About prior saturation points for affine control...

About prior saturation points for affine control systems

Terence Bayen (Universite de Montpellier)

(collaboration avec O. Cots)

JOURNEES MODE 2018 - AUTRANS

28 mars 2018

T. Bayen (Universite de Montpellier) 1 / 25

Outline

1 Preliminary results for affine controlled systems

2 Saturation phenomenon in a fed-batch model

3 Determination of prior saturation points and the notion of bridge


Preliminary results for affine controlled systems

Outline






Minimal time control problem

Given smooth vector fields f ,g : Rn ! Rn and a closed subset T ⇢ Rn, consider

v(x0) := infu2U Tu s.t. xu(Tu) 2 T8><>:

x = f (x) + u(t)g(x), |u| 1,x(0) = x0.

GoalSynthesize an optimal feedback control x0 7! u[x0] by:

u[x0] := u⇤(0, x0).

We suppose classical hypotheses ensuring existence of an optimal control.



Application of the Pontryagin Maximum Principle

• Hamiltonian condition: H =⌦p, f (x)

↵+ u

⌦p,g(x)

↵+ p0 ) for a.e. t 2 [0,Tu ]

8>>><>>>:

�(t) > 0 ) u(t) = +1�(t) < 0 ) u(t) = �1�(t) = 0, 8t 2 [t1, t2] ) Singular arc

where t 7! p(t) satisfies p(t) = �rx H(x(t),p(t),�1,u(t)).

• By differentiating the switching function t 7! �(t) := ⌦p(t),g(x(t))

↵:

�(t) =⌦p(t), [f ,g](x(t))

↵,

�(t) =⌦p(t), [f , [f ,g]](x(t))

↵+ u(t)

⌦p(t), [g, [f ,g]](x(t))

↵

• The singular control us : [t1, t2]! R is

us(t) := �⌦p(t), [f , [f ,g]](x(t))

↵⌦p(t), [g, [f ,g]](x(t))

↵

(for a singular arc of order 1).



Saturation phenomenon

QuestionConsider a singular arc of order 1 that satisfies Legendre-Clebsch optimality condition@@u

d2

dt2 Hu > 0 (i.e. it is a turnpike), and such that the singular arc becomes non-admissible i.e.

9t 2 [t1, t2], |us(t)| > 1,

• Should we follow the singular arc until the saturation point such that us(t ⇤) = 1 ?• How does it affect the synthesis ?



Related works about saturation and prior-saturation

A. Rapaport, T. B., M. Sebbah, A. Donoso, A. Torrico, Dynamical modelling and optimal control of landfills,Mathematical Models and Methods in Applied Sciences, issue No 05, vol. 26, pp. 901-929, 2016.

T. B., F. Mairet, M. Mazade, Analysis of an optimal control problem connected to bioprocesses involving asaturated singular arc, DCDS, series B, vol. 20, 1, pp.39–58, 2015.

B. Bonnard, O. Cots, S. Glaser, M. Lapert, D. Sugny, Y. Zhang, Geometric optimal control of the contrastimaging problem in nuclear magnetic resonance, IEEE Trans. Automat. Control, 57 (2012), no 8, pp.1957–1969.

H. Schaettler and M. Jankovic, A synthesis of time-optimal controls in the presence of saturate singulararcs, Forum Mathematicum, 5, (1993), pp. 203–241.

U. Ledzewicz, H. Schattler, Anti-angiogenic therapy in cancer treatment as an optimal control problem,SIAM J. on Control and Optimization, 46, 3, pp. 1052–1079, 2007.

U. Ledzewicz, H. Schattler, Geometric Optimal Control : Theory, Methods and Examples, Springer, 2012.

B. Bonnard, M. Chyba, Singular Trajectories and their role in Control Theory, Springer, SMAI, vol. 40, 2002.



Existence of a prior saturation point in the plane

x = f (x) + u(t)g(x)

Assumption

• The target T is reachable from every point x0 2 R2 and det(f (x),g(x)) < 0 for any x 2 R2 (i.e.�0 = ;).

• Along �SA, the strict Legendre-Clebsch condition @@u

d2

dt2 Hu > 0 is satisfied.

