Lyapunov-like functions and Lie bracketsmarson/TS16/Slides/Rampazzo.pdf · Lyapunov-like functions...

Lyapunov-like functions and Lie brackets

Franco RampazzoMonica Motta

11th Meeting on Nonlinear Hyperbolic PDEs andApplications

On the occasion of the 60th birthday of Alberto BressanTrieste, June 2016

Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

A control system and a target Cx(t) = f (x(t), u(t)) u(t) ∈ Ux(0) = zlimt→tu x(t) ∈ C

system pict1.pdf

TARGET C

f(x,u) u ϵ U

A control system and a target Cx(t) = f (x(t), u(t)) u(t) ∈ Ux(0) = zlimt→tu x(t) ∈ C

system pict1.pdf

TARGET C

f(x,u) u ϵ U

Globally Asymptoticly Controllable (GAC) systems

The system x = f (x(t), u(t))is (GAC) to the target Cif, 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 limt→Tu

β(r , t) = 0 and r 7→ β(r , t) is increasing.

d(xz ,u(t), C)→ 0,

d(xz,u(t), C) ≤ β(d(z , C), t)

where limt→Tu

d(xz ,u(t), C)→ 0,

d(xz,u(t), C) ≤ β(d(z , C), t)

where limt→Tu

Globally Asymptoticly Controllable systems

TARGET

Figure: Globally Asymptoticly Controllable systems

TARGET

Exit time!

(Possibly = +)

Goal: find sufficient conditions for (GAC)

A simple idea originally due to A. Lyapunov, in the case ofdynamical 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 foreach control, one needs to say:

”V decreases along suitable trajectories”

Such a V is called a Control Lyapunov Function

Two Lyapunovs

Lyapunov

Two Lyapunovs

Liapunov,Serjei, Russian MUSICIAN

Liapunov,Aleksandr, MATHEMATICIAN,Serjei's brother

A third Lyapunov

Figure: Aleksey Lyapunov (range of vector measures)

TARGET

Level sets of a Lyapunov function V

TARGET

(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

whereH(x,p) := min

u∈U〈p , f (x , u)〉

Why control Lyapunov functions are important?

Lyapunov-like Theorem:

If there exists a Control LyapunovFunction

then the system is GloballyAsymptotically Controllable

Why control Lyapunov functions are important?

Lyapunov-like Theorem:

Smooth Lyapunov control functions?

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

This might be somehow unexpected, for the differentialinequality

H(x,DV(x)) < 0

seems far less demandingthan the corresponding equation

H(x,DV(x)) = 0....

Instead , even in some geometrically promising cases, itmay happen that

H(x,DV(x))(= infu〈DV, f(z,u))〉≤ 0

H(x,DV(x)) < 0

H(x,DV(x)) = 0....

H(x,DV(x)) < 0

H(x,DV(x)) = 0....

H(x,DV(x)) < 0

H(x,DV(x)) = 0....

H(x ,DV (x)) = infu〈DV , f (z , u)〉≤ 0

(instead of infu〈DV , f (z , u)〉< 0).

TARGET

Level sets of a Liapunov function V

DVf(x,u)

f(x,u)

H(x ,DV (x)) = infu〈DV , f (z , u)〉≤ 0

(instead of infu〈DV , f (z , u)〉< 0).

TARGET

Level sets of a Liapunov function V

DVf(x,u)

f(x,u)

Zooming, we get:

TARGET

F(z,u)

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

An example in R3: the Nonholonomic Integrator.

Brockett’s nonholonomic integrator.

10−x2

x = f (x , u) := u1f1 + u2f2 |u| = 1

Target: C = 0.

(By Chow-Rashewsky Th., this system is locallycontrollable )

10−x2

x = f (x , u) := u1f1 + u2f2 |u| = 1

Target: C = 0.

10−x2

x = f (x , u) := u1f1 + u2f2 |u| = 1

Target: C = 0.

Brockett’s nonholonomic integrator:

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

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

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

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

⟩<0 ?

No, it doesn’t! In fact, on the vertical axis one has

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

⟩<0 ?

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

⟩<0 ?

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

⟩<0 ?

No, it doesn’t!

In fact, on the vertical axis one has

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

⟩<0 ?

H(x ,DV (x)) = infu

⟨DV (x), u1f1(x) + u2f2(x)

So the distance V (x) = d(x , 0) is not aLyapunov function

because the system dynamics is in thekernel of DV (x).

Maybe another smooth function wouldwork?

