Optimal control of parabolic PDEs with state constraints Francesco - - PowerPoint PPT Presentation

Optimal control of parabolic PDEs with state constraints

Francesco Ludovici Joint work with Ira Neitzel and Winnifried Wollner

WIAM2016 Hamburg, 31.08 - 02.09

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Problematics in high-quality glass cooling

◮ Avoid damages due to thermal stress

Consequences on the model

◮ Keep track of the temperature gradient inside the glass

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Problematics in high-quality glass cooling

◮ Avoid damages due to thermal stress ◮ Preserve the quality monitoring the chemical reactions

Consequences on the model

◮ Keep track of the temperature gradient inside the glass ◮ Provides a temperature profile

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Problematics in high-quality glass cooling

◮ Avoid damages due to thermal stress ◮ Preserve the quality monitoring the chemical reactions ◮ Reduce the energy cost of the furnace

Consequences on the model

◮ Keep track of the temperature gradient inside the glass ◮ Provides a temperature profile ◮ Control the cooling process

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Problematics in high-quality glass cooling

◮ Avoid damages due to thermal stress ◮ Preserve the quality monitoring the chemical reactions ◮ Reduce the energy cost of the furnace ◮ High temperature of the process, ca. 1500K

Consequences on the model

◮ Keep track of the temperature gradient inside the glass ◮ Provides a temperature profile ◮ Control the cooling process ◮ Transport equation for the radiation intensity I = I(x, t, s, ν)

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For q, u furnace and glass temperature, B(u, ν) Planck function for black body radiation in glass, [Clever, Lang ’12]

Augmented Objective Functional

J(u, q) = 1 2

T

• u − ud2 + δu∇u2 + δq(t)(q − qd)
• dt

Equation Constraints

∂tu − ∇ · (κc∇u) = −

• ν0
• S

κν

• B(u, ν) − I
• dsdν

s · ∇I + (κν + σν)I = σν

• S

Ids + κνB(u, ν)

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Boundary and Initial Condition

κcn · ∇u = hc(q − u) + F(B(u, ν)) I = r(n · s) ¯ I + (1 − n · s)B(q, ν)

u(x, 0) = u0

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Boundary and Initial Condition

κcn · ∇u = hc(q − u) + F(B(u, ν)) I = r(n · s) ¯ I + (1 − n · s)B(q, ν)

u(x, 0) = u0 The list of possible applications is wider:

◮ Crystal Growth by sublimation ◮ Cancer treatment by local hypothermia ◮ Material failure

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Model Problem

For I = (0, T) and Ω ⊂ Rn, n = {2, 3} smooth domain min(u,q)∈U×Qad

1 2

• I
• Ω(u(x, t) − ud(x, t))2dxdt + α

2

• I q(t)2dt

subject to

∂tu(t, x) − ∆u(t, x) + d(t, x, u) = q(t)g(x)

in I × Ω, u(t, x) = 0

• n I × ∂Ω,

u(0, x) = u0 in Ω, and control and state constraints qmin ≤ q(t) ≤ qmax F(u) ≤ b, ∀t ∈ [0, T],

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Model Problem

with state constraints F1(u) =

• Ω

|∇u(x, t)|2ω(x)dx ≤ b, ∀t ∈ [0, T],

F2(u) =

• Ω

u(x, t)ω(x)dx ≤ b,

∀t ∈ [0, T],

the former for the the linear state equation, the latter for the semi-linear.

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State Equation Control and State Space

Qad = {q ∈ L2(I, Rm) | qmin ≤ q(t) ≤ qmax, a.e. in I} W(0, T) = {u ∈ L2(I, V), ∂tu ∈ L2(I, V ∗)} U = {L2(I, H2(Ω) ∩ H1

0) ∩ L∞(I × Ω) ∩ H1(I, L2(Ω))}

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State Equation Control and State Space

Qad = {q ∈ L2(I, Rm) | qmin ≤ q(t) ≤ qmax, a.e. in I} W(0, T) = {u ∈ L2(I, V), ∂tu ∈ L2(I, V ∗)} U = {L2(I, H2(Ω) ∩ H1

0) ∩ L∞(I × Ω) ∩ H1(I, L2(Ω))}

Linear Case

For q ∈ Qad, there exists u ∈ U solution of the state equation. U ⊂ C(¯ I, V) F1 : U → C(¯ I)

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State Equation Control and State Space