• The singular locus �SA := {x 2 R2 ; det(g(x), [f ,g](x) = 0)} has exactly one saturation pointx? s.t. ✓(x?) = 1 where for x 2 �SA

✓(x) := �⌦g(x)?, [f , [f ,g]](x)

↵⌦g(x)?, [g, [f ,g]](x)

↵

• Moreover, there exists a neighborhoodV of x? s.t. ✓ is increasing inV.• If �� := {xm(t , x?) ; t � 0} where xm(·, x?) is the unique solution of the system starting from

x? at time 0 with the control u = �1, then

T \ �� = ;.



Existence of a prior saturation point

TheoremUnder the previous assumptions:• there exists a prior-saturation point xa 2 �SA : any admissible singular arc losses its

optimality at xa (say at time ta) such that ✓(xa) 2] � 1,1[.• an optimal control satisfies u = +1 in a right neigborhood of ta.

Sketch of proof. The switching function t 7! �(t) := ⌦p(t),g(x(t))

↵satisfies the ODE:

�(t) = �u(t)�(t) + ↵(xu(t)) a.e. t 2 [0,Tu ]

where↵(x) := �det(g(x), [f ,g](x))

det(f (x),g(x))and �(x) :=

det(f (x), [f ,g](x))det(f (x),g(x))

,

and�u(t) := �(xu(t)) � ↵(xu(t))u(t), t 2 [0,Tu ].

• Switching rule : switching points in {x 2 R2 ; det(f (x), [f ,g](x)) > 0} occur from �1 to 1.• If xu(·) is optimal until x? at t = t? ) �(t) < 0, 8t > t? ) contradiction with the PMP.


Saturation phenomenon in a fed-batch model

Outline






Fed-batch reactor with one species

X,S

Outlet pump when V = vmaxFeed pump

Sin

Agitation system,

X, S, V

Temperature �,...

sensors

• Biological reaction : S ! X8>>><>>>:

x = µ(s)x � uv x ,

s = �µ(s)x + uv (sin � s),

v = u,

• X , S : micro-organisms, substrateconcentrations ;V : volume ;u: dilution ratesin: input substrate concentration

• Objective : transformation of sin intosref s.t.

sref ⌧ sin.



The two-dimensional optimal control problem

• Target set:

T := {(s, v) 2 [0, sin] ⇥R⇤+ ; s sref and v = v } = [0, sref ] ⇥ {v }.

• Notice that M := v(x + s � sin) is conserved) x = Mv + sin � s

• Admisible control setU := {u : [0,+1)! [0,1] ; u meas.}

Optimal control problem

minu2U

Tu ,

8>><>>:s = �µ(s)

hMv + sin � s

i+ u

v (sin � s),v = u,

s.t.

((s(Tu), v(Tu)) 2 T ,

(s(0), v(0)) = (s0, v0).

• Kinetics of Haldane type : µ(s) = µsk+s+k 0s2 with a

unique maximum s 2 [0, sin] s.t.

µ0(s) = 0.



Pontryagin Maximum Principle

• Hamiltonian : H := ��sµ(s)h

Mv � (s � sin)

i+ u

"�s(sin � s)

v+ �v

#

| {z }�

+�0.

• Switching function t 7! �(t)

8><>:�(t) < 0 =) u(t) = 0 (No feeding),�(t) > 0 =) u(t) = 1 (Maximal feeding),

Lemma(i) The singular locus is s = s and the singular control is

us[v ] := µ(s)"v +

Msin � s

#

(ii) In the set {s > s}, resp. {s < s}, only a switching from 0 to 1, resp. from 1 to 0 is possible



Synthesis without saturation

TheoremIf the singular arc is admissible i.e. 8v 2 [0, v ], us[v ] 1, the optimal feedback control is:

u[s, v ] =

8>>><>>>:

1 if s < s and v < v ,us[v ] if s = s and v v ,

0 if s > s or v = v .

0 2 4 61 3 5

0

2

1

3

0.5

1.5

2.5

Substrate

Vo

lum

e



Synthesis with saturation

Assumption : 9 v ⇤ < v s.t. us[v ⇤] = 1 :

v ⇤ :=1µ(s)

� Msin � s

Corollary

• Any singular optimal trajectory leaves the singular arc before v ⇤ with u = 1 i.e. there existsva < v ⇤ such that a singular arc is not optimal for v > va.

• A switching curve emanates from (s, va).

Sketch of proof. verify the assumptions of the previous proposition.