NO!Actually, by algebraic topologicalarguments (essentially the hairy balltheorem) one can prove that

No (smooth) Lyapunov functions exist

So the distance V (x) = d(x , 0) is not aLyapunov functionbecause the system dynamics is in thekernel of DV (x).

Actually, by algebraic topologicalarguments (essentially the hairy balltheorem) one can prove that

”What to do?”

Nonsmooth Answer:”Avoid bad points where H(x ,DV ) = 0 byallowing ... Lyapunov functions which arenonsmooth at those bad points”(this rules out the distance function in theexample)

”What to do?”

Nonsmooth Answer:

”Avoid bad points where H(x ,DV ) = 0 byallowing ... Lyapunov functions which arenonsmooth at those bad points”(this rules out the distance function in theexample)

”What to do?”

Nonsmooth Answer:”Avoid bad points where H(x ,DV ) = 0 byallowing ... Lyapunov functions which arenonsmooth at those bad points”

(this rules out the distance function in theexample)

”What to do?”

Nonsmooth Answer:”Avoid bad points where H(x ,DV ) = 0 byallowing ... Lyapunov functions which arenonsmooth at those bad points”(this rules out the distance function in theexample)

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;

It verifies the partial differential inequality :

H(x ,D∗V (x)) = minu∈U〈D∗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

kDV (xk), lim

kzk = z

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

D∗V (x).

=w : w = lim

kDV (xk), lim

kzk = z

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

D∗V (x).

=w : w = lim

kDV (xk), lim

kzk = z

Remark: In general D∗V (x) is not convex.Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

a non-homogeneous special case:

H`(x ,D∗V (x)) = min

u∈U(〈D∗V (x) , f (x , u)〉+ p0`(x , u)) < 0

for some p0 ≥ 0, where `(x , u) ≥ 0 is a current cost.V is called p0-Minimum Restraint Function: if p0 > 0 its existenceguarantees (Motta-Rampazzo 2013):

Global Asymptotic Controllability,

A bound on the value function W = inf∫ Tx

0 `(x(t), u(t))dx

W ≤ V /p0

I am NOT speaking of this special case today

u∈U(〈D∗V (x) , f (x , u)〉+ p0`(x , u)) < 0

0 `(x(t), u(t))dx

W ≤ V /p0

I am NOT speaking of this special case today

u∈U(〈D∗V (x) , f (x , u)〉+ p0`(x , u)) < 0

0 `(x(t), u(t))dx

W ≤ V /p0

I am NOT speaking of this special case todayFranco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

In the case of Brockett’s nonholonomic integrator

one can try V = max

√x21 + x22 , |x3| −

√x21 + x22

which has singularities on the vertical axis:

V(x)= cost.

In the case of Brockett’s nonholonomic integrator

one can try: V = max

√x21 + x22 , |x3| −

√x21 + x22

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

V(x)= cost.D*V

Notice:NO VERTICALGRADIENTS

Why nonsmooth Lyapunov control functions areimportant?

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

Nonsmooth Lyapunov-like Theorem:

Many important results since the 80’s, with various notions ofnonsmooth gradients and/or generalized notions of ODE solutions.

Quite incomplete list of authors includes : Sontag,Artstein,Bacciotti,Clarke, Subbotin,Malisoff, Rifford

What if we insist with smooth functions ?

For instance, some function V such that

it is useful as a Lyapunov function (i.e., aLyapunov-likeTheorem holds true)

it has more chances to be smooth ???

What if we insist with smooth functions ?

For instance, some function V such that

it is useful as a Lyapunov function (i.e., aLyapunov-likeTheorem holds true)

it has more chances to be smooth ???

IDEA:USE NON-COMMUTATIVITY

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

10−x2

x = u1f1 + u2f2

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

10−x2

x = u1f1 + u2f2

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

Φtf2 Φt

Φ−tf1 Φt

f2 Φt

Φ−tf2 Φ−tf1

Φtf2 Φt

f1(x) 6= x

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)

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)

Remind:The asymptotics of a Lie bracket:

Ψ[f1,f2](t)(x) := Φ−tf2 Φ−tf1

Φtf2 Φt

Asymptotic formula:

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

Ψ[f1,f2](t)(x) := Φ−tf2 Φ−tf1

Φtf2 Φt

Asymptotic formula:

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

Ψ[f1,f2](t)(x) := Φ−tf2 Φ−tf1

Φtf2 Φt

Asymptotic formula:

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

Ψ[f1,f2](t)(x) := Φ−tf2 Φ−tf1

Φtf2 Φt

Asymptotic formula:

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

Sophus Lie

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

Observe:

Lie brackets show up in

higher order necessary conditions for minima

controllability

boundary conditions of HJ equations, toguarantee continuity of time optimal functions

they are not included in HJ equations orHJ inequalities

Observe:

Lie brackets show up in

higher order necessary conditions for minima

controllability

boundary conditions of HJ equations, toguarantee continuity of time optimal functions

BUTthey are not included in HJ equations orHJ inequalities

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

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) := infv∈F(j)(x)

〈p, v〉

Notice that

H (1) = H andH = H (1) ≥ H (2) ≥ . . .H (k−1) ≥ H (k)

x =m∑i=1

uigi (x) u = ±e1, . . . ,±em

F(1) :=±f1, . . . ,±fm

F(2) := F(1)∪

[fi , fj ] i , j = 1, . . . ,m

〈p, v〉

Notice that

H (1) = H andH = H (1) ≥ H (2) ≥ . . .H (k−1) ≥ H (k)

x =m∑i=1

uigi (x) u = ±e1, . . . ,±em

F(1) :=±f1, . . . ,±fm

F(2) := F(1)∪

[fi , fj ] i , j = 1, . . . ,m. . .

〈p, v〉

Notice that

H (1) = H andH = H (1) ≥ H (2) ≥ . . .H (k−1) ≥ H (k)

x =m∑i=1

uigi (x) u = ±e1, . . . ,±em

F(1) :=±f1, . . . ,±fm

F(2) := F(1)∪

[fi , fj ] i , j = 1, . . . ,m

〈p, v〉

Notice that

H (1) = H andH = H (1) ≥ H (2) ≥ . . .H (k−1) ≥ H (k)

x =m∑i=1

uigi (x) u = ±e1, . . . ,±em

F(1) :=±f1, . . . ,±fm

F(2) := F(1)∪

[fi , fj ] i , j = 1, . . . ,m

〈p, v〉

Notice that

H (1) = H andH = H (1) ≥ H (2) ≥ . . .H (k−1) ≥ H (k)

x =m∑i=1

uigi (x) u = ±e1, . . . ,±em

F(1) :=±f1, . . . ,±fm

F(2) := F(1)∪

[fi , fj ] i , j = 1, . . . ,m

〈p, v〉

Notice that

H (1) = H andH = H (1) ≥ H (2) ≥ . . .H (k−1) ≥ H (k)

x =m∑i=1

uigi (x) u = ±e1, . . . ,±em

F(1) :=±f1, . . . ,±fm

F(2) := F(1)∪

[fi , fj ] i , j = 1, . . . ,m

〈p, v〉

Notice that

H (1) = H and

H = H (1) ≥ H (2) ≥ . . .H (k−1) ≥ H (k)

x =m∑i=1

uigi (x) u = ±e1, . . . ,±em

F(1) :=±f1, . . . ,±fm

F(2) := F(1)∪

[fi , fj ] i , j = 1, . . . ,m

〈p, v〉

Notice that

H (1) = H andH = H (1) ≥ H (2) ≥ . . .H (k−1) ≥ H (k)

Definition

Let U : Rn \ C → R be continuous function, locallysemiconcave, positive definite, and proper. If

H (h)(x ,D∗U(x)) < 0

we say that U is a degree-h Control LyapunovFunction

Remark:if h1 ≤ h2,U is a degree-h1 CLF =⇒ U is a degree-h2 CLFIndeed: H = H (1) ≥ H (2) ≥ . . .H (k−1) ≥ H (k)

Definition

H (h)(x ,D∗U(x)) < 0

Definition

H (h)(x ,D∗U(x)) < 0

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)

Definition

H (h)(x ,D∗U(x)) < 0

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

Because a ”Lyapunov-like” theorem holdstrue!

degree-h Lyapunov-like Theorem:

If there exists a degree-h ControlLyapunov Function

Moreover, degree-h Control LyapunovFunctions are likely smoother than stan-dard CLFs

Application to the nonholonomic integrator

x = u1f1 + u2f2

10−x2

, f2 =

, [f1, f2] =

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):

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

Istead Dd(x) = (0, 0, 1) IS NOT A BAD GRADIENT FORH2(x , p):

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

= − 1

x = u1f1 + u2f2

10−x2

, f2 =

, [f1, f2] =

H1(x , p) = H(x , p) = −√

(p1 − x2p3)2 + (p2 + x2p3)2

H2(x , p) = min−√

(p1 − x2p3)2 + (p2 + x2p3)2 , − |p3|16

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

= − 1

x = u1f1 + u2f2

10−x2

, f2 =

, [f1, f2] =

H1(x , p) = H(x , p) = −√

(p1 − x2p3)2 + (p2 + x2p3)2

H2(x , p) = min−√

(p1 − x2p3)2 + (p2 + x2p3)2 , − |p3|16

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

= − 1

x = u1f1 + u2f2

10−x2

, f2 =

, [f1, f2] =

H1(x , p) = H(x , p) = −√

(p1 − x2p3)2 + (p2 + x2p3)2

H2(x , p) = min−√

(p1 − x2p3)2 + (p2 + x2p3)2 , − |p3|16

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

= − 1

x = u1f1 + u2f2

10−x2

, f2 =

, [f1, f2] =

H1(x , p) = H(x , p) = −√

(p1 − x2p3)2 + (p2 + x2p3)2

H2(x , p) = min−√

(p1 − x2p3)2 + (p2 + x2p3)2 , − |p3|16

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

= − 1

x = u1f1 + u2f2

10−x2

, f2 =

, [f1, f2] =

H1(x , p) = H(x , p) = −√

(p1 − x2p3)2 + (p2 + x2p3)2

H2(x , p) = min−√

(p1 − x2p3)2 + (p2 + x2p3)2 , − |p3|16

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

= − 1

So the distance function -while NOT being a standardLyapunov function-

is a (C∞!) Lyapunov function ofdegree-2

In other words, the new Partial Differential Inequality

H2 < 0

does have a C∞ solution (while H < 0 has no C 1 solutions)

So the distance function -while NOT being a standardLyapunov function- is a (C∞!) Lyapunov function ofdegree-2

H2 < 0

Two Variationson the Nonholonomic Integrator

Variation 1 on the Nonholonomic Integrator:higher order

x = u1f1 + u2f2

, f2 =

, [f1, f2] =

2x1 − 2x2

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

[f1, [f1, f2]] =

H(1)(x ,Dd(x)) ≤ 0

H(2)(x , d(x)) ≤ 0

with H(2) = 0 on the x3-axisBUT

H(3)(x , d(x)) =< 0 ! Hence the distance d = d(x) is a degree-3control Lyapunov function.

x = u1f1 + u2f2

, f2 =

, [f1, f2] =

2x1 − 2x2

[f1, [f1, f2]] =

H(1)(x ,Dd(x)) ≤ 0

H(2)(x , d(x)) ≤ 0

x = u1f1 + u2f2

, f2 =

, [f1, f2] =

2x1 − 2x2

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

[f1, [f1, f2]] =

H(1)(x ,Dd(x)) ≤ 0

H(2)(x , d(x)) ≤ 0

x = u1f1 + u2f2

, f2 =

, [f1, f2] =

2x1 − 2x2

[f1, [f1, f2]] =

H(1)(x ,Dd(x)) ≤ 0

H(2)(x , d(x)) ≤ 0

x = u1f1 + u2f2

, f2 =

, [f1, f2] =

2x1 − 2x2

[f1, [f1, f2]] =

H(1)(x ,Dd(x)) ≤ 0

H(2)(x , d(x)) ≤ 0

x = u1f1 + u2f2

, f2 =

, [f1, f2] =

2x1 − 2x2

[f1, [f1, f2]] =

H(1)(x ,Dd(x)) ≤ 0

H(2)(x , d(x)) ≤ 0

H(3)(x , d(x)) =< 0 !

Hence the distance d = d(x) is a degree-3control Lyapunov function.

x = u1f1 + u2f2

, f2 =

, [f1, f2] =

2x1 − 2x2

[f1, [f1, f2]] =

H(1)(x ,Dd(x)) ≤ 0

H(2)(x , d(x)) ≤ 0

Variation 2 of the Nonholonomic Integrator:Lipschitz fields

x = u1f1 + u2f2

|x2| − 2x2

, f2 =

|x1|+ 2x1

, [f1, f2]set =

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 toLipschitz vector fields and a Lyapunov-like theorem holdtrue.The Hamiltonian H(2) is now a inf-sup, intead of a inf. In thiscase:

H(2)(x , p) = infH(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

x = u1f1 + u2f2

|x2| − 2x2

, f2 =

|x1|+ 2x1

, [f1, f2]set =

H(2)(x , p) = infH(1)(x , p) , sup

w∈I (x)w p3 , sup

w∈−I (x)w p3

x = u1f1 + u2f2

|x2| − 2x2

, f2 =

|x1|+ 2x1

, [f1, f2]set =

H(2)(x , p) = infH(1)(x , p) , sup

w∈I (x)w p3 , sup

w∈−I (x)w p3

x = u1f1 + u2f2

|x2| − 2x2

, f2 =

|x1|+ 2x1

, [f1, f2]set =

H(2)(x , p) = infH(1)(x , p) , sup

w∈I (x)w p3 , sup

w∈−I (x)w p3

x = u1f1 + u2f2

|x2| − 2x2

, f2 =

|x1|+ 2x1

, [f1, f2]set =

H(2)(x , p) = infH(1)(x , p) , sup

w∈I (x)w p3 , sup

w∈−I (x)w p3

x = u1f1 + u2f2

|x2| − 2x2

, f2 =

|x1|+ 2x1

, [f1, f2]set =

H(2)(x , p) = infH(1)(x , p) , sup

w∈I (x)w p3 , sup

w∈−I (x)w p3

degree-2 Lyapunov function, i.e. H(2)(x ,Dd(x)) < 0Franco RampazzoMonica Motta Lyapunov-like functions and Lie brackets

The idea behind the new PDI:

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

1 For any u let x(·) be such a trajectory issuing from x withderivative x(0) = < 0

2 Choose (if possible!) u so that the derivative of t 7→ V (x(t))at t = 0 is strictly negative:

dtU(x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

m∑i=1

ui fi (x)〉 < 0

3 This gives the classical inequality:

H(x ,DV (x)) = H1(x ,DV (x)) = infu〈DV (x),

m∑i=1

ui fi (x)〉 < 0

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

THE STANDARD INEQUALITY:

dtU(x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

m∑i=1

ui fi (x)〉 < 0

m∑i=1

ui fi (x)〉 < 0

dtU(x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

m∑i=1

ui fi (x)〉 < 0

m∑i=1

ui fi (x)〉 < 0

dtU(x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

m∑i=1

ui fi (x)〉 < 0

m∑i=1

ui fi (x)〉 < 0

dtU(x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

m∑i=1

ui fi (x)〉 < 0

m∑i=1

ui fi (x)〉 < 0

dtU(x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

m∑i=1

ui fi (x)〉 < 0

m∑i=1

ui fi (x)〉 < 0

dtU(x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

m∑i=1

ui fi (x)〉 < 0

m∑i=1

ui fi (x)〉 < 0

What are we doing if 〈DV (x),∑m

i=1 ui fi (x)〉 = 0 for all u, as inthe case of the x3-axis in the nonholonomic integrator ?

Solvethe 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:

ddtV (x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

∑mi=1 ui fi (x)〉 =

H(x ,DV (x)) = H1(x ,DV (x)) = infu〈DV (x),∑m

i=1 ui fi (x)〉= 0

d2tU(x(t))t=0 = 〈x(0)D2V (x), x(0)〉+ 〈DV (x), x(0)〉 =

= 0 + 〈DV (x), [fi , fj ]〉 ≤ H2(x ,DV (x))< 0

i=1 ui fi (x)〉 = 0 for all u, as inthe case of the x3-axis in the nonholonomic integrator ?Solvethe equation

x(t) = [fi , fj ]t(x(t)) x(0) = x x(0) = 0

ddtV (x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

∑mi=1 ui fi (x)〉 =

i=1 ui fi (x)〉= 0

= 0 + 〈DV (x), [fi , fj ]〉 ≤ H2(x ,DV (x))< 0

i=1 ui fi (x)〉 = 0 for all u, as inthe case of the x3-axis in the nonholonomic integrator ?Solvethe equation

x(t) = [fi , fj ]t(x(t)) x(0) = x x(0) = 0

ddtV (x(t))t=0 = 〈DV (x), x(0)〉 = 〈DV (x),

∑mi=1 ui fi (x)〉 =

i=1 ui fi (x)〉= 0

= 0 + 〈DV (x), [fi , fj ]〉 ≤ H2(x ,DV (x))< 0

[1] Motta, F. R., (2013) Asymptotic controllability and optimalcontrol JDE[2] E.Feleqi, F.R. (2015) Integral representation for bracketgenerating multi-flows DCDS(A)[3] Motta, F. R., (2016) Lyapunov-like functions involving Liebrackets

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

Definition

(Semiconcavity). Let Ω ⊂ Rn. A continuous function F : Ω→ Ris said to be semiconcave on Ω if there exist ρ > 0 such that

F (z1) + F (z2)− 2F

(z1 + z2

)≤ ρ|z1 − z2|2,

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

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