Qad = {q ∈ L2(I, Rm) | qmin ≤ q(t) ≤ qmax, a.e. in I} W(0, T) = {u ∈ L2(I, V), ∂tu ∈ L2(I, V ∗)} U = {L2(I, H2(Ω) ∩ H1

0) ∩ L∞(I × Ω) ∩ H1(I, L2(Ω))}

Semi-Linear Case

For q ∈ Qad, there exists u ∈ W(0, T) solution of the state equation. W(0, T) ⊂ C(¯ I, H) F2 : W(0, T) → C(¯ I)

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Optimality Conditions

Denoting the concatenation of the control-to-state map S : L∞(I, R) → W(0, T) ∩ L∞(I × Ω) and the state constraint F = (u(t, x), ω(x)) with G := (F ◦ S) : L∞(I, R) → R, we rely on the following linearized Slater’s condition

∃ qγ ∈ Qad s.t. G(¯

q) + G

′(¯

q)(qγ − ¯ q) ≤ −γ < 0 for some γ ∈ R+, where ¯ q is a local solution in the sense of L2(I, R).

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Optimality Conditions

Denoting the concatenation of the control-to-state map S : L∞(I, R) → W(0, T) ∩ L∞(I × Ω) and the state constraint F = (u(t, x), ω(x)) with G := (F ◦ S) : L∞(I, R) → R, we rely on the following linearized Slater’s condition

∃ qγ ∈ Qad s.t. G(¯

q) + G

′(¯

q)(qγ − ¯ q) ≤ −γ < 0 for some γ ∈ R+, where ¯ q is a local solution in the sense of L2(I, R). The condition above ensures with standard argument the well-posedness of the

• ptimal control problem as well as first order necessary conditions in KKT-form.

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Optimality Conditions

Issue with state constraints, the adjoint equation reads b(ϕ, ¯ z) + (ϕ, ∂ud(·, ·, ¯ u)¯ z) = (¯ u − ud, ϕ)I + ¯

µ, F(ϕ)C(¯

I)∗,C(¯ I)

where µ ∈ C(¯ I)∗.

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Optimality Conditions

Issue with state constraints, the adjoint equation reads b(ϕ, ¯ z) + (ϕ, ∂ud(·, ·, ¯ u)¯ z) = (¯ u − ud, ϕ)I + ¯

µ, F(ϕ)C(¯

I)∗,C(¯ I)

where µ ∈ C(¯ I)∗. For the derivation of convergence rate in the linear setting, one uses

F(¯

u), ¯

µ = 0, ¯ µ ≥ 0, F(¯

u) ≤ 0 to circumvent low regularity of adjoint variable. In our setting, the presence of the semi-linear term requires another approach.

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Optimality Conditions

For the discussion of SSC, we introduce the Hamiltonian and Lagrangian H(q, u, z) = 1 2(u − ud)2 + α 2 q2(t) + z

• m
• i=1

qi(t)gi(x) − d(·, ·, u)

• ,

L(q, µ) = j(q) + µ, F(u),

and the cone of critical directions: p ∈ L2(I, R) such that pi(t) =

   ≥ 0

if ¯ qi = qmin,

≤ 0

if ¯ qi = qmax, = 0 if

• Ω ∂q ¯

Hidx = 0, for all i = 1, ..., m

∂F ∂u (¯

u)vp ≤ 0 if F(¯ u) = 0,

• Ω

∂F ∂u (¯

u)vpd ¯

µ = 0

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Optimality Conditions

We rely on a weak SSC [Casas et al. ’07, De los Reyes et al. ’08] Assumption: let ¯ q be a feasible control fulfilling first-order optimality conditions. We assume the existence of positive constants ν, ξ such that there holds

∂2 ¯ L ∂2q p2 > 0 ∀p ∈ C¯

q \ {0},

∂2

q ¯

Hi,i ≥ ξ

∀t ∈ I \ Eν

i , ∀i = 1, ..., m,

where Eν

i =

• t ∈ I
• |
• Ω

∂q ¯

Hidx| ≥ ν

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. Ludovici | 13

Quadratic growth condition

Under the weak SSC and first order necessary conditions, for constants δ, η > 0, there holds j(¯ q) + δq − ¯ q2

L2(I,Rm) ≤ j(q),

q − ¯

qL2(I,R) ≤ η

◮ The proof moves by contradiction. ◮ Two-norm discrepancy removed thanks to

j′′(¯ q)p2 ≤ lim inf

k→∞ j′′(qk)p2 k

if p = 0 then Λ lim inf

k→∞ pk2 L2(I,Rm) ≤ lim inf k→∞ j′′(qk)p2 k

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Discretization

Time discretization via discontinuous Galerkin method:

· Partitioning of ¯

I = [0, T] in subintervals In = (tn−1, tn] of size kn, with maximum size k.

