the fast-switching phenomenon in time-optimal control problemstrelat/gdt/confs/... · ;r 1;1sq,...

The fast-switching phenomenon in time-optimalcontrol problems

Francesco Boarotto

Joint work with Mario Sigalotti

LJLL, UPMC Paris 6 and Team CaGe, Inria Paris

LJLL GdT Controle - December 1st, 2017

The Bang-Bang principle in the linear case

Consider the finite-dimensional control system

9x “ Ax ` Bu, x P Rn, A P Rn2, B P Rnˆm,

and U Ă Rm a bounded polyhedron.

Problem (Linear time-optimal problem)

Given x0, x1 P Rn, find u P L2pr0,T s,Rmq such that

xp0, x0; uq “ x0 and xpT , x0; uq “ x1;

uptq P U for a.e. t P r0,T s;

T be as small as possible.

If there exist any pair rT ą 0 and ru P L2pr0, rT s,Uq that solvethe problem, then there exist a minimal T ą 0 and an optimalcontrol u˚ P L2pr0,T s,Uq that solve it.

The Bang-Bang principle in the linear case

Taking the dual curve t ÞÑ λptq P Rn˚zt0u, that satisfies theadjoint equation

9λptq “ ´ATλptq, t P r0,T s,

the optimal control u˚ solves, for a.e. t P r0,T s, the linearprogramming problem

xλptq,Bu˚ptqy “ minuPUxλptq,Buy, a.e. t P r0,T s,

Under appropriate normality assumptions, we findnon-overlapping intervals I1, . . . , IN Ă r0,T s such that

r0,T s “Nď

i“1

Ii , u˚ptq “ vi if t P intpIi q.

and vi a vertex of U. The control bangs around vertexes of U.LaSalle, 1961

The Bang-Bang principle in infinite dimensions

The Bang-Bang phenomenon appears if we want to controlparabolic PDE’s.

For hyperbolic equations: “There is no true Bang-Bangprinciple at all” Russell.

Depending on the initial and on the final state, the controlsgoverning a controlled wave equation

B2w

Bt2´B2w

Bx2“ gpxquptq,

may possess any degree of differentiability.


Zuazua, Wang et al. Let Ω Ă Rn be a bounded domain, BΩ aC 2 boundary, ω Ă Ω be an open set. Consider

$

&

%

yt ´∆y ` apx , tqy “ χωu in Ωˆ R`y “ 0 on BΩˆ R`ypx , 0q “ y0pxq in Ω

where

a P L8pΩˆ R`q and y0 P L2pΩqzt0u,

It is of the form 9y “ Ay ` Bu, with

Apx , tqy “ ∆y ´ apx , tqy ,Bpxq “ χωpxq.


For M ą 0, we choose the control u in

UM :“ tu : R` Ñ L2pΩq | upt, ¨qL2pΩq ď M a.e. t ą 0u.

We say that a control u belongs to UMadm if u P UM and there

exists t ą 0 such that ypt, x ; uq “ 0.

Problem (Time-optimal problem)

T pMq :“ infuPUM

adm

tt ą 0 | ypt, x ; uq “ 0u,


For T ą 0 define the norm-optimal control problem pNPqT

NpT q :“ inftuL8pr0,T s,L2pΩqq | ypT , x ; uq “ 0u, N :“ limTÑ8

NpT q.

NpT q ă `8 because heat equations are null-controllable

N exists.

Theorem (Wang, Xu, Zhang, 2014)

Let y0 P L2pΩqzt0u. Then the time-optimal problem admits

time-optimal controls iff M ą N.

The crucial point of the proof is the Bang-Bang property(Wang, Phung 2013) for the pNPqT problem: any optimalcontrol u˚ for pNPqT verifies

u˚pt, ¨qL2pΩq “ NpT q, a.e. t P r0,T s.


Consider another example, studied by Fattorini, Russell,Egorov et al. in the 60’s - 70’s. For w “ wpx , tq, consider

Bw

Bt´B2w

Bx2` rpxqw “ gpxquptq, t ě 0, 0 ă x ă 1,

with boundary conditions

a0wp0, tq ` b0Bw

Bxp0, tq “ 0, a1wp1, tq ` b1

Bw

Bxp1, tq “ 0

a2i ` b2

i ‰ 0, i “ 0, 1.

It is of the form 9w “ Aw ` Bu, with

Apxqw “ Lw “ B2wBx2 ´ rpxqw a Sturm-Liouville operator,

Bpxq “ gpxq.


Problem (Time-optimal control problem)

Assuming wp¨, 0q “ 0 and u P L8pR`, r´1, 1sq, reach in minimaltime τ , the final state wp¨, τq “ w1.

If u have no bounds, the problem has a solution only if w1 isin the domain of exppA1p´Lq

12q, A1 ą 0 a suitable constant.

No solution to the time-optimal control problem is granted forL2 controls.

A bang-bang principle can be established only for statesw1 P B known to be reachable. The subspace B Ă L2pr0, 1sqdepends in general on the eigensystem associated to L and thecoefficients of g in this basis.


Theorem (Russell, 1978)

Suppose that a final state w1 P B is reached from w0 “ 0 at timeτ ą 0 with a control u P L8pr0, τ s, r´1, 1sq. Suppose that τ isminimal with respect to all u P L8pR`, r´1, 1sq. Then

|uptq| “ 1 for almost every t P r0, τ s.

There exists t ÞÑ ηptq P R such thatuptq “ sgnpηptqq, if ηptq ‰ 0.

η is determined by the data of our problem, is real-analytic onr0, τq, and has discrete zeros accumulating at τ .

Notice that the situation is different between u P L2 oru P L8, in general.

Typically we can avoid oscillations with unbounded controls.

Back to the finite-dimensional case

Fuller, 1963

In R2, among the trajectories of the differential system

"

9x1 “ x2,9x2 “ u, u P r´1, 1s,

with initial conditions x1p0q “ x1, x2p0q “ x2, and the origin p0, 0qas final state, minimize the cost functional

ż 8

0|x1ptq|

2dt,

among all trajectories from px1, x2q to p0, 0q.

This problem can be reformulated as a (non-linear) time-optimalproblem in dimension 3.It can be generalized to higher dimensions.


Fuller, 1963

In R2, among the trajectories of the differential system

"

9x1 “ x2,9x2 “ u, u P r´1, 1s,

with initial conditions x1p0q “ x1, x2p0q “ x2, and the origin p0, 0qas final state, minimize the cost functional

ż 8

0|x1ptq|

2dt,

among all trajectories from px1, x2q to p0, 0q.This problem can be reformulated as a (non-linear) time-optimalproblem in dimension 3.It can be generalized to higher dimensions.


Optimal trajectories exist. They exhibit this strange behavior...

Two different phenomena

ÝÑ The trajectory is a join of a singular and a bang bangpart. It makes an infinite number of switchings to “escape”from the singular trajectory.

ÝÑ The optimal control is bang-bang but switches wildlynear a singular trajectory.


This situation is unpleasant for various reasons:

Theoretically it is difficult to investigate these trajectories

Has important drawbacks on the optimal synthesis

Practically chattering can destroy machines

There are engineering reports observing that theimplementation of optimal controls on a machine may lead toits break down.

Regularizing the chattering

Attempts have been made to regularize fast-oscillationsCaponigro, Ghezzi, Piccoli, Trelat, 2017

Works for more general optimal control problems

C puq “

ż tpuq

0Lps, xpsq, upsqqds Ñ min,

with u P U an admissible control and L P C 0pRˆRN ˆRmq aLagrangian function. Optimal controls u˚ exist under standardhypotheses (Lie Algebra condition and STLC at the origin)

For ε ą 0, consider a new cost Cεpuq :“ C puq ` εTV puq. Thecorrection excludes chattering controls (TV puq “ `8).Optimal controls u˚ε exist

These controls u˚ε are quasi-optimal for the original problem,but don’t chatter

Single-input control affine systems and regularity oftime-optimal trajectories

9q “ f0pqq ` uf1pqq, u P r´1, 1s

q P M smooth n-dimensional manifold, f0, f1 P VecpMq (i.e.,smooth vector fields on M)

Time optimal problem: qp0q “ q0, qpT q “ q1, T Ñ min

Regularity of a time-optimal trajectory q : r0,T s Ñ M measured interms of

O “ď

ω open, q|ω smooth

ω, Σ “ r0,T szO

Is Σ empty? finite? countable? of finite measure? of emptyinterior?

