Characterization of quadratic growth for strong minimain the optimal control of semi-linear equations

J.F. Bonnans 1

INRIA Saclay and CMAP

Joint work withT. Bayen (U. Montpellier II)

F. J. Silva (U. Paris VII)

Groupe de Travail Controle, U. Paris VI, 13 avril 2012

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1 Preliminaries

2 First order expansions for the cost and the PMP

3 Standard second order conditions for semi-linear problems

4 A decomposition result for the second order expansion of the cost

5 Extensions of the standard results to the strong case

6 An improved result

7 Future work2 / 32

Outline

1 Preliminaries

2 First order expansions for the cost and the PMP

3 Standard second order conditions for semi-linear problems

4 A decomposition result for the second order expansion of the cost

5 Extensions of the standard results to the strong case

6 An improved result

7 Future work

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Strong and weak minima in calculus of variation

Let ` : [a, b]× Rn × Rn of class C 1, and consider the problem ofminimizing the functional (say with fixed end-points):

miny(·)∈C1

J(y) :=

∫ b

a`(t, y(t), y(t))dt

Definition

We say that y0 is a local weak minimum iff J(y) ≥ J(y0) whenever‖y − y0‖C1([a,b]) ≤ ε and that y0 is a local strong minimum iffJ(y) ≥ J(y0) whenever ‖y − y0‖C0([a,b]) ≤ ε

Example: J(y) :=∫ 1

0 [y 2 − y 4]dt : y = 0 is a weak minimum and nota strong minimum.Necessary condition: If y0 ∈ C 2([a, b]) is a weak minimum, then itsatisfies Euler-Lagrange equation and Legendre condition.Known characterization of strong optimality with quadratic growth(QG): “Weierstrass+QG” + second order optimality condition + QG

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State equation

Let Ω ⊆ Rn bounded with C 1,1 boundary and ϕ : Ω×R×R→ R be C 1

(H1) The function ϕ satisfies:

(i) D(y ,u)ϕ(x , 0, 0) is bounded,(ii) D(y ,u)ϕ(x , ·, ·) is locally Lipschitz uniformly. on x ∈ Ω.

(iii) We have ϕy (x , y , u) ≥ 0.

Proposition

Under (H1), for every u ∈ L∞(Ω) and s ∈ (n2 ,∞), the equation−∆y(x) + ϕ(x , y(x), u(x)) = 0 a.e. x in Ω,

y(x) = 0 a.e. x in ∂Ω,

has a unique solution yu ∈W 1,s0 (Ω) ∩ C (Ω). Moreover, if K ⊂ L∞ is a

bounded set, ∃Cs > 0 such that

||yu||∞ + ||yu||1,s ≤ Cs , for all u ∈ L∞(Ω) ∩ K.5 / 32

Optimal control problem

Let ` : Ω× R× R→ R and suppose that:

(H2) ` satisfies (H1) except for (iii).

Define the cost function J : L∞(Ω)→ R by

J(u) :=

∫Ω`(x , yu(x), u(x))dx .

For a, b ∈ C (Ω) with a ≤ b define

K = u ∈ L∞(Ω) | a(x) ≤ u(x) ≤ b(x), a.e. x in Ω

Consider the optimal control problem

min J(u) subject to u ∈ K. (CP)

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Optimal heat source with distributed control

Let Ω ⊂ R3 heated by electromagnetic induction or by microwaves.

Assume that the boundary temperature vanishes.

The optimal control problem becomes (with N > 0):

min J(u) :=1

2

∫Ω|y(x)− yd(x)|2 +

N

2

∫Ω|u(x)|2dx

subject to −∆y(x) = β(x)u(x) a.e. x in Ω,

y(x) = 0 a.e. x in ∂Ω,

and u ∈ K, i.e.:

a(x) ≤ u(x) ≤ b(x), a.e. x in Ω.

