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 ∂ t θ − div( κ ( θ ) ∇ θ ) = ( σ ( θ ) ∇ ϕ ) · ∇ ϕ in Q := Ω × ( 0 , T ) ν · κ ( θ ) ∇ θ + αθ = αθ l on Σ := ∂ Ω × ( 0 , T ) θ ( 0 ) = θ 0 in Ω (T) − div( σ ( θ ) ∇ ϕ ) = 0 in Q ν · σ ( θ ) ∇ ϕ = u on Σ N := Γ N × ( 0 , T ) ϕ = 0 on Σ D := Γ D × ( 0 , 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 J s ( θ, ϕ ) + J c ( u ) (P T ) s.t. ( θ, ϕ, u ) satisfy (T) where J c is coercive on a suitable control space U Theorem Suppose that the control space U embeds compactly in L ∞ (( 0 , T ); W − 1 , q (Ω)) with Γ d q > 3 = dim(Ω) . Then, under suitable (mild) assumptions on the data, there exists an optimal solution of (P T ) . 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 (Ω)) requires at least a little bit time Γ d differentiability of the control � T � Naive choice: U = W 1 , p (( 0 , T ); L p (Γ N )) , J c ( u ) = � ∂ t u � p L p (Γ N ) + � u � p L p (Γ n ) dt 0 with p large enough so that the trace is compact in W − 1 , q (Ω) Γ d BUT: • Derivative of � · � p L p (Γ N ) is highly nonlinear ⇒ J c will lead to a rather complicated ODE as gradient equation for the control • Similar for any other exponent � ∂ t u � α L p (Γ N ) , α > 1 Idea: split time and space regularity Choose � T | ∂ t u | 2 dt + | u | p dt U = W 1 , 2 (( 0 , T ); L 2 (Γ N )) ∩ L p (( 0 , T ); L p (Γ N )) and J c ( u ) = 0 ⇒ 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 J s ( y ) + J c ( u ) s.t. e ( y , u ) = 0 (OCP) and u ∈ U ad Notation: J := ( 0 , T ) Assumptions on the State Equation � U and Y are a Banach spaces and r ∈ [ 1 , ∞ ] . � For every u ∈ L r ( J ; U ) , there exists a unique solution y ∈ Y so that e ( y , u ) = 0. � The associated solution mapping S : L r ( 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 J s ( y ) + J c ( u ) s.t. e ( y , u ) = 0 (OCP) and u ∈ U ad Assumptions on the Control Space → d H . � H is a Hilbert space and X is a reflexive Banach space with X ֒ � J c ( u ) := β � H dt + γ � � ∂ t u ( t ) � 2 � u ( t ) � p X dt with β, γ > 0 and p > 1 2 p J J � 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 = W 1 , 2 p ( X ; H ) := W 1 , 2 ( J ; H ) ∩ L p ( 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 = W 1 , 2 p ( X ; H ) = W 1 , 2 ( J ; H ) ∩ L p ( J ; X ) with H Hilbert space, → d H X reflexive Banach space, and X ֒ � Control-to-state map S : L r ( 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 = L 2 (Γ N ) , X = L p (Γ N ) , U = W − 1 , q E = tr ∗ (Ω) , Γ D p > 2 η = 3 / ( 2 q ) − 1 / p r = ∞ , q > 3 , 3 q > 2 , 1 / 2 − 1 / p � Existence of η follows from interpolation theory (Riesz-Thorin) � E = tr ∗ is compact from L p (Γ N ) to W − 1 , q (Ω) , but even not continuous from Γ D L 2 (Γ N ) to W − 1 , q (Ω) Γ D 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: W 1 , q p ( X 1 ; X 3 ) := { u ∈ L q ( J ; X 1 ) : ∂ t u ∈ L p ( J ; X 3 ) } Aubin-Lions Lemma If X 1 , X 2 , X 3 are three Banach spaces with X 1 ֒ − ֒ → X 2 ֒ → X 3 . Then the embedding of W 1 , q p ( X 1 ; X 3 ) with p , q ∈ [ 1 , ∞ ] in L p ( J ; X 2 ) , if p < ∞ , and in C ( J ; X 2 ) , if p = ∞ , is compact. Theorem → d X 3 , X 1 ֒ Let X 1 , X 2 , X 3 be three Banach spaces with X 1 ֒ − ֒ → X 2 , and → X 3 for some η ∈ ( 0 , 1 ) . Then W 1 , 2 ( X 1 , X 2 ) η, 1 ֒ p ( X 1 ; X 3 ) embeds compactly L r ( J ; X 2 ) in with r < 2 p / ( 2 − ( p + 2 ) η ) , if 0 < η ≤ 2 / ( p + 2 ) , C ̺ ( J ; X 2 ) and in with ̺ < η/ 2 − ( 1 − η ) / p , if 2 / ( p + 2 ) < η < 1 . � No direct relation between X 2 and X 3 , i.e., X 2 need not be embedded in X 3 ! � 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 J s ( y ) + J c ( u ) (OCP) s.t. e ( y , u ) = 0 and u ∈ U ad with U = W 1 , 2 p ( X ; H ) and J c ( u ) := β J � ∂ t u ( t ) � 2 H dt + γ J � u ( t ) � p � � X dt 2 p Theorem Suppose in addition to the above assumptions that J s : Y → R is bounded from below and lower semi-continuous and that U ad 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 X 1 = X , X 2 = U , and X 3 = 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 ( E u )( t ) := Eu ( t ) f.a.a. t ∈ J � Thermistor setting: X = L p (Γ N ) , H = L 2 (Γ N ) , E = tr ∗ E : W 1 , 2 → L ∞ ( J ; U ) with U = W − 1 , q = ⇒ p ( X ; H ) ֒ − (Ω) ֒ q > 3 , p > 2 Γ D 3 q � Continuous controls: X = W 1 , q (Ω) , H = L 2 (Ω) , E = id E : W 1 , 2 = ⇒ p ( X ; H ) ֒ − → C ( J ; U ) with U = C (Ω) qd ֒ q > d := dim(Ω) , p > q − d � Controls in Lebesgue spaces: X = W 1 , q (Ω) , H = L 2 (Ω) , E = id � → L r ( J ; U ) with U = L s (Ω) E : W 1 , 2 = ⇒ p ( X ; H ) ֒ − ֒ q ≤ d := dim(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
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