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 TUD | F . Ludovici | 1

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 TUD | F . Ludovici | 2

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 TUD | F . Ludovici | 2

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 TUD | F . Ludovici | 2

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 ) TUD | F . Ludovici | 2

TUD | F . Ludovici | 3

For q , u furnace and glass temperature, B ( u , ν ) Planck function for black body radiation in glass, [Clever, Lang ’12] Augmented Objective Functional � T J ( u , q ) = 1 � u − u d � 2 + δ u �∇ u � 2 + δ q ( t )( q − q d ) � � dt 2 0 Equation Constraints � � � � ∂ t u − ∇ · ( κ c ∇ u ) = − B ( u , ν ) − I κ ν dsd ν S ν 0 � s · ∇I + ( κ ν + σ ν ) I = σ ν I ds + κ ν B ( u , ν ) S TUD | F . Ludovici | 4

Boundary and Initial Condition κ c n · ∇ u = h c ( q − u ) + F ( B ( u , ν )) I = r ( n · s ) ¯ I + (1 − n · s ) B ( q , ν ) u ( x , 0) = u 0 TUD | F . Ludovici | 5

Boundary and Initial Condition κ c n · ∇ u = h c ( q − u ) + F ( B ( u , ν )) I = r ( n · s ) ¯ I + (1 − n · s ) B ( q , ν ) u ( x , 0) = u 0 The list of possible applications is wider: ◮ Crystal Growth by sublimation ◮ Cancer treatment by local hypothermia ◮ Material failure TUD | F . Ludovici | 5

Model Problem For I = (0, T ) and Ω ⊂ R n , n = { 2, 3 } smooth domain 1 � � Ω ( u ( x , t ) − u d ( x , t )) 2 dxdt + α � I q ( t ) 2 dt min ( u , q ) ∈ U × Q ad 2 I 2 subject to ∂ t u ( t , x ) − ∆ u ( t , x ) + d ( t , x , u ) = q ( t ) g ( x ) in I × Ω , on I × ∂ Ω , u ( t , x ) = 0 u (0, x ) = u 0 in Ω , and control and state constraints q min ≤ q ( t ) ≤ q max F ( u ) ≤ b , ∀ t ∈ [0, T ], TUD | F . Ludovici | 6

Model Problem with state constraints � |∇ u ( x , t ) | 2 ω ( x ) dx ≤ b , ∀ t ∈ [0, T ], F 1 ( u ) = Ω � F 2 ( u ) = u ( x , t ) ω ( x ) dx ≤ b , ∀ t ∈ [0, T ], Ω the former for the the linear state equation, the latter for the semi-linear. TUD | F . Ludovici | 7

State Equation Control and State Space Q ad = { q ∈ L 2 ( I , R m ) | q min ≤ q ( t ) ≤ q max , a . e . in I } W (0, T ) = { u ∈ L 2 ( I , V ), ∂ t u ∈ L 2 ( I , V ∗ ) } U = { L 2 ( I , H 2 ( Ω ) ∩ H 1 0 ) ∩ L ∞ ( I × Ω ) ∩ H 1 ( I , L 2 ( Ω )) } TUD | F . Ludovici | 8

State Equation Control and State Space Q ad = { q ∈ L 2 ( I , R m ) | q min ≤ q ( t ) ≤ q max , a . e . in I } W (0, T ) = { u ∈ L 2 ( I , V ), ∂ t u ∈ L 2 ( I , V ∗ ) } U = { L 2 ( I , H 2 ( Ω ) ∩ H 1 0 ) ∩ L ∞ ( I × Ω ) ∩ H 1 ( I , L 2 ( Ω )) } Linear Case For q ∈ Q ad , there exists u ∈ U solution of the state equation. U ⊂ C (¯ F 1 : U → C (¯ I , V ) I ) TUD | F . Ludovici | 8

State Equation Control and State Space Q ad = { q ∈ L 2 ( I , R m ) | q min ≤ q ( t ) ≤ q max , a . e . in I } W (0, T ) = { u ∈ L 2 ( I , V ), ∂ t u ∈ L 2 ( I , V ∗ ) } U = { L 2 ( I , H 2 ( Ω ) ∩ H 1 0 ) ∩ L ∞ ( I × Ω ) ∩ H 1 ( I , L 2 ( Ω )) } Semi-Linear Case For q ∈ Q ad , there exists u ∈ W (0, T ) solution of the state equation. W (0, T ) ⊂ C (¯ F 2 : W (0, T ) → C (¯ I , H ) I ) TUD | F . Ludovici | 9

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 ′ (¯ ∃ q γ ∈ Q ad s . t . G (¯ q ) + G q )( q γ − ¯ q ) ≤ − γ < 0 for some γ ∈ R + , where ¯ q is a local solution in the sense of L 2 ( I , R ). TUD | F . Ludovici | 10