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Weak and strong minimum

Definition

(i) We say that u ∈ K is a strong minimum if ∃ε > 0

J(u) ≥ J(u) for all u ∈ K with ‖yu − yu‖∞ ≤ ε

(ii) For s ∈ (1,∞), u ∈ K is a Ls -weak minimum if ∃ε > 0

J(u) ≥ J(u) for all u ∈ K with ‖u − u‖s ≤ ε

(iii) We say that u ∈ K is a weak minimum if ∃ε > 0

J(u) ≥ J(u) for all u ∈ K with ‖u − u‖∞ ≤ ε

Remark:

We have (i)⇒ (ii)⇒ (iii). For proving (i)⇒ (ii), we use‖yu − yu‖∞ = O(‖u − u‖s)Since K ⊆ L∞(Ω) is bounded, Lp ∩ K and Lq ∩ K have the sameopen sets. Thus, every Ls -weak minimum is a L1-weak minimum.

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Weak and strong minimum with quadratic growth

Definition

(i) u ∈ K is a strong minimum with quadratic growth if ∃ α, ε > 0

J(u) ≥ J(u) + α‖u − u‖22 for all u ∈ K with ‖yu − yu‖∞ ≤ ε

(ii) For s ∈ (1,∞), u ∈ K is a Ls -weak minimum with quadratic growth if∃ α, ε > 0

J(u) ≥ J(u) + α‖u − u‖22 for all u ∈ K with ‖u − u‖s ≤ ε

(iii) u ∈ K is a weak minimum with quadratic growth if ∃ α, ε > 0

J(u) ≥ J(u) + α‖u − u‖22 for all u ∈ K with ‖u − u‖∞ ≤ ε

We have an analogous remark to the previous one.

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Outline

1 Preliminaries

2 First order expansions for the cost and the PMP

3 Standard second order conditions for semi-linear problems

4 A decomposition result for the second order expansion of the cost

5 Extensions of the standard results to the strong case

6 An improved result

7 Future work

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Adjoint system

The Hamiltonian H : Ω× R3 → R for (CP) is

H(x , y , p, u) = `(x , y , u)− pϕ(x , y , u).

Set y := yu. The adjoint state p, associated to u, is the unique solution ofthe linear equation:

−∆p = Hy (x , y , p, u) in Ω,

p = 0 on ∂Ω.

For a nominal u ∈ K:

Set `(x) := `(x , y(x), u(x)). Similar conventions for ϕ(x), H(x).

Given u ∈ K, set δu := u − u, δy := yu − y , and

δH(x) := H(x , y(x), p(x), u(x))− H(x , y(x), p(x), u(x)).

with similar conventions for δ`, δϕ and its derivatives.11 / 32

First order estimates

The first order Pontryagin linearization z1[u] of u → yu in the directionu − u is the unique solution of

−∆z1 + ϕy (x)z1 + δϕ(x) = 0 in Ω,

z1 = 0 on ∂Ω.

Lemma

Under assumptions (H1)-(H2), for every s ∈ (n/2,+∞), we have:‖δy‖1 = O(‖δu‖1), ‖δy‖2 = O(‖δu‖2), ‖δy‖∞ = O(‖δu‖s),

‖z1‖1 = O(‖δu‖1), ‖z1‖2 = O(‖δu‖2), ‖z1‖∞ = O(‖δu‖s),

‖z1 − δy‖1 = O(‖δu‖1‖δu‖s), ‖z1 − δy‖2 = O(‖δy‖∞‖δu‖2).

Idea of proof. Use standard regularity results for elliptic equations: ifα ≥ 0 and z satisfies −∆z + α(x)z = f , in Ω with Dirichlet condition,then ‖z‖2,s ≤ cs‖f ‖s , ‖z‖1 ≤ c1‖f ‖1.

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Pontryagin maximum principle

Lemma

Under (H1)-(H2), for all u ∈ K and s ∈ (n/2,∞) we have

J(u)− J(u) =∫

Ω δH(x)dx + O(‖δy‖∞‖δu‖2),J(u)− J(u) =

∫Ω δH(x)dx + O(‖δu‖1‖δu‖s).

Theorem [Raitum ’86, BonCas ’89]

Let u a L1-weak minimum of J. Then we have:

u(x) ∈ argminv∈[a(x),b(x)]H(x , y(x), p(x), v) a.e. x ∈ in Ω.