Genericity

As pointed out by Sussmann in 1986, for any t ÞÑ uptqmeasurable and any M, q0, there exist f0, f1 P VecpMq such thatthe admissible trajectory driven by u and starting at q0 istime-optimal.

The natural reflex is to look for generic properties: properties thathold for all time-optimal trajectories of the single-input controlaffine system, provided that pf0, f1q belongs to a residual set ofVecM ˆVecM for the C8 Whitney topology. (Open problem 1 inAgrachev 2014, G. Stefani et al. editors)

Time-extremal trajectories and the switching function

By the Pontryagin maximum principle, if q : r0,T s Ñ M istime-optimal, then there exists an extremal liftλ : r0,T s Ñ T ˚Mzt0u of qp¨q such that, for every X P VecM,

d

dtxλptq,X pqptqqy “ xλptq, rf0 ` uptqf1,X spqptqqy a.e. t P r0,T s

and the switching function

h1ptq “ xλptq, f1pqptqqy

satisfies uptq “ sgnph1ptqq whenever h1ptq ‰ 0.

Let f˘ “ f0 ˘ f1 and, for I “ pi1 ¨ ¨ ¨ idq a word with letters int`,´, 0, 1ud ,

fI “ rfi1 , . . . , rfid´1, fid s ¨ ¨ ¨ s, hI ptq “ xλptq, fI pqptqqy.

In particular, ddt h1ptq “ h01ptq for every t P r0,T s, and

d2

dt2 h1ptq “ h001ptq ` uptqh101ptq for a.e. t P r0,T s.

Time-extremal trajectories and the switching function

By the Pontryagin maximum principle, if q : r0,T s Ñ M istime-optimal, then there exists an extremal liftλ : r0,T s Ñ T ˚Mzt0u of qp¨q such that, for every X P VecM,

d

dtxλptq,X pqptqqy “ xλptq, rf0 ` uptqf1,X spqptqqy a.e. t P r0,T s

and the switching function

h1ptq “ xλptq, f1pqptqqy

satisfies uptq “ sgnph1ptqq whenever h1ptq ‰ 0.Let f˘ “ f0 ˘ f1 and, for I “ pi1 ¨ ¨ ¨ idq a word with letters int`,´, 0, 1ud ,

fI “ rfi1 , . . . , rfid´1, fid s ¨ ¨ ¨ s, hI ptq “ xλptq, fI pqptqqy.

In particular, ddt h1ptq “ h01ptq for every t P r0,T s, and

d2

dt2 h1ptq “ h001ptq ` uptqh101ptq for a.e. t P r0,T s.

Previous results

For n “ 2, Σ is generically finite Lobry 1970, Sussmann1982, 1987.

Finiteness of Σ close to points at which some suitablenon-dependence condition between Lie brackets holdsAgrachev, Bressan, Gamkrelidze, Krener, Schattler, Sigalotti,Sussmann,. . . .

For n large enough time-extremal trajectories of genericsystems might exhibit Fuller phenomenon (#Σ “ 8) Kupka1990, Zelikin–Borisov 1994

Generically, for every extremal trajectory q : r0,T s Ñ M, theset O is open and dense in r0,T s Agrachev 1995

Generically, for any extremal triple pqp¨q, up¨q, λp¨qq on r0,T ssuch that h1|r0,T s ” 0, the set Ω “ tt P r0,T s | h101ptq ‰ 0uis of full measure in r0,T s and uptq “ ´h001ptqh101ptqalmost everywhere on Ω Bonnard–Kupka 1997,Chitour–Jean–Trelat 2008

Bang and singular arcs

Definition (Bang and singular arcs)

An arc ω is a connected component of O.

bang if u (a.e.) constant and equal to ˘1 on ω,

singular otherwise.

