Optimization of Time Delays in a Parabolic Delay Equation Fredi - PowerPoint PPT Presentation

Optimization of Time Delays in a Parabolic Delay Equation Fredi Tröltzsch Technische Universität Berlin New trends in PDE constrained optimization Linz, October 2019 Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 1 / 41

Joint work with Eduardo Casas (Santander, Spain) Martin Gugat (Erlangen, Germany) Mariano Mateos (Gijón, Spain) Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 2 / 41

Outline Introduction 1 Control-to-state mapping 2 Optimization problem 3 Numerical Discretization 4 Numerical examples 5 Nonlocal Pyragas type feedback 6 The problem of stability 7 Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 3 / 41

A linear ODE with time delay y ′ ( t ) = κ y ( t − 1 ) , t > 0 y ( t ) = y 0 ( t ) , − 1 ≤ t ≤ 0 . T. Erneux, Applied delay differential equations, Springer, 2009 κ = − 1 . 8 , y 0 ( 0 ) = 1 , y 0 ( t ) = 0 , t < 0 κ = − 1 . 1 κ = − π/ 2 Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 4 / 41

Nonlinear ODE with delay We consider nonlinear equations with cubic nonlinearity, e.g. y ′ ( t ) + y 3 ( t ) = κ y ( t − τ ) , t > 0 y ( t ) = y 0 ( t ) , − 1 ≤ t ≤ 0 . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 5 / 41

Nonlinear ODE with delay We consider nonlinear equations with cubic nonlinearity, e.g. y ′ ( t ) + y 3 ( t ) = κ y ( t − τ ) , t > 0 y ( t ) = y 0 ( t ) , − 1 ≤ t ≤ 0 . Find κ and τ generating a desired solution, say one with a desired oscillation. Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 5 / 41

Nonlinear ODE with delay We consider nonlinear equations with cubic nonlinearity, e.g. y ′ ( t ) + y 3 ( t ) = κ y ( t − τ ) , t > 0 y ( t ) = y 0 ( t ) , − 1 ≤ t ≤ 0 . Find κ and τ generating a desired solution, say one with a desired oscillation. We will control the weight κ and the time delay τ as real numbers. Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 5 / 41

Nonlinear ODE with delay We consider nonlinear equations with cubic nonlinearity, e.g. y ′ ( t ) + y 3 ( t ) = κ y ( t − τ ) , t > 0 y ( t ) = y 0 ( t ) , − 1 ≤ t ≤ 0 . Find κ and τ generating a desired solution, say one with a desired oscillation. We will control the weight κ and the time delay τ as real numbers. Instead of R ( y ) = y 3 , consider more general reaction terms like R ( y ) = ( y − y 1 )( y − y 2 )( y − y 3 ) and y ′ ( t ) + R ( y ( t )) = κ y ( t − τ ) . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 5 / 41

Nonlinear ODE with delay We consider nonlinear equations with cubic nonlinearity, e.g. y ′ ( t ) + y 3 ( t ) = κ y ( t − τ ) , t > 0 y ( t ) = y 0 ( t ) , − 1 ≤ t ≤ 0 . Find κ and τ generating a desired solution, say one with a desired oscillation. We will control the weight κ and the time delay τ as real numbers. Instead of R ( y ) = y 3 , consider more general reaction terms like R ( y ) = ( y − y 1 )( y − y 2 )( y − y 3 ) and y ′ ( t ) + R ( y ( t )) = κ y ( t − τ ) . Pyragas feedback control: y ′ ( t ) + R ( y ( t )) = κ ( y ( t − τ ) − y ( t )) . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 5 / 41

PDE case So far, we had y : [ 0 , T ] → R . Let y also depend on a spatial variable x ∈ Ω ⊂ R n , y = y ( x , t ) , and consider Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 6 / 41

PDE case So far, we had y : [ 0 , T ] → R . Let y also depend on a spatial variable x ∈ Ω ⊂ R n , y = y ( x , t ) , and consider ( ∂ t y − ∆ x y + R ( y ))( x , t ) = κ y ( x , t − τ ) in Ω × ( 0 , T ) y = y 0 , in Ω × [ − τ, 0 ] ∂ n y = 0 in ∂ Ω × ( 0 , T ) . Reaction term: R ( y ) = ρ ( y − y 1 )( y − y 2 )( y − y 3 ) , ρ > 0 , y 1 ≤ y 2 ≤ y 3 . Let R ′ ( y ) . m R := min y Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 6 / 41

Multiple time delays More general are multiple time delays 0 ≤ τ 1 < τ 2 . . . < τ m ≤ T , � m ( ∂ t y − ∆ y + R ( y ))( x , t ) = i = 1 κ i y ( x , t − τ i ) ( x , t ) ∈ Q = Ω × ( 0 , T ) y = y 0 in Q − = Ω × [ − T , 0 ] ∂ n y = 0 in Σ = ∂ Ω × ( 0 , T ) . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 7 / 41

Multiple time delays More general are multiple time delays 0 ≤ τ 1 < τ 2 . . . < τ m ≤ T , � m ( ∂ t y − ∆ y + R ( y ))( x , t ) = i = 1 κ i y ( x , t − τ i ) ( x , t ) ∈ Q = Ω × ( 0 , T ) y = y 0 in Q − = Ω × [ − T , 0 ] ∂ n y = 0 in Σ = ∂ Ω × ( 0 , T ) . This is feedback with control m � u ( x , t ) = κ i y ( x , t − τ i ) . i = 1 Application: Laser technology, research in treatment of Parkinson’s disease, ... Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 7 / 41

Multiple time delays More general are multiple time delays 0 ≤ τ 1 < τ 2 . . . < τ m ≤ T , � m ( ∂ t y − ∆ y + R ( y ))( x , t ) = i = 1 κ i y ( x , t − τ i ) ( x , t ) ∈ Q = Ω × ( 0 , T ) y = y 0 in Q − = Ω × [ − T , 0 ] ∂ n y = 0 in Σ = ∂ Ω × ( 0 , T ) . This is feedback with control m � u ( x , t ) = κ i y ( x , t − τ i ) . i = 1 Application: Laser technology, research in treatment of Parkinson’s disease, ... We will optimize the weights κ i and the delays τ i for fixed m . Set τ := ( τ 1 , . . . , τ m ) , κ := ( κ 1 , . . . , κ m ) , u := ( τ, κ ) . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 7 / 41

Outline Introduction 1 Control-to-state mapping 2 Optimization problem 3 Numerical Discretization 4 Numerical examples 5 Nonlocal Pyragas type feedback 6 The problem of stability 7 Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 8 / 41

Existence and uniqueness Hale and Ladeira (1991) proved existence and uniqueness of y , locally in time. Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 9 / 41

Existence and uniqueness Hale and Ladeira (1991) proved existence and uniqueness of y , locally in time. Joint with E. Casas and M. Mateos, we considered the more general Nonlocal problem with Borel measure µ ∈ M [ 0 , T ] � T ∂ t y ( x , t ) − ∆ y ( x , t ) + R ( y ( x , t )) = 0 y ( x , t − s ) d µ ( s ) ( x , t ) ∈ Q y = y 0 in Q − ∂ n y = 0 in Σ . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 9 / 41

Existence and uniqueness Hale and Ladeira (1991) proved existence and uniqueness of y , locally in time. Joint with E. Casas and M. Mateos, we considered the more general Nonlocal problem with Borel measure µ ∈ M [ 0 , T ] � T ∂ t y ( x , t ) − ∆ y ( x , t ) + R ( y ( x , t )) = 0 y ( x , t − s ) d µ ( s ) ( x , t ) ∈ Q y = y 0 in Q − ∂ n y = 0 in Σ . µ = � m Particular case of interest: i = 1 κ i δ τ i . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 9 / 41

Existence and uniqueness Hale and Ladeira (1991) proved existence and uniqueness of y , locally in time. Joint with E. Casas and M. Mateos, we considered the more general Nonlocal problem with Borel measure µ ∈ M [ 0 , T ] � T ∂ t y ( x , t ) − ∆ y ( x , t ) + R ( y ( x , t )) = 0 y ( x , t − s ) d µ ( s ) ( x , t ) ∈ Q y = y 0 in Q − ∂ n y = 0 in Σ . µ = � m Particular case of interest: i = 1 κ i δ τ i . Then � T m � y ( x , t − s ) d µ ( s ) = κ i y ( x , t − τ i ) . 0 i = 1 Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 9 / 41

Existence and uniqueness Hale and Ladeira (1991) proved existence and uniqueness of y , locally in time. Joint with E. Casas and M. Mateos, we considered the more general Nonlocal problem with Borel measure µ ∈ M [ 0 , T ] � T ∂ t y ( x , t ) − ∆ y ( x , t ) + R ( y ( x , t )) = 0 y ( x , t − s ) d µ ( s ) ( x , t ) ∈ Q y = y 0 in Q − ∂ n y = 0 in Σ . µ = � m Particular case of interest: i = 1 κ i δ τ i . Then � T m � y ( x , t − s ) d µ ( s ) = κ i y ( x , t − τ i ) . 0 i = 1 Assume y 0 ∈ C ( Q − ) . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 9 / 41

The nonlocal problem with measures Theorem (Casas, Mateos, Tr. 2017) For all T > 0 and every µ ∈ M [ 0 , T ] , the nonlocal problem has a unique solution y µ ∈ Y = W ( 0 , T ) ∩ C ( ¯ Q ) . We have � � � y µ � L 2 ( 0 , T ; H 1 (Ω)) ≤ C � y 0 � L 2 ( Q − ) � µ � M [ 0 , T ] + � y 0 ( · , 0 ) � L 2 (Ω) + | R ( 0 ) | � � � y µ � C (¯ Q ) ≤ C � y 0 � C (¯ Q − ) � µ � M [ 0 , T ] + � y 0 ( · , 0 ) � C (¯ Ω) + | R ( 0 ) | , where C depends on � µ � M [ 0 , T ] , but can be taken fixed on bounded subsets of M [ 0 , T ] . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 10 / 41

The nonlocal problem with measures Theorem (Casas, Mateos, Tr. 2017) For all T > 0 and every µ ∈ M [ 0 , T ] , the nonlocal problem has a unique solution y µ ∈ Y = W ( 0 , T ) ∩ C ( ¯ Q ) . We have � � � y µ � L 2 ( 0 , T ; H 1 (Ω)) ≤ C � y 0 � L 2 ( Q − ) � µ � M [ 0 , T ] + � y 0 ( · , 0 ) � L 2 (Ω) + | R ( 0 ) | � � � y µ � C (¯ Q ) ≤ C � y 0 � C (¯ Q − ) � µ � M [ 0 , T ] + � y 0 ( · , 0 ) � C (¯ Ω) + | R ( 0 ) | , where C depends on � µ � M [ 0 , T ] , but can be taken fixed on bounded subsets of M [ 0 , T ] . = ⇒ Theorem (Casas, Mateos, Tr. 2018) To each weight κ ∈ R m and delay τ ∈ R m , there exists a unique solution y τ,κ ∈ Y . The mapping ( τ, κ ) �→ y τ,κ is continuous from R 2 m to Y . Fredi Tröltzsch (TU Berlin) Time delays 18.10.2019 10 / 41