Idea of proof : Combine lemma and a needle perturbation(Pontryagin-McShane perturbation):

uε(x) =

v , x ∈ B(x , ε),

u(x), x ∈ Ω\B(x , ε),

One has: ‖uε − u‖1 = O(ε). 13 / 32

Outline

1 Preliminaries

2 First order expansions for the cost and the PMP

3 Standard second order conditions for semi-linear problems

4 A decomposition result for the second order expansion of the cost

5 Extensions of the standard results to the strong case

6 An improved result

7 Future work

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Standard second order conditions for semi-linear problems

We suppose

(H3) For ψ = ϕ, `, we have ψ(x , ·, ·) is C 2, D2(y ,u)2ψ(x , 0, 0) is bounded,

and D2(y ,u)2ψ(x , ·, ·) is locally Lipschitz uniformly on x ∈ Ω.

For v ∈ L2(Ω), define “weak” linearization ζ[v ] by

−∆ζ + ϕy (x)ζ + ϕu(x)v = 0, in Ω,ζ = 0, on ∂Ω.

and the quadratic form Q2[u] :L2(Ω)→ R as

Q2[u](v) =

∫Ω

[Hyy (x)(ζ[v ])2 + 2Hyu(x)ζ[v ]v + Huu(x)v 2

]dx .

The tangent cone to K at u is

TK(u) :=

v ∈ L2(Ω) ; v(x) ≥ 0 if u(x) = a(x),v(x) ≤ 0 if u(x) = b(x) .

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Second order necessary conditions for semi-linear problems

The critical cone to K at u is

CK(u) := v ∈ TK(u) ; Hu(x)v(x) = 0 a.e. in Ω .We have the following standard necessary conditions (Bonnans ’98, Casas-Troltzsch-Unger ’96)

Proposition

Under (H1)-(H3), If u is a weak minimum, then:

(i) Hu(x)v(x) ≥ 0 a.e. in Ω, for all v ∈ TK(u).(ii) Q2[u](v) ≥ 0 for all v ∈ CK(u).

and (Bonnans ’98)

Proposition

Under (H1)-(H3), If u is a weak minimum with quadratic growth, then:

(i) Hu(x)v(x) ≥ 0 a.e. in Ω, for all v ∈ TK(u).(ii) Q2[u](v) ≥ α‖v‖2 for all v ∈ CK(u).

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Sufficient conditions for semi-linear problems

Definition

Given a Hilbert space H, a quadratic form Q : H → R is a Legendre formif it is sequentially w.l.s.c. and if hk converges weakly to h in H andQ(hk)→ Q(h) then hk conv. strongly to h.

Example: Q weakly continuous quadratic form =⇒ x 7−→ ‖x‖2 + mQ(x)Legendre form (Bonnans&Shapiro)We have the following standard sufficient condition for weak quadraticgrowth (Bonnans ’98)

Theorem

Suppose (H1)-(H3) and that :

(i) Hu(x)v(x) ≥ 0 a.e. in Ω, for all v ∈ TK(u).(ii) Q2[u](v) ≥ α‖v‖2 for all v ∈ CK(u).

(iii) The quadratic for Q2[u] is a Legendre form.

Then u is a weak minimum with quadratic growth.

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For τ > 0 define the strongly active set

Aτ (u) := x ∈ Ω ; |Hu(x)| > τ ,

and the τ -critical cone

C τK(u) := v ∈ TK(u) ; v(x) = 0 for x ∈ Aτ (u) .

Evidently CK(u) ( C τK(u). We have ( Casas-Troltzsch-Unger ’96)

Theorem

Suppose (H1)-(H3) and that :

(i) Hu(x)v(x) ≥ 0 a.e. in Ω, for all v ∈ TK(u).(ii) There exists τ such that Q2[u](v) ≥ α‖v‖2 for all v ∈ C τ

K(u).

Then u is a weak minimum with quadratic growth.

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Outline

1 Preliminaries

2 First order expansions for the cost and the PMP

3 Standard second order conditions for semi-linear problems

4 A decomposition result for the second order expansion of the cost

5 Extensions of the standard results to the strong case

6 An improved result

7 Future work

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A decomposition result for the second order expansion ofthe cost

Let uk ∈ K, set δku = uk − u and suppose that ‖δku‖2 → 0. Defineyk := yuk and a sequence of measurable sets Ak ⊂ Ω and Bk ⊂ Ω suchthat

|Ak ∪ Bk | = |Ω|, |Ak ∩ Bk | = 0 and |Bk | → 0.

