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Optimal Control of Parabolic Equations in Tailored Control Spaces

Christian Meyer

TU Dortmund, Faculty of Mathematics joint work with

Hannes Meinlschmidt (RICAM Linz) and Joachim Rehberg (WIAS Berlin) lawoc 2018, Septempber 3–8, 2018, Quito, Ecuador

Outline An Application Problem Abstract Setting Existence of Optimal Controls First-Order Optimality Conditions Improved Regularity Conclusion and Outlook

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

The Thermistor Problem Application: Heating of conducting materials by means of direct current

Thermistor Problem

∂tθ − div(κ(θ)∇θ) = (σ(θ)∇ϕ) · ∇ϕ in Q := Ω × (0, T) ν · κ(θ)∇θ + αθ = αθl

• n Σ := ∂Ω × (0, T)

θ(0) = θ0 in Ω − div(σ(θ)∇ϕ) = 0 in Q ν · σ(θ)∇ϕ = u

• n ΣN := ΓN × (0, T)

ϕ = 0

• n ΣD := ΓD × (0, T)

                       (T) with: θ – temperature, ϕ – electric potential, u – induced current density (control)

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Optimal Control of the Thermistor Problem

Optimal Control Problem

min Js(θ, ϕ) + Jc(u) s.t. (θ, ϕ, u) satisfy (T)

• (PT)

where Jc is coercive on a suitable control space U

Theorem

Suppose that the control space U embeds compactly in L∞((0, T); W −1,q

Γd

(Ω)) with q > 3 = dim(Ω). Then, under suitable (mild) assumptions on the data, there exists an

• ptimal solution of (PT).

Question

How to choose the control space such that

• 1. the compact embedding is guaranteed

and at the same time

• 2. “handy” optimality conditions can be derived?

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Choice of the Control Space

Compact embedding in L∞((0, T); W −1,q

Γd

(Ω)) requires at least a little bit time differentiability of the control

Naive choice: U = W 1,p((0, T); Lp(ΓN)), Jc(u) =

T ∂tup

Lp(ΓN) + up Lp(Γn) dt

with p large enough so that the trace is compact in W −1,q

Γd

(Ω) BUT:

• Derivative of · p

Lp(ΓN) is highly nonlinear

⇒ Jc will lead to a rather complicated ODE as gradient equation for the control

• Similar for any other exponent ∂tuα

Lp(ΓN), α > 1

Idea: split time and space regularity Choose U = W 1,2((0, T); L2(ΓN)) ∩ Lp((0, T); Lp(ΓN)) and Jc(u) = T |∂tu|2dt + |u|p dt = ⇒ Compact embedding and “nice” derivative

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Outline An Application Problem Abstract Setting Existence of Optimal Controls First-Order Optimality Conditions Improved Regularity Conclusion and Outlook

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Abstract Optimal Control Problem

Optimal Control Problem

min Js(y) + Jc(u) s.t. e(y, u) = 0 and u ∈ Uad      (OCP) Notation: J := (0, T)

Assumptions on the State Equation

U and Y are a Banach spaces and r ∈ [1, ∞]. For every u ∈ Lr(J; U), there exists a unique solution y ∈ Y so that e(y, u) = 0. The associated solution mapping S : Lr(J; U) ∋ u → y ∈ Y is continuous, but not

weakly continuous.

As far as the existence results are concerned, also other settings are possible, e.g. S : C(J; U) → Y

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Abstract Optimal Control Problem

Optimal Control Problem

min Js(y) + Jc(u) s.t. e(y, u) = 0 and u ∈ Uad      (OCP)

Assumptions on the Control Space

H is a Hilbert space and X is a reflexive Banach space with X ֒

→d H.