Two arcs are concatenated if they share one endpoint.

The time-instant between two concatenated arcs (i.e., anisolated point of Σ) is a switching time.

Fuller times

Definition (Fuller times)

Let Σ0 be the set of isolated points in Σ (switching times).The elements of ΣzΣ0 are Fuller times.By recurrence, let

Σk set of isolated points of ΣzpYk´1j“0 Σjq, k P NY t8u.

If t P Σk then t is a Fuller time of order k .

Σ1 are the isolated points of ΣzΣ0 ÝÑ every point in Σ1

admits a converging sequence in Σ0 but not in ΣzΣ0.

Σ2 are the isolated points of ΣzpΣ0 Y Σ1q ÝÑ every point inΣ2 admits a converging sequence in pΣ0 Y Σ1q but not inΣzpΣ0 YΣ1q. Actually it admits a converging sequence in Σ1,otherwise it would be a point in Σ1. And so on...

Main result

Theorem (F. B., M. Sigalotti, submitted)

There exists K pnq P N such that, for a generic pair pf0, f1q, everyextremal trajectory qp¨q of the time-optimal control problem

9q “ f0pqq ` uf1pqq, q P M, u P r´1, 1s,

has at most Fuller times of order K pnq, i.e.,

Σ “ Σ0 Y ¨ ¨ ¨ Y ΣKpnq.

In particular, u can be taken smooth out of a finite union ofdiscrete sets (hence, out of a countable set).

Strategy of the proof

At switching times h1 “ 0 (hence by continuity h1|Σ ” 0)

Elements of ΣzΣ0 are accumulations of switching times andone easily deduces that d

dt h1 “ h01 vanishes on ΣzΣ0

(between two zeros of h1, ddt h1 vanishes)

Higher order Fuller times are accumulations of accumulationsand new relations between λptq and the brackets of f0pqptqqand f1pqptqq can be derivedInitialization on ΣzΣ0

Recursion Σz Ykj“0 Σj ÝÑ Σz Yk`1

j“0 Σj

At high order Fuller times pλptq, jNpf0, f1qpqptqqq belong to aset of large codimension

The projection of sets of large codimension has largecodimension

Fuller times of too large order can be ruled out by standardtransversality arguments

Initialization: dependence conditions on ΣzΣ0

Proposition (No genericity assumption here)

Let t P ΣzΣ0. Then h1ptq “ h01ptq “ 0 and, in addition, eitherh`01ptq “ 0 or h´01ptq “ 0.

This proposition does not require genericity. Can we deriveother conditions without it? This will improve the bounds onK pnq in our main theorem

Heuristic of the proof: let t P Σ. The difficult case is when wecan find arbitrarily close to t an infinite sequence of bang arcs.Technical considerations permit to conclude in all the othercases.

Key point: assuming h˘01ptq ‰ 0 allows to suppose thatalong all these arcs the second derivative of switching functionh1ptq remains bounded from below in absolute value.

A technical lemma

Lemma

Assume that there exists an infinite sequence of concatenated bangarcs converging to τ P r0,T s. Then either h`01pτq “ 0 orh´01pτq “ 0.

Denote the lengths of the subsequent bang arcs by tτiuiPN.The proof works by contradiction: if h`01pτq ‰ 0 and h´01pτq ‰ 0,then a simple computation shows that τi`1 “ Opτi q and

τi`2 “ τi ` Opτ2i q.

Hence8ÿ

i“1

τi “ `8.

Remark: I1 “ p1q and I2 “ p01q would have not worked as astarting point for the recurrence. We need this third wordI3 “ p˘01q.

Recursion Σz Ykj“0 Σj ÝÑ Σz Yk`1

j“0 Σj

Let I1 and I2 be two words with letters in t`,´, 0, 1ud , tn asequence of times converging to t such that, along q : r0,T s Ñ M

hI1ptnq “ 0 “ hI2ptnq.

Up to subsequences

1

t ´ tn

ż t

tn

upsqds Ñ u P r´1, 1s.