Decompose uk into uAkand uBk

defined by :uAk

= uk on Ak , uBk= u on Ak ,

uAk= u on Bk , uBk

= uk on Bk .

We set

δAku := uAk

− u, δBku := uBk

− u and hence δku = δAku + δBk

u.

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A decomposition result for the second order expansion ofthe cost

Let δHk(x) := H(x , y(x), p(x), uk(x))− H(x , y(x), p(x), u(x)).

We have the following decomposition result

Theorem

Under (H1)-(H3) we have that as ‖δku‖2 → 0, ‖δAku‖∞ → 0

J(uk)− J(u) =∫BkδHk(x)dx +

∫Ak

Hu(x)δAku(x)dx + 1

2 Q2[u](δAku) + o(‖δku‖2

2)

= J(uBk)− J(u) + J(uAk

)− J(u) + o(‖δku‖22).

Proof based on fundamental estimates from the Ls - theory for linearelliptic equations.

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Outline

1 Preliminaries

2 First order expansions for the cost and the PMP

3 Standard second order conditions for semi-linear problems

4 A decomposition result for the second order expansion of the cost

5 Extensions of the standard results to the strong case

6 An improved result

7 Future work

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Strict Pontryagin condition

Set K(x) := [a(x), b(x)], assume that min(b − a) > 0.

Definition

(i) We say that u satisfies the strict PMP condition if for all x ∈ Ω,

H(x , y(x), p(x), u(x)) < H(x , y(x), p(x), v) for all v ∈ K(x), v 6= u(x).

(ii) H satisfies the pointwise global quadratic growth property at u if∃ α > 0 such that for all x ∈ Ω, v ∈ K(x), we have

H(x , y(x), p(x), u(x)) + α|v − u(x)|2 ≤ H(x , y(x), p(x), v)

Lemma

H satisfies the pointwise global quadratic growth property at u iff usatisfies the strict PMP condition and Q2[u]v ≥ α‖v‖2

2 for all v ∈ CK(u).

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Extension of the standard result to the strong case

We have the following extension of Bonnans ’98.

Theorem

Suppose (H1)-(H3) and that u ∈ K satisfies :

(i) The strict PMP condition.

(ii) Q2[u](v) ≥ α‖v‖2 for all v ∈ CK(u).

(iii) The quadratic for Q2[u] is a Legendre form.

Then u is a strong minimum with quadratic growth.

Idea of the proof: Suppose that exists uk ∈ K with ‖δky‖∞ → 0 andJ(uk)− J(u) ≤ o(||δku||22), then we easily get that ‖δku‖2 → 0. Choose

Ak :=

x ∈ Ω | |uk(x)− u(x)| ≤√‖δku‖1

and Bk := Ω\Ak .

Verify that |Bk | → 0, set σAk:= ‖δAk

u‖2, σBk:= ‖δBk

u‖2. IfσAk

= o(σBk) we get a contradiction with the global growth of H.

Otherwise, up to subsequence, σBk= O(σAk

) and we can proceed as inBonnans ’98.

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Extension of the standard result to the strong case

End of the proof:

Recall σAk:= ‖δAk

u‖2, σBk:= ‖δBk

u‖2, ‖δku‖22 = σ2

Ak+ σ2

Bk.

Applying decomposition result yields:∫Bk

δHk(x)dx +

∫Ak

Hu(x)δAku(x)dx + 1

2 Q2[u](δAku) ≤ o(‖δku‖2

2).

1) If σAk= o(σBk

), then, Q2[u](δAku) = O(σ2

Ak) = o(σ2

Bk) and the

previous inequality implies

σ2Bk≤ o(σ2

Bk),

which is a contradiction.2) If σBk

= o(σAk), we can proceed as previously to obtain a contradiction.

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Example

Let f , yd ∈ C (Ω), g ∈ C 2(R) s.t. gy ≥ 0 and gyy local Lipschitz.