Jc(u) := β

2

• J

∂tu(t)2

H dt + γ

p

• J

u(t)p

Xdt

with β, γ > 0 and p > 1

There is a linear operator E, which is compact from X to U and bounded from

(X, H)η,1 to U with η ∈ (0, 1) and 1/r > (1 − η)/p − η/2 Notation: control space U = W1,2

p (X; H) := W 1,2(J; H) ∩ Lp(J; X)

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Overview over the Spaces In summary:

Control space U = W1,2

p (X; H) = W 1,2(J; H) ∩ Lp(J; X) with H Hilbert space,

X reflexive Banach space, and X ֒ →d H

Control-to-state map S : Lr(J; U) → Y continuous, but not weakly continuous E : X → U compact and E : (X, H)η,1 → U continuous

In the thermistor example: H = L2(ΓN), X = Lp(ΓN), U = W −1,q

ΓD

(Ω), E = tr∗ r = ∞, q > 3, p > 2 3 q > 2, η = 3/(2q) − 1/p 1/2 − 1/p

Existence of η follows from interpolation theory (Riesz-Thorin) E = tr∗ is compact from Lp(ΓN) to W −1,q

ΓD

(Ω), but even not continuous from L2(ΓN) to W −1,q

ΓD

(Ω)

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Outline An Application Problem Abstract Setting Existence of Optimal Controls First-Order Optimality Conditions Improved Regularity Conclusion and Outlook

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Crucial Compact Embedding Result Notation: W1,q

p (X1; X3) := {u ∈ Lq(J; X1) : ∂tu ∈ Lp(J; X3)}

Aubin-Lions Lemma

If X1, X2, X3 are three Banach spaces with X1 ֒ − ֒ → X2 ֒ → X3. Then the embedding of W1,q

p (X1; X3) with p, q ∈ [1, ∞] in Lp(J; X2), if p < ∞, and in C(J; X2), if p = ∞, is

compact.

Theorem

Let X1, X2, X3 be three Banach spaces with X1 ֒ →d X3, X1 ֒ − ֒ → X2, and (X1, X2)η,1 ֒ → X3 for some η ∈ (0, 1). Then W1,2

p (X1; X3) embeds compactly

in Lr(J; X2) with r < 2p/(2 − (p + 2)η), if 0 < η ≤ 2/(p + 2), and in C̺(J; X2) with ̺ < η/2 − (1 − η)/p, if 2/(p + 2) < η < 1.

No direct relation between X2 and X3, i.e., X2 need not be embedded in X3! Instead of an embedding, the theorem also holds with a linear operator E with the

respective properties (e.g. tr∗).

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Existence of Optimal Solutions Recall: min Js(y) + Jc(u) s.t. e(y, u) = 0 and u ∈ Uad

• (OCP)

with U = W1,2

p (X; H) and Jc(u) := β 2

• J ∂tu(t)2

H dt + γ p

• J u(t)p

Xdt

Theorem

Suppose in addition to the above assumptions that Js : Y → R is bounded from below and lower semi-continuous and that Uad is a nonempty, closed and convex subset of

• U. Then there exists at least one optimal solution of (OCP).

Proof: Apply the above compactness theorem with X1 = X, X2 = U, and X3 = H and standard arguments from the direct method of calculus of variations.

• Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Possible Settings

Notation: define E by (Eu)(t) := Eu(t) f.a.a. t ∈ J

Thermistor setting:

X = Lp(ΓN), H = L2(ΓN), E = tr∗ q > 3, p > 2 3 q      = ⇒ E : W1,2

p (X; H) ֒

− ֒ → L∞(J; U) with U = W −1,q

ΓD

(Ω)

Continuous controls:

X = W 1,q(Ω), H = L2(Ω), E = id q > d := dim(Ω), p > qd q − d      = ⇒ E : W1,2

p (X; H) ֒

− ֒ → C(J; U) with U = C(Ω)

Controls in Lebesgue spaces:

X = W 1,q(Ω), H = L2(Ω), E = id q ≤ d := dim(Ω)

• =

⇒ E : W1,2

p (X; H) ֒

− ֒ → Lr (J; U) with U = Ls(Ω) with s > 2 and r ≥ 1 depending on p and q

In all cases, the space U is no “intermediate” space between X and H, i.e., U ֒ → H Compactness in a large variety of control spaces (depending what is needed for the discussion of the state equation)

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Outline An Application Problem Abstract Setting Existence of Optimal Controls First-Order Optimality Conditions Improved Regularity Conclusion and Outlook

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Derivative of Jc

Question

What is the advantage of U = W1,2

p (X; H) with H being a Hilbert space?

Recall: Jc(u) := β 2

• J

∂tu(t)2

H dt + γ

p

• J

u(t)p

Xdt

Assumption

X is smooth, i.e., nX : X ∋ x → xX ∈ R is Gâteaux-differentiable on X \ {0}.

Lemma

Under the smoothness assumption on X, Jc is Gâteaux-differentiable on W1,s

p (X; H):

J′

c(u)h = β

• J

(∂tu(t), ∂th(t))Hdt + γ

• J

u(t)p−1

X

ϕX(u(t)), h(t)X∗,Xdt with the support mapping ϕX(u) := {x′ ∈ X ∗ : x′X∗ ≤ 1, x′, u = uX}.

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

First-Order Optimality Conditions Reduced optimal control problem: min

u∈Uad Js(S(Eu)) + Jc(u)

with E : W1,2

p (X; H) → Lr(J; U), (Eu)(t) := Eu(t)

Assumption

Js : Y → R is continuously differentiable and U satisfies the Radon-Nikodýn property.

Theorem

Let u ∈ W1,2

p (X; H) be locally optimal for (OCP). Then there exists an adjoint state

ψ ∈ Lr′(J; (H∗, X ∗)η,∞) such that ψ(t) = E∗ S′(Eu)∗J′

s(S(Eu))

• (t),

(AD)

• J

β(∂tu, ∂t(u − u))H + γ up−1

X

ϕX(u), u − uX∗,X + ψ, u − u(X,H)η,1 dt ≥ 0 ∀ u ∈ Uad (GRAD)

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Optimality System ψ(t) = E∗ S′(Eu)∗J′

s(S(Eu))

• (t),

(AD)

• J

β(∂tu, ∂t(u − u))H + γ up−1

X

ϕX(u), u − uX∗,X + ψ, u − u(X,H)η,1 dt ≥ 0 ∀ u ∈ Uad (GRAD)

Proof is straight forward based on the Radon-Nikodýn property of U, which yields

(Lr(J; U))∗ = Lr′(J; U∗) and the duality of real interpolation functors, which implies E∗ : U∗ → (X; H)∗

η,1 = (H∗, X ∗)η,∞.

Interpretation of the adjoint equation in (AD) as (weak form) of a linear parabolic

equation depends on precise form of S and Y and may allow to improve the regularity of p. Major advantage of the control space W1,2

p (X; H) = W 1,2(J; H) ∩ Lp(J; X):

Distributional time derivative ∂tu enters the gradient equation just linearly.

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

The Case without Control Constraints If Uad = W1,2

p (X; H), then

(GRAD) ⇐ ⇒

• J

β(∂tu, ∂th)H + γup−1

X

ϕX(u), hX∗,X + ψ, h(X,H)η,1 dt = 0 for all h ∈ W1,2

p (X; H)

Tailored formula of integration by parts in intersected spaces gives ∂ttu ∈ Lp′(J; X ∗) + Lr′(J′, (H∗, X ∗)η,∞) β ∂ttu(t) = γu(t)p−1

X

ϕX(u(t)) + ψ(t) in X ∗ Optimal control solves a Banach space valued ODE

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Outline An Application Problem Abstract Setting Existence of Optimal Controls First-Order Optimality Conditions Improved Regularity Conclusion and Outlook

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Method of Proof Recall gradient equation:

• J

β(∂tu, ∂t(u − u))H + γ up−1

X

ϕX(u), u − uX∗,X + ψ, u − u(X,H)η,1 dt ≥ 0 ∀ u ∈ Uad (GRAD)

Question

Does the optimal control admit a second weak time derivative in the case with control constraints? Similarly to higher regularity for the obstacle problem,

• cf. e.g. Stampacchia/Kinderlehrer:

Consider a regularized version of (GRAD) Prove that the regularized solutions admit a second weak time derivative, whose

norm is bounded uniformly in the regularization parameter

Pass to the limit in the regularized version of (GRAD) Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Regularization Define: s := min{p′, r ′}, g := γup−1

X

ϕX(u) + ψ ∈ Ls(J; X ∗) so that (GRAD) ⇐ ⇒ min

u∈Uad

• J

1 2 ∂tu2

H − g, uX′,Xdt

Regularization

Consider a fixed, but arbitrary local minimizer u of (OCP). min

u∈W1,2

p

(X;H)

• J

1 2 ∂tu2

H − g, uX′,X + 1

p

• u − up

X + 1

λu − PX(u)p

X

• dt

(R) with λ > 0 (regularization parameter) and PX(z) = arg min

ζ∈Uad

ζ − zX

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Sharpened Assumptions

Assumption

(i) Uad = {u ∈ W1,2

p (X; H): u(t) ∈ Uad for a.e. t ∈ J} with Uad ⊆ X closed convex

(ii) PH(z) := arg minζ∈Uad ζ − zH continuously maps X to X (iii) Growth condition: PH(u)X ≤ muX + b (iv) PH is directionally differentiable from H to H (restrictive!) (v) PH : X → X is Lipschitz continuous (very restrictive!)