Then, for j “ 1, 2,

0 “hIj ptq ´ hIj ptnq

t ´ tn“

şttnph0Ij psq ` upsqh1Ij psqqds

t ´ tnÑ h0Ij ptq`uh1Ij ptq.

If |u| “ 1 then we get a new word J of longer length such thathJptq “ 0.

In any case det

ˆ

h0I1ptq h1I1ptqh0I2ptq h1I2ptq

˙

“ 0.

Tree of dependence conditions

Along a branch of the tree

Along a branch (continued)

and independence of theconditions

Two reasons for losing independence:

h0Ik “ 0 “ h1Ik : all determinants Q1, . . . ,Ql`1 vanish

f0 ^ f1 “ 0: conditions on lower order jetse.g., if f0pqq “ 0 then rf0, rf0, . . . , rf0, f1s . . . spqq only dependson f1pqq and Df0pqq

Along a branch (continued) and independence of theconditions

Two reasons for losing independence:

h0Ik “ 0 “ h1Ik : all determinants Q1, . . . ,Ql`1 vanish

f0 ^ f1 “ 0: conditions on lower order jetse.g., if f0pqq “ 0 then rf0, rf0, . . . , rf0, f1s . . . spqq only dependson f1pqq and Df0pqq

Collecting conditions in tf0 ^ f1 ‰ 0u

If hI1 “ 0, . . . , hIk “ 0,Q1 “ 0, . . . ,Ql “ 0 are independentconditions at λptq then pλptq, jNpf0, f1qpqptqqq is in a codimensionk ` l set (N large enough).For k ` l larger than 2n´ 1 we get a condition on jNpf0, f1q that isgenerically nowhere satisfied on M.Any higher Fuller order gives rise to one of the following moves

pk , 0q Ñ pk ` 1, 0q

pk , lq Ñ pk , l ` 1q

pk , lq Ñ pk ` 2, 0q

To find K pnq we computethe longest sequence ofmoves staying in tk ` l ď 2n ´ 1u

b b

(3, 0) (4, 0)

x1 + x2 = 2n− 1

Inside the collinearity set tf0 ^ f1 “ 0u

This set is already of codimension n ´ 1

We need to find just one more condition to conclude

We may work with accumulation points qptq that lie in the set

tf1 ^ rf0, f1s ‰ 0u.

The two conditions single out an embedded hypersurfacetransverse to f1.

Write, up to subsequences, 9qptq “ f0pqptqq ` uf1pqptqq.

On the right-hand side we are parallel to f1pqptqq, on the leftwe are transverse to it. Both sides are zero.

The limit u exists (does not depend upon subsequences), andis forced by the pair pf0, f1q.

We obtained a new smooth field to use in our algorithm alsoin this case.

Including estimates for the set tf0^ f1 “ 0u, we can show thatK pnq ď pn ´ 1q2.

Better bounds using optimality

We have not used optimality

For time-optimal trajectories we have 2nd order conditions thatmay rule out trajectories with too many concatenated arcs.

They are usable in low dimensions (n “ 3, 4). After this theybecome difficult to handle.

In dimension 3 we have the following result

Proposition

For a generic pair pf0, f1q P VecpMq2, time-optimal trajectories

have at most Fuller times of order two. Notice thatK p3q “ 2 ď p3´ 1q2 “ 4.

This follows collecting previous results of Agrachev,Gamkrelidze, Krener, Schattler, Sigalotti et al.

Perspectives and open problems

What is the minimal K pnq (for time-extremal andtime-optimal trajectories)? We showed K pnq ď pn ´ 1q2. Canwe at least bound K pnq by a sub-quadratic function in n?

What can be said for M, f0, f1 analytic?

What about the multi-input case? Chattering phenomenon forU polytope structurally stable for extremal trajectoriesZelikin–Lokutsievskiy–Hildebrand, 2012.Switching studied for U ball in Agrachev–Biolo.

Is optimality of (iterated) Fuller extremals structurally stable?Nobody has proved yet that the extremals found in theexample of Kupka are time-optimal (but everybody believesso).

Thank you for the attention!

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