Optimization of nonlocal distributed feedback controllers with time - PowerPoint PPT Presentation

Optimization of nonlocal distributed feedback controllers with time delay F . Tröltzsch Technische Universität Berlin International Workshop ”From Open to Closed Loop Control” Mariatrost, June 22-26, 2015 F. Tröltzsch (TU Berlin) 1 / 35

Joint work with Peter Nestler and Eckehard Schöll F. Tröltzsch (TU Berlin) 2 / 35

1D Schlögl model (Nagumo equation) 1D Schlögl model ∂ t y − ∂ xx y + R ( y ) = u ( x , t ) ∈ Q :=( a , b ) × ( 0 , T ) . y ( x , 0 ) = y 0 ( x ) , x ∈ ( a , b ) ∂ x y ( a , t ) = ∂ x y ( b , t ) = 0 , t ∈ ( 0 , T ) . with control function (”forcing”) u = u ( x , t ) and cubic reaction term R ( y ) = ρ ( y − y 1 )( y − y 2 )( y − y 3 ) , ρ > 0 , y 1 ≤ y 2 ≤ y 3 . F. Tröltzsch (TU Berlin) 3 / 35

Optimal (open loop) control By optimal control, the state y is controlled in a desired way. One might be interested in approximating a desired state � y by an optimal control u : Optimal control problem �� u J ( y u ) := 1 y ( x , t )) 2 dxdt ( y u ( x , t ) − � min 2 Q where y u is the unique solution of ∂ t y − ∂ xx y + R ( y ) = u subject to given initial - and boundary conditions and certain constraints on u . This is a problem of open loop control that some theoretical physicists call ”optimal forcing”. F. Tröltzsch (TU Berlin) 4 / 35

Uncontrolled ”natural” wave front, u = 0 √ √ R = 1 3 y 3 − y = 1 3 ( y + 3 ) y ( y − 3 ) , ( a , b ) = ( 0 , L ) = ( 0 , 20 ) � √ 1 . 2 3 , x ∈ [ 9 , 11 ] y 0 ( x ) = 0 , else. Two propagating fronts F. Tröltzsch (TU Berlin) 5 / 35

Different visualization Initial state Uncontrolled wave fronts F. Tröltzsch (TU Berlin) 6 / 35

Time delayed feedback control In theoretical physics, in particular related to lasers, a nonlocal coupling of the control u with the state y by time-delayed feedback is popular. Two particular options: u ( x , t ) = κ ( y ( x , t − τ ) − y ( x , t )) ”Pyragas type” �� ∞ � u ( x , t ) = κ g ( τ ) y ( x , t − τ ) d τ − y ( x , t ) ”Nonlocal time-delayed”. 0 F. Tröltzsch (TU Berlin) 7 / 35

Time delayed feedback control In theoretical physics, in particular related to lasers, a nonlocal coupling of the control u with the state y by time-delayed feedback is popular. Two particular options: u ( x , t ) = κ ( y ( x , t − τ ) − y ( x , t )) ”Pyragas type” �� ∞ � u ( x , t ) = κ g ( τ ) y ( x , t − τ ) d τ − y ( x , t ) ”Nonlocal time-delayed”. 0 We concentrate on a finite interval of time and the nonlocal version : �� T � u ( x , t ) = κ g ( τ ) y ( x , t − τ ) d τ − y ( x , t ) . 0 F. Tröltzsch (TU Berlin) 7 / 35

Time delayed feedback control In theoretical physics, in particular related to lasers, a nonlocal coupling of the control u with the state y by time-delayed feedback is popular. Two particular options: u ( