Escola de Matemática Aplicada, FGV-Riow3.impa.br/~aronna/slides_oktobermat.pdf · Escola de Matemática Aplicada, FGV-Rio OktobermatXV,19e20deOutubrode2017,PUC-Rio. Introduction

Introduction & motivation Limit solutions Commutative case Noncommutative case Concluding remarks & References

Limit solutions for control systems

M. Soledad Aronna

Escola de Matemática Aplicada, FGV-Rio

Oktobermat XV, 19 e 20 de Outubro de 2017, PUC-Rio


1 Introduction & motivation

2 Limit solutions

3 Commutative case

4 Noncommutative case

5 Concluding remarks & References


Optimal Control, example 1: Fishing problem

[Clark, 1976]

State variable: x is the size of the fish population

Control variable: u is the fishing effort

Cost: net revenue in a fixed time interval

max

∫ T

0

(Eu(t)− c

x(t)u(t)

)dt,

x(t) = r x(t) (1− x(t)/k) − u(t),

0 ≤ u(t) ≤ Umax, a.e. on [0, T ],

x(t) ≥ 0 on [0, T ],

x(0) = x0, x(T ) free.

E : selling cost, c/x(t) : fishing cost, r : reproduction rater/k : mortality due to space competition


Optimal Control, example 2: the Goddard problem in 1D(vertical motion of a rocket)

[Goddard, 1919] [Seywald & Cliff, 1993]

State variables: (h, v,m) are the position, speed and mass.Control variable: u is the normalized thrust (proportion of Tmax)Cost: final mass of the rocket (or minimize fuel consumption)

max m(T ),

s.t. h(t) = v(t),

v(t) = −1/h(t)2 + 1/m(t)(Tmax u(t)−D(h(t), v(t))

),

m(t) = −b Tmax u(t),

0 ≤ u(t) ≤ 1, a.e. on [0, T ],

h(0) = 0, v(0) = 0, m(0) = 1, h(T ) = 1,

T : free final time, b : fuel consumption coefficient,Tmax is the maximal thrust, D(h, v) is the drag


Standard Optimal control problem

A standard optimal control problem is generally written as

max

∫ T

0

L(t, x(t), u(t))dt+ φ(T, x(T )), [cost]

x(t) = f(t, x(t), u(t)), p.c.t. t ∈ [0, T ], [dynamics]x(0) = x0, [initial condition]

(T, x(T )) ∈ S ⊆ IRn+1, [final constraints]u(t) ∈ U ⊆ IRm. [control constraint]

(OCP)

In general, u ∈ L1([0, T ];U) and x ∈W 1,1([0, T ]; IRn).


Example 3: Impulsive Control System

[Bressan & Piccoli, 2007]Boy riding swing: changes the radius of oscillationθ(t) : angle, r(t) : radius of oscillationKinetic and potential energies:

T (r, θ, r, θ) =m

2

(r2 + r2θ2), V (r, θ) = mgr cos θ.

Set L := T − V for the Lagrangian, then equation of motion for θ(t) :

d

dt

∂L

∂θ=∂L

∂θ


Example 3: equations of motion

We get2mrθr +mr2θ = −mgr sin θ.

Setting ω := θ for the angular velocity, u(t) := r(t) for the control,

θ = ω,

ω = −g sin θ

u− 2

ω

uu.

If we consider the problem of maximizing the angle

max |θ(T )|

we find that the optimal control u(t) jumps at the position θ = 0!


Example 3: equations of motion

We get2mrθr +mr2θ = −mgr sin θ.

Setting ω := θ for the angular velocity, u(t) := r(t) for the control,

θ = ω,

ω = −g sin θ

u− 2

ω

uu.

If we consider the problem of maximizing the angle

max |θ(T )|

we find that the optimal control u(t) jumps at the position θ = 0!


AIM

We aim at giving an appropriate interpretation of these impulsive controlequations, for control functions u that may jump or even have

unbounded variation!


The impulsive control system

For a control equation of the form

x = f(t, x, u, v) +

m∑α=1

gα(x)uα, for t ∈ [a, b],

with x : [a, b]→ IRn being the state variable and u : [a, b]→ U ⊂ IRm,v : [a, b]→ V ⊂ IRl the control variables, we propose a notion ofsolution that

is defined for L1 inputs u (here v is a standard, bounded control),

subsumes former concepts of solution.


The impulsive control system

For a control equation of the form

x = f(t, x, u, v) +

m∑α=1


with x : [a, b]→ IRn being the state variable and u : [a, b]→ U ⊂ IRm,v : [a, b]→ V ⊂ IRl the control variables, we propose a notion ofsolution that

is defined for L1 inputs u (here v is a standard, bounded control),

subsumes former concepts of solution.


Some remarks

When u ∈ AC([a, b]; IRm), the system is standard and the solutionis interpreted in the sense of Carathéodory.Some reasons for looking at discontinuous trajectories:1) lack of coercivity in the cost,2) large change of value in a very short amount of time

[Azimov & Bishop, New trends in astrodynamics and applications:Optimal trajectories for space guidance. Annals of the NY Academy ofSciences, 2005][Catllá et al., On Spiking models for synaptic activity and impulsivedifferential equations, SIAM Review, 2005][Gajardo, Ramirez & Rapaport, Minimal time sequential batch reactorswith bounded and impulse controls, SIAM J. Control Opt., 2007]


Crucial features

Some basic issues are nowadays consolidated. For instance:a notion of solution in distributional-theoretical sense is wrong,[Hájek, 1985]; [Miller, 2003]

Lie brackets [gα, gβ ] := g′β gα − g′α gβ matter.


“Good” notions of solution already existed in two cases.

CASE I: The commutative case

[gα, gβ ] ≡ 0.

One defines the solution via density of AC in L1

(we adopt this definition here).

It exploits the commutativity via the Flow-box theorem.

References:

[A. Bressan & F. Rampazzo, 1991]

[A.V. Sarychev, 1991]


CASE II: The noncommutative case

[gα, gβ ] 6= 0,

and controls u of bounded variation.

The concept of graph completion.

References:

[R.W. Rishel, 1965]

[A. Bressan and F. Rampazzo, 1988]

[G.N. Silva and R.B. Vinter, 1996]

[F.L. Pereira and G.N. Silva, 2000]



2 Limit solutions

3 Commutative case




Limit solutions - Definitions

Consider the Cauchy problem

x = f(t, x, u, v) +

m∑α=1


x(a) = x.

Given x ∈ IRn, u ∈ L1([a, b];U) and v ∈ L1([a, b];V ) :

1 x : [a, b]→ IRn is a LIMIT SOLUTION if, for every τ ∈ [a, b],there exists (uτk) ⊂ AC([a, b];U) such that, the correspondingCarathéodory solutions (xτk) are uniformly bounded and satisfy

|(xτk, uτk)(τ)− (x, u)(τ)|+‖(xτk, uτk)− (x, u)‖1 → 0, when k →∞.

2 A limit solution x : [a, b]→ IRn is a SIMPLE limit solution if (uτk)can be chosen independently of τ.

3 A simple limit solution x : [a, b]→ IRn is a BV-SIMPLE limitsolution if the approximating inputs uk have equibounded variation.


Outline of the results

Consistency with standard (Carathéodory) solutions

Some existence results

Uniqueness in the commutative case

Comparison with “graph completion” solutions


Example 1[H.J. Sussmann & W. Liu, 1991]

g1(x) := (1, 0, x2)>, g2(x) := (0, 1,−x1)>, [g1, g2] = (0, 0,−2)>,

x = g1(x)u1 + g2(x)u2, t ∈ [0, 1], x(0) = (0, 0, 0)>.

For u ≡ (0, 0)>, the Carathéodory solution is xC ≡ (0, 0, 0)>.Approximating u by

uk(t) = (k−1/2(cos kt− 1), k−1/2 sin kt)>,

one gets

xk(t) = (k−1/2(cos kt− 1), k−1/2 sin kt,−t+ k−1 sin kt)>,

uk → (0, 0)>, xk → x(t) := (0, 0,−t)>, uniformly on [0, 1].

u ∈ AC, and x ∈ AC is a simple limit solution (but not BV simple!)Lack of uniqueness: the solution depends on (uk)

(we’ll see this holds in general whenever [gα, gβ ] 6= 0)


Example 2

x = xu, x(0) = 1, t ∈ [0, 1].

For every bounded u ∈ L1([0, 1]; IR), the map

x(t) = eu(t)−u(a)

is a limit solution.Actually, since m = 1, we will prove that x is the unique limit solution.Yet, x is not a simple limit solution in general, since for instance, if

u := 1Q∩[0,1]

is the Dirichlet function, there is no way of pointwise approximating uwith one sequence of absolutely continuous uk.



2 Limit solutions

3 Commutative case




The commutative case, [gα, gβ] ≡ 0

Theorem 1 (Characterization): The limit solution is characterized bythe solution of a change of coordinates (given by the Flow-box theorem)as follows.

Change of coordinates: Extend the vector fields setting

gα :=

(gαeα

), f :=

(f0

).

Then [gα, gβ ] = 0.

Let Π: IRn × IRm → IRn be the canonical projection on the first factor:

Π(x, z) := x,

and ϕ : IRn × IRm → IRn be defined by

ϕ(x, z) := Π(

(x, z)e(−zmgm) . . . e(−z1g1))

= Π(

(x, z)e−∑mα=1 zαgα

).

Consider the diffeomorphism φ : IRn × IRm → IRn × IRm given by

φ(x, z) := (ϕ(x, z), z).


Commutative case: change of coordinates (continuation)

Then G := Dφ g, F = Dφ f are such that

Gα =∂

∂zα=

(0eα

), F =

n∑i,j=1

(∂φi∂xj

fj

)∂

∂xi=

(F0

).

Hence the differential system after the change of variables φ reads

ξ = F (t, ξ, z, v),

z = u.

With the change of coordinates we can reduce the system to anon-impulsive one!


Theorem 2 (Continuity w.r.t. data): For every x ∈ IRn, u ∈ L1,v ∈ L1 there exists a unique limit solution of

x = f(t, x, u, v) +

m∑α=1


x(a) = x,

that depends continuously on initial data and control:

|x1(t)− x2(t)|+ ‖x1 − x2‖1 ≤

M[|x1 − x2|+ |u1(a)− u2(a)|+ |u1(t)− u2(t)|+ ‖u1 − u2‖1

].

and also w.r.t. v in the L1-norm.

Corollary (Consistency): When u ∈ AC, the Carathéodory solutioncoincides with the unique limit solution.


Example 3

x = g1(x)u1 + g2(x)u2, x1(0) = x2(0) = 1,

with g1(x) := (x1, x2)t, g2(x) := (0, x1)t, then [g1, g2] ≡ 0.

Case I:

u1(t) :=

{0 on

[0, 12[

1 on[12 , 1] , u2(t) :=

{0 on

[0, 12]

1 on]12 , 1] .

uk,1(t) :=

0 on

[0, 12 −

1k+1

](k + 1)

(t− 1

2 + 1k+1

)on]12 −

1k+1 ,

12

]1 on

]12 , 1] ,

uk,2(t) :=

0 on

[0, 12]

(k + 1)(t− 1

2

)on]12 ,

12 + 1

k+1

]1 on

]12 + 1

k+1 , 1] .


Example 3 - continuation

x1(t) :=

{1 on

[0, 12[

e on[12 , 1] , x2(t) :=

1 on

[0, 12[

e on t = 12

2e on]12 , 1]

Case II: We invert the controls, u1 := u2, u2 := u1,

(u1, u2) = (u1, u2) on [0, 1]\{1/2} uk,1 := uk,2, uk,2 := uk,1,

x1(t) :=

{1 on

[0, 12]

e on]12 , 1] , x2(t) :=

1 on

[0, 12[

2 on t = 12

2e on]12 , 1] .


Example 3 - continuation

x1(t) :=

{1 on

[0, 12[

e on[12 , 1] , x2(t) :=

1 on

[0, 12[

e on t = 12

2e on]12 , 1]

Case II: We invert the controls, u1 := u2, u2 := u1,

(u1, u2) = (u1, u2) on [0, 1]\{1/2} uk,1 := uk,2, uk,2 := uk,1,

x1(t) :=

{1 on

[0, 12]

e on]12 , 1] , x2(t) :=

1 on

[0, 12[

2 on t = 12

2e on]12 , 1] .


Solutions coincide almost everywhere

In general:

Proposition: If the system is commutative and u, u ∈ L1([a, b];U)coincide a.e. in [a, b] (this is, u = u in L1) and verify u(a) = u(a), onehas

x(t) = x(t)e∑mα=1(uα(t)−uα(t))gα , ∀t ∈ [a, b].

Hence, x(t) = x(t), a.e. on [a, b].


Example 3 - With non-commutative vector fields!

Set now g2(x) := (0, x21)t, then [g1, g2] = −(0, x21)t, and

x1(t) :=

{1 on

[0, 12[

e on[12 , 1] , x2(t) :=

1 on

[0, 12[

e on t = 12

e+ e2 on]12 , 1] ,

while

x1(t) :=

{1 on

[0, 12]

e on]12 , 1] , x2(t) :=

1 on

[0, 12[

2 on t = 12

2e on]12 , 1] .

So, in the general noncommutative case, the solutions do not coincidealmost everywhere even if the control do.



2 Limit solutions

3 Commutative case




The noncommutative case, u ∈ BV - Space-time system

The existing notion of solution in the noncommutative case:

• For regular (absolutely continuous) controls u ∈ AC([a, b];U), onecan consider

s : [a, b]→ [0, 1], s(t) :=t+ Var[a,t](u)

b− a+ Var[a,b](u),

and reparametrize time ϕ0(·) := s−1(·) with ϕ0 : [0, 1]→ [a, b], and setϕ(s) := u ◦ ϕ0(s), y := x ◦ ϕ0.

• (ϕ0, ϕ, y) is solution of the space-times systemy′0(s) = ϕ′0(s),

y′(s) = f(y0(s), y(s), ϕ(s), ψ(s))ϕ′0(s) +

m∑α=1

gα(y(s))ϕα′(s),

(y0, y)(0) = (a, x) ,

s ∈ [0, 1].


u ∈ BV - Graph completions

• For BV controls u, let (ϕ0, ϕ) be a graph completion of u :

(ϕ0, ϕ) : [0, 1]→ [a, b]× U Lipschitz continuous such that,∀t ∈ [a, b], there exists s ∈ [0, 1] verifying (t, u(t)) = (ϕ0, ϕ)(s).


Graph completion solutions

Given u ∈ BV ([a, b];U) and (ϕ0, ϕ) : [0, 1]→ [a, b]× U a graphcompletion of u, we let y be the solution ofy′0(s) = ϕ′0(s),

y′(s) = f(y0(s), y(s), ϕ(s), ψ(s))ϕ′0(s) +

m∑α=1

gα(y(s))ϕα′(s),

(y0, y)(0) = (a, x) ,

s ∈ [0, 1].

Graph completion solution: (possibly) set-valued map x : [a, b] ⇒ IRn,

t Z=⇒ x(t) := y ◦ ϕ−10 (t).

Single-valued graph completion solution:x : [a, b]→ IRn, σ : [a, b]→ [0, 1] a right-inverse of ϕ0 such that

(t, u(t)) = (ϕ0, ϕ) ◦ σ(t), x(t) = y ◦ σ(t), for all t ∈ [a, b].


Representation of BV simple limit solutions

Theorem 3 (Equivalence BV simple limit solution andsingle-valued graph completion solution): Given x ∈ IRn, u ∈ BV,v ∈ L1 : x is a BV simple limit solution if and only if x is a single-valuedgraph completion solution.

Ingredients of the proof:“if”: one has (t, u(t), x(t)) = (ϕ0, ϕ, y) ◦ σ(t), with σ ∈ BVincreasing. We prove pointwise approximation of σ with ACfunctions of uniform BV.

“only if”: compactness (Ascoli-Arzelà theorem)


Existence of limit solutions for u ∈ BV

An arcwise connected set U has the Whitney property isd(x, y) ≤M |x− y|, for some M > 0, where d is the geodesic distance.

Theorem 4 (Existence of limit solution): If U has the Whitneyproperty then, for any x ∈ IRn, u ∈ BV, v ∈ L1, there exists a BV simplelimit solution.


Consistency with Carathéodory solutions

Theorem 5 (Consistency): If x ∈ IRn, u ∈ AC, v ∈ L1, and x ∈ AC isa BV simple limit solution, then x is the Carathéodory solution.

Recall Example 1!



2 Limit solutions

3 Commutative case




Concluding remarks

The same concept of solution is used both for the commutative andnoncommutative cases.

In the commutative case, the limit solutions are represented via achange of coordinates.

In the noncommutative case, there is a representation of the BVsimple limit solutions via the graph completion.

When u ∈ BV : “it is equivalent to approximate by AC functionswith uniform BV, than to complete the graph.”


References

H.J. Sussmann. Ann. Probability, 1978.A. Bressan and F. Rampazzo. Boll. Un. Mat. Ital., 1988.A. Bressan and F. Rampazzo. J. Optim. Theory Appl., 1991.A.V. Sarychev. In Nonlinear synthesis (Sopron, 1989), volume 9 ofProgr. Systems Control Theory, Birkhäuser Boston, Boston, MA,1991.M.S. Aronna and F. Rampazzo. A note on systems with ordinaryand impulsive controls. IMA J. Math. Control Inform., 2016M.S. Aronna and F. Rampazzo. L1 limit solutions for controlsystems. J. Differential Equations, 2015M.S. Aronna, M. Motta and F. Rampazzo. Necessary conditionsinvolving Lie brackets for impulsive optimal control problems, arXiv& in progress


OBRIGADA!

Documents

Escola de Matemática Aplicada, FGV-Riow3.impa.br/~aronna/slides_oktobermat.pdf · Escola de Matemática Aplicada, FGV-Rio OktobermatXV,19e20deOutubrode2017,PUC-Rio. Introduction