Lemma (Projection Error)

For every λ > 0, there exists a unique solution uλ ∈ W1,2

p (X; H) of (R). Under

Assumptions (i)–(iii), there holds uλ − PX(uλ) → 0 in Lp(J; X), uλ − PH(uλ) → 0 in Lp(J; H), uλ − PH(uλ) ⇀ 0 in Lp(J; X) as λ ց 0.

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Convergence of the Regularization

Lemma (Generalized Stampacchia lemma)

Under Assumption (i)–(iv), PH maps W1,2

p (X; H) to itself with

• ∂tPH(u)
• (t) = P′

H(u(t); ∂tu(t))

f.a.a. t ∈ J. Proof: Immediate consequence of directional differentiability of PH and the Radon-Nikodýn property of the Hilbert space H

• Lemma

Under Assumption (i)–(iv), there holds uλ → u in W1,2

p (X; H).

Proof: test VI for u with PH(uλ) ∈ Uad = {u ∈ W1,2

p (X; H): u(t) ∈ Uad f.a.a. t ∈ J} and

variational equation for uλ with u − uλ

• Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Improved Regularity As in the case without control constraints, the variational equation for uλ can be rewritten as an ODE: ∂ttuλ = −g(t)+uλ −up−1

X

ϕX(uλ −u)+λ−1uλ − PX(uλ)p−1

X

ϕX(uλ −PX(uλ) in X ∗

Lemma

Under Assumption (i)–(v),

• λ−1uλ − PX(uλ)p−1

Lp(J;X)

• λ>0 is bounded.

Proof uses the general structure of directional derivatives of projection operators involving the second weak subderivative of the indicator functional of Uad.(1),(2) Only in this proof, the restrictive assumption on the Lipschitz continuity of PH in X is needed.

Theorem

Under Assumption (i)–(v), there holds ∂ttu ∈ Lp′(J; X ∗) and ∂tu(0) = ∂tu(T) = 0.

(1) Do, C. N., Generalized second-order derivatives of convex functions in reflexive Banach spaces. Trans.

• Amer. Math. Soc., 334(1):281–301, 1992

(2) Christof, C. and Wachsmuth, G., Differential sensitivity analysis of variational inequalities with locally Lipschitz continuous solution operators. Preprint, arXiv:1711.02720, 2017.

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

The Restrictive Lipschitz Assumption Assumption (v), i.e., PH : X → X is Lipschitz continuous, is very restrictive:

Example 1

Let Ω = (0, 1), X = H1

0(Ω), H = L2(Ω) and Uad = {v ∈ H1 0(Ω): v ≥ 0 a.e. in Ω}. Then

PH is not locally Lipschitz continuous from X to X.

Example 2, thermistor

Let X = Lp(ΓN), H = L2(ΓN) and Uad = {v ∈ Lp(ΓN): 0 ≤ v ≤ umax a.e. on ΓN}. Then, PH = PX so that the Assumption (v) is trivially fulfilled. Question: Are there other practically relevant (and non-trivial) settings fulfilling the Lipschitz assumption on PH?

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Conclusion and Outlook

Tailored control spaces lead to “handy” ODEs and variational inequalities (VIs)

as gradient equation for the control, even if the parabolic state equation requires fairly regular controls.

Improved regularity:

Under additional restrictive assumptions, the optimal control can be shown to be twice weakly time differentiable. To do:

Improved regularity allows to reformulate the VI corresponding to the gradient

equation as complementarity system (pointwise in time) involving multipliers in Bochner-Lebesgue-spaces

Use the reformulation to design a semi-smooth Newton method Superlinear convergence? Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018

Thank you for your attention!

Christian Meyer (TU Dortmund) · Optimal Control of Parabolic Equations in Tailored Control Spaces · lawoc 2018