Consider the following data for (CP),

`(x , y , u) = 12 |u|

2 + 12 |y − yd(x)|2, ϕ(x , y , u) = g(y) + u + f .

We have H(x , y , p, u) = 12 |u|

2 + 12 |y − yd(x)|2−p(g(y) + u + f ) and:

* Q2[u] is a Legendre form: Q2[u](v) = ‖v‖22 + Q(v)

* H is strictly convex with respect to u.

Thus:

(I) For all v ∈ TK(u) we have Hu(x)v(x) ≥ 0 a.e. in Ωand

(II) Q2[u](v) ≥ α‖v‖2 for all v ∈ CK(u)

are a sufficient condition for a strong minimum with quadratic growth.

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Extension to the strong case without Legendre condition

Using analogous arguments we have the following extension of Casas-Troltzsch-Unger ’96.

Theorem

Suppose (H1)-(H3) and that u ∈ K satisfies :

(i) The strict PMP condition.

(ii) There exists τ such that Q2[u](v) ≥ α‖v‖2 for all v ∈ C τK(u)

Then u is a strong minimum with quadratic growth.

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Outline

1 Preliminaries

2 First order expansions for the cost and the PMP

3 Standard second order conditions for semi-linear problems

4 A decomposition result for the second order expansion of the cost

5 Extensions of the standard results to the strong case

6 An improved result

7 Future work

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An improved result

Theorem

Suppose (H1)-(H3) and that u ∈ KThen u is a strong minimum with quadratic growth iff the two conditionsbelow hold:

(i) The Hamiltonian satisfies the global quadratic growth at u.

(ii) We have Q2[u](v) ≥ α‖v‖2 for all v ∈ CK(u)

Idea of the proof: Suppose that exists uk ∈ K with ‖δky‖∞ → 0 andJ(uk)− J(u) ≤ o(||δku||22), then we easily get that ‖δku‖2 → 0. Let εk ↓ 0,Bk = B1

k ∪ B2k and Ak := Ω \ Bk where

B1k :=

x ∈ Ω : |uk(x)− u(x)| ≥

√‖δku‖1

,

B2k := x ∈ Ω : |Hu(x)| ≤ εk ,

Set σAk:= ‖δAk

u‖2, σBk:= ‖δBk

u‖2. If σAk= o(σBk

) we get a contradiction with

the global growth of H. Otherwise,up to subsequence, σBk= O(σAk

). Setting

hk := δAku/σAk

we get α‖hk‖22 ≤ Q2[u](hk) ≤ o(1) which is impossible.

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Outline

1 Preliminaries

2 First order expansions for the cost and the PMP

3 Standard second order conditions for semi-linear problems

4 A decomposition result for the second order expansion of the cost

5 Extensions of the standard results to the strong case

6 An improved result

7 Future work

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Extensions ?

- Finitely many integral constraints (non unique multiplier)

Gi (y , u) :=

∫Ω

gi (x , y(x), u(x))dx ≤ 0, i = 1, . . . , q.

- Mixed state and control constraints

gi (x , y(x), u(x))dx ≤ 0, for a.a. x ∈ Ω, i = 1, . . . , q.

- Parabolic problems- Sensitivity analysis

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References

T. Bayen, J.F. Bonnans, F.J. Silva, Strong second order optimalityconditions for semilinear elliptic equations optimal controlproblems, Inria report RR-7765, Oct. 2011.

J.F. Bonnans, Second-order analysis of optimal controlproblems with control and initial-final state constraints,Appl. Math. Optim. 38-3:303–325, 1998.

J.F. Bonnans, N.P. Osmolovski, Second-order analysis of optimalcontrol problems with control and initial-final stateconstraints, J. Convex analysis 17-3 (2010), 885–913.

E. Casas, F. Troelzsch, A. Unger, Second order sufficientoptimality conditions for some state-constrainted controlproblems of semi linear elliptic equations, SIAM J. ControlOptim., 38:369–391, 2000.

X. Li, J. Yong, Optimal Control Theory For Infinite Dimensional Systems,Birkhauser, 1994

A.A. Milyutin, N. P. Osmolovski, Calculus of Variations andOptimal Control, AMS, 1998. 32 / 32