DefinitionThe point (s, va) is a prior saturation point ; it is a frame point of type (CS)2 at the intersection of asingular arc and a switching curve emanating from this point.

U. Boscain, B. Piccoli, Optimal Syntheses for Control Systems on 2-D Manifolds, Springer SMAI, vol. 43,2004.



Switching curve and frame point of type (CS)2

• The switching curve is computed by minimization of the time of trajectories B� � B+ � B�starting at (sin, v0).

0 2 4 61 3 5

0

2

4

6

1

3

5

7

Substrate

Vo

lum

e

Figure: Singular arc until va < v ⇤ (in red) and switching curve emanating from (s, va) (in green).

Lemma

(i) The switching curve connects (s, va) and (s, v).(ii) The switching curve C can be parameterized by a curve v 2 [va, v ] 7�! sc(v).



Synthesis with saturation

TheoremThe optimal feedback control is:

u[s, v ] =

8>>><>>>:

1 if s < sc(v) and v < v ,us[v ] if s = s and v va,

0 if s > sc(v) or v = v .

0 2 4 61 3 5

0

2

4

6

1

3

5

7

Substrate

Vo

lum

e

Remark : The singular arc is indeed necessarily optimal if v is small !T. Bayen (Universite de Montpellier) 17 / 25


Synthesis with saturation (with v ⇤ < 0)

0 20102 4 6 8 12 14 16 18

0

2

4

6

1

3

5

7

Substrate

Volu

me


Determination of prior saturation points and the notion of bridge

Outline






Synthesis 1 : from D to (sref , v)

0 1 2 3 4 5 6

s

0

1

2

3

4

5

6

7

8

v

The Bang arc u = 1 is tangent to the switching curve at (s, va).



Synthesis 2 : from (s, va) to D

0 1 2 3 4 5 6

s

0

1

2

3

4

5

6

7

8

v

The switching curve is tangent to the singular locus at (s, v ⇤)



Bridge between a singular arc and the extended target

DefinitionWe call bridge the bang arc u = +1 which connects the prior saturation point �SA to the extendedtarget set T := [0, sin] ⇥ {v }Write the system x = f (x) + u(t)g(x). Finding va amounts to solve

8>>>>><>>>>>:

⌦p(0),g(x(0))

↵= 0,⌦

p(0), [f ,g](x(0))↵

= 0,⌦p(tc),g(x(tc))

↵= 0 and x(tc) 2 T ,

H = 0

with unknown: (va,p(0), tc) 2 R4.

0 1 2 3 4 5 6

s

0

1

2

3

4

5

6

7

8

v

• Synthesis 1: in blue : initial condition in D ;target point (sref , v)

• Synthesis 2: in green : initial condition(s(tc), va) ; “free” terminal condition



Bridge between two singular arcs

DefinitionWe call bridge the bang arc u = +1 which connects two singular arcs (from a prior-saturationpoint).

B. Bonnard, O. Cots, J. Rouot, T. Verron, Working Notes on the Time Minimal Saturation of a Pair of Spinsand Application in Magnetic Resonance Imaging https://hal.archives-ouvertes.fr/hal-01721845/



Example in MRI with a single spin

min|u|1 Tu s.t. (y(Tu), z(Tu)) = (0,0) with

(y = ��y � u(t)zz = �(1 � z) + u(t)y

y(0)z(0)

!2 R2

• �SA = {z = zs} [ {y = 0}• Determination of the prior-saturation point:

8>>>>>><>>>>>>:

⌦�(0),g(x(0))

↵= 0,⌦

�(0), [f ,g](x(0))↵

= 0,⌦�(tc),g(x(tc))

↵= 0,⌦

�(tc), [f ,g](x(tc))↵

= 0,H = 0

where x = (y , z) and unknown:(z(0),�(0), tc , y(tc)) 2 R5.• A bridge (u=+1) connects the two singular arcs.



Concluding remarks

In the setting of affine systems in the plane with a single input:

• Synthesis in blue- A switching curve emanates from the prior saturation point xa.- The switching curve connects two singular arcs or a singular arc to T .- The bridge is tangent to the switching curve at the prior saturation point (to be done...)

• Synthesis in green- A switching curve emanates from xsat .- The switching curve connects two singular arcs or a singular arc to T .- The switching curve is tangent to the singular locus at the saturation point (to be done...)

and in higher dimension?

Merci pour votre attention


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