· For V = H1

0(Ω), semi-discrete state and test space:

Uk = {vk ∈ L2(I, V) | vk|In ∈ P0(In, V)} Reasons for dG(0):

· Admits a variational formulation · Equations for each time intervals · Galerkin ortogonality

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Discretization

With w, ˆ w solutions to backward uncontrolled state equation, the dG(0)-method requires estimates [Meidner et al. ’11]

w − ˆ

wL1(I,L2(Ω)),

w − ˆ

wH−2(Ω),

• I

(T − t)∂tw(t)2

H−1(Ω)dt,

for the error at the nodal points tn and in the interior of In.

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Discretization

With w, ˆ w solutions to backward uncontrolled state equation, the dG(0)-method requires estimates [Meidner et al. ’11]

w − ˆ

wL1(I,L2(Ω)),

w − ˆ

wH−2(Ω),

• I

(T − t)∂tw(t)2

H−1(Ω)dt,

for the error at the nodal points tn and in the interior of In. To extend the estimates to the semilinear case, [Nochetto ’88] ˜ d =

• d(u(t,x))−d(uk(t,x))

u(t,x)−uk(t,x)

if u(t, x) = uk(t, x) else. exploiting ˜ d bounded in L∞(I × Ω).

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Discretization

Space discretization with standard conforming finite elements: Ukh = {ϕkh ∈ L2(I, Vh) | ϕkh|In ∈ P0(In, Vh)} where Vh piecewise linear functions. Control variable is discretized implicitly by the optimality conditions. In our case means, qk is piecewise constant.

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Discretization

Space discretization with standard conforming finite elements: Ukh = {ϕkh ∈ L2(I, Vh) | ϕkh|In ∈ P0(In, Vh)} where Vh piecewise linear functions. Control variable is discretized implicitly by the optimality conditions. In our case means, qk is piecewise constant. Negative-norm estimates also for the fully discrete problem linearized with ˜ d =

• d(uk(t,x))−d(ukh(t,x))

uk(t,x)−ukh(t,x)

if uk(t, x) = ukh(t, x) else.

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Error Analysis - Linear Case

In the linear case we obtain a clear separation of the spatial and temporal error introducing an intermediate time-discrete problem. By means of a duality argument we obtain

¯

q − ¯ qk2

L2(I,Rm) ≤ ck

• log T

k + 1

1

2 ,

¯

qk − ¯ qkh2

L2(I,Rm) ≤ ch

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Error Analysis - Linear Case

In the linear case we obtain a clear separation of the spatial and temporal error introducing an intermediate time-discrete problem. By means of a duality argument we obtain

¯

q − ¯ qk2

L2(I,Rm) ≤ ck

• log T

k + 1

1

2 ,

¯

qk − ¯ qkh2

L2(I,Rm) ≤ ch

thus

¯

q − ¯ qkh2

L2(I) ≤ C

• k
• log T

k + 1

1

2 + h

• .

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Error Analysis - Semi-Linear Case

We use the two-way feasibility approach [Falk ’73, Meyer ’08)] together with a localization argument [Casas et al. ’13]. Two-way feasibility: construct feasible competitors close to a solution of (Pkh), feasible for (P) and viceversa. Localization argument: To deal with local solutions, we use the two-way feasibility in a neighbourhood of ¯ q Qr = {q ∈ Qad s.t q − ¯ qL2(I,Rm) ≤ r} introducing auxiliary problems (Pr), (Pr

kh) with control, respectively, from

Qr

feas = Qfeas ∩ Qr,

Qr

kh,feas = Qkh,feas ∩ Qr.

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Error Analysis - Semi-Linear Case

Without intermediate step

¯

q − ¯ qkh2

L2(I) ≤ C

• k2

log T k + 1

1

2 + h2

log T k + 1

• ,

a clear separation of spatial/temporal error would requires a QGC for the time-discrete problem. Problem of stability unless we assume stronger SSCs.

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Numerics

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Numerics

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For the future

• Boundary controls
• Non-smooth domains
• Gradient state constraint pointwise in time and space

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Thank you for your attention! Vielen Dank für Ihre Aufmerksamkeit! Grazie per la vostra attenzione!

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