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 ′ (¯ ∃ q γ ∈ Q ad s . t . G (¯ q ) + G q )( q γ − ¯ q ) ≤ − γ < 0 for some γ ∈ R + , where ¯ q is a local solution in the sense of L 2 ( I , R ). The condition above ensures with standard argument the well-posedness of the optimal control problem as well as first order necessary conditions in KKT-form. TUD | F . Ludovici | 10

Optimality Conditions Issue with state constraints, the adjoint equation reads b ( ϕ , ¯ z ) + ( ϕ , ∂ u d ( · , · , ¯ u )¯ z ) = (¯ u − u d , ϕ ) I + � ¯ µ , F ( ϕ ) � C (¯ I ) ∗ , C (¯ I ) where µ ∈ C (¯ I ) ∗ . TUD | F . Ludovici | 11

Optimality Conditions Issue with state constraints, the adjoint equation reads b ( ϕ , ¯ z ) + ( ϕ , ∂ u d ( · , · , ¯ u )¯ z ) = (¯ u − u d , 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. TUD | F . Ludovici | 11

Optimality Conditions For the discussion of SSC, we introduce the Hamiltonian and Lagrangian m H ( q , u , z ) = 1 2( u − u d ) 2 + α � � 2 q 2 ( t ) + z � q i ( t ) g i ( x ) − d ( · , · , u ) , i =1 L ( q , µ ) = j ( q ) + � µ , F ( u ) � , and the cone of critical directions: p ∈ L 2 ( I , R ) such that  ≥ 0 if ¯ q i = q min ,  p i ( t ) = ≤ 0 if ¯ q i = q max , for all i = 1, ..., m Ω ∂ q ¯ � = 0 if H i dx � = 0,  ∂ F � ∂ F u ) v p ≤ 0 if F (¯ µ = 0 ∂ u (¯ u ) = 0, ∂ u (¯ u ) v p d ¯ Ω TUD | F . Ludovici | 12

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 ∂ 2 q p 2 > 0 ∀ p ∈ C ¯ q \ { 0 } , ∂ 2 q ¯ H i , i ≥ ξ ∀ t ∈ I \ E ν i , ∀ i = 1, ..., m , where � � � � ∂ q ¯ E ν i = t ∈ I � | H i dx | ≥ ν � Ω TUD | F . Ludovici | 13

Quadratic growth condition Under the weak SSC and first order necessary conditions, for constants δ , η > 0, there holds q � 2 j (¯ q ) + δ � q − ¯ L 2 ( I , R m ) ≤ j ( q ), � q − ¯ q � L 2 ( I , R ) ≤ η ◮ The proof moves by contradiction. ◮ Two-norm discrepancy removed thanks to q ) p 2 ≤ lim inf j ′′ (¯ k →∞ j ′′ ( q k ) p 2 k k →∞ � p k � 2 k →∞ j ′′ ( q k ) p 2 if p = 0 then Λ lim inf L 2 ( I , R m ) ≤ lim inf k TUD | F . Ludovici | 14

Discretization Time discretization via discontinuous Galerkin method: · Partitioning of ¯ I = [0, T ] in subintervals I n = ( t n − 1 , t n ] of size k n , with maximum size k . · For V = H 1 0 ( Ω ), semi-discrete state and test space: U k = { v k ∈ L 2 ( I , V ) | v k | I n ∈ P 0 ( I n , V ) } Reasons for dG(0): · Admits a variational formulation · Equations for each time intervals · Galerkin ortogonality TUD | F . Ludovici | 15

Discretization With w , ˆ w solutions to backward uncontrolled state equation, the dG (0)-method requires estimates [Meidner et al. ’11] � ( T − t ) � ∂ t w ( t ) � 2 � w − ˆ w � L 1 ( I , L 2 ( Ω )) , � w − ˆ w � H − 2 ( Ω ) , H − 1 ( Ω ) dt , I for the error at the nodal points t n and in the interior of I n . TUD | F . Ludovici | 16

Discretization With w , ˆ w solutions to backward uncontrolled state equation, the dG (0)-method requires estimates [Meidner et al. ’11] � ( T − t ) � ∂ t w ( t ) � 2 � w − ˆ w � L 1 ( I , L 2 ( Ω )) , � w − ˆ w � H − 2 ( Ω ) , H − 1 ( Ω ) dt , I for the error at the nodal points t n and in the interior of I n . To extend the estimates to the semilinear case, [Nochetto ’88] � d ( u ( t , x )) − d ( u k ( t , x )) if u ( t , x ) � = u k ( t , x ) ˜ u ( t , x ) − u k ( t , x ) d = 0 else. exploiting ˜ d bounded in L ∞ ( I × Ω ). TUD | F . Ludovici | 16