   2   Trivial calculations give: H(1)(x, Dd(x)) ≤ 0 H(2)(x, d(x)) ≤ 0 with H(2) = 0 on the x3-axis BUT H(3)(x, d(x)) =< 0 !

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 114

Variation 1 on the Nonholonomic Integrator:higher order

˙ x = u1f1 + u2f2 f1 =   1 x2

2

  , f2 =   1 x2

1

  , [f1, f2] =   2x1 − 2x2  

So : [f1, f2] = 0 on x1 = x2, in particular on the x3-axis. BUT [f1, [f1, f2]] =   2   Trivial calculations give: H(1)(x, Dd(x)) ≤ 0 H(2)(x, d(x)) ≤ 0 with H(2) = 0 on the x3-axis BUT H(3)(x, d(x)) =< 0 ! Hence the distance d = d(x) is a degree-3 control Lyapunov function.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 115

Variation 2 of the Nonholonomic Integrator:Lipschitz fields

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 116

Variation 2 of the Nonholonomic Integrator:Lipschitz fields

˙ x = u1f1 + u2f2

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 117

Variation 2 of the Nonholonomic Integrator:Lipschitz fields

˙ x = u1f1 + u2f2 f1 =   1 |x2| − 2x2   , f2 =   1 |x1| + 2x1   , [f1, f2]set =   I(x)  

where I(x) is a compact interval.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 118

Variation 2 of the Nonholonomic Integrator:Lipschitz fields

˙ x = u1f1 + u2f2 f1 =   1 |x2| − 2x2   , f2 =   1 |x1| + 2x1   , [f1, f2]set =   I(x)  

where I(x) is a compact interval. [h, k]set is a set-valued Lie bracket for Lipschitz vector field (Rampazzo-Sussmann,2001)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 119

Variation 2 of the Nonholonomic Integrator:Lipschitz fields

˙ x = u1f1 + u2f2 f1 =   1 |x2| − 2x2   , f2 =   1 |x1| + 2x1   , [f1, f2]set =   I(x)  

where I(x) is a compact interval. [h, k]set is a set-valued Lie bracket for Lipschitz vector field (Rampazzo-Sussmann,2001) The notion of degree-2 Lyapunov function can be extended to Lipschitz vector fields and a Lyapunov-like theorem hold true.The Hamiltonian H(2) is now a inf-sup, intead of a inf. In this case: H(2)(x, p) = inf

  • H(1)(x, p) ,

sup

w∈I(x)

w p3 , sup

w∈−I(x)

w p3

  • Franco RampazzoMonica Motta

Lyapunov-like functions and Lie brackets

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SLIDE 120

Variation 2 of the Nonholonomic Integrator:Lipschitz fields

˙ x = u1f1 + u2f2 f1 =   1 |x2| − 2x2   , f2 =   1 |x1| + 2x1   , [f1, f2]set =   I(x)  

where I(x) is a compact interval. [h, k]set is a set-valued Lie bracket for Lipschitz vector field (Rampazzo-Sussmann,2001) The notion of degree-2 Lyapunov function can be extended to Lipschitz vector fields and a Lyapunov-like theorem hold true.The Hamiltonian H(2) is now a inf-sup, intead of a inf. In this case: H(2)(x, p) = inf

  • H(1)(x, p) ,

sup

w∈I(x)

w p3 , sup

w∈−I(x)

w p3

  • The distance d = d(x) (from the origin) turns out to be a C ∞

degree-2 Lyapunov function, i.e. H(2)(x, Dd(x)) < 0

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 121

The idea behind the new PDI:

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 122

The idea behind the new PDI:

Work the monotonicity issue: ”A Lyapunov functions is decreasing to zero along (suitable) trajectories”

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 123

The idea behind the new PDI:

Work the monotonicity issue: ”A Lyapunov functions is decreasing to zero along (suitable) trajectories” THE STANDARD INEQUALITY:

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 124

The idea behind the new PDI:

Work the monotonicity issue: ”A Lyapunov functions is decreasing to zero along (suitable) trajectories” THE STANDARD INEQUALITY:

1 For any u let x(·) be such a trajectory issuing from x with

derivative ˙ x(0) = < 0

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 125

The idea behind the new PDI:

Work the monotonicity issue: ”A Lyapunov functions is decreasing to zero along (suitable) trajectories” THE STANDARD INEQUALITY:

1 For any u let x(·) be such a trajectory issuing from x with

derivative ˙ x(0) = < 0

2 Choose (if possible!) u so that the derivative of t → V (x(t))

at t = 0 is strictly negative:

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 126

The idea behind the new PDI:

Work the monotonicity issue: ”A Lyapunov functions is decreasing to zero along (suitable) trajectories” THE STANDARD INEQUALITY:

1 For any u let x(·) be such a trajectory issuing from x with

derivative ˙ x(0) = < 0

2 Choose (if possible!) u so that the derivative of t → V (x(t))

at t = 0 is strictly negative: d dt U(x(t))t=0 = DV (x), ˙ x(0) = DV (x),

m

  • i=1

uifi(x) < 0

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 127

The idea behind the new PDI:

Work the monotonicity issue: ”A Lyapunov functions is decreasing to zero along (suitable) trajectories” THE STANDARD INEQUALITY:

1 For any u let x(·) be such a trajectory issuing from x with

derivative ˙ x(0) = < 0

2 Choose (if possible!) u so that the derivative of t → V (x(t))

at t = 0 is strictly negative: d dt U(x(t))t=0 = DV (x), ˙ x(0) = DV (x),

m

  • i=1

uifi(x) < 0

3 This gives the classical inequality:

H(x, DV (x)) = H1(x, DV (x)) = inf

u DV (x), m

  • i=1

uifi(x) < 0

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 128

The idea behind the new PDI:

What are we doing if DV (x), m

i=1 uifi(x) = 0 for all u, as in

the case of the x3-axis in the nonholonomic integrator ?

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 129

The idea behind the new PDI:

What are we doing if DV (x), m

i=1 uifi(x) = 0 for all u, as in

the case of the x3-axis in the nonholonomic integrator ?Solve the equation ¨ x(t) = [fi, fj]t(x(t)) x(0) = x ˙ x(0) = 0 where [fi, fj]t is the integrating bracket as defined in (Ermal-Rampazzo 2015 ) ([fi, fj]t = [fi, fj] for t = 0)

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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SLIDE 130

The idea behind the new PDI:

What are we doing if DV (x), m

i=1 uifi(x) = 0 for all u, as in

the case of the x3-axis in the nonholonomic integrator ?Solve the equation ¨ x(t) = [fi, fj]t(x(t)) x(0) = x ˙ x(0) = 0 where [fi, fj]t is the integrating bracket as defined in (Ermal-Rampazzo 2015 ) ([fi, fj]t = [fi, fj] for t = 0) By differentiating one gets:

d dt V (x(t))t=0 = DV (x), ˙

x(0) = DV (x), m

i=1 uifi(x) =

H(x, DV (x)) = H1(x, DV (x)) = infuDV (x), m

i=1 uifi(x)= 0 d2 d2t U(x(t))t=0 = ˙

x(0)D2V (x), ˙ x(0) + DV (x), ¨ x(0) = = 0 + DV (x), [fi, fj] ≤ H2(x, DV (x))< 0

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[1] Motta, F. R., (2013) Asymptotic controllability and optimal control JDE [2] E.Feleqi, F.R. (2015) Integral representation for bracket generating multi-flows DCDS(A) [3] Motta, F. R., (2016) Lyapunov-like functions involving Lie brackets

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SLIDE 132

Thank you for your attention! Happy birthday dear friend Alberto!

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

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Definition (Semiconcavity). Let Ω ⊂ Rn. A continuous function F : Ω → R is said to be semiconcave on Ω if there exist ρ > 0 such that F(z1) + F(z2) − 2F z1 + z2 2

  • ≤ ρ|z1 − z2|2,

for all z1, z2 ∈ Ω such that [z1, z2] ⊂ Ω. The constant ρ above is called a semiconcavity constant for F in Ω. Fis said to be locally semiconcave on Ω if it semiconcave on every compact subset of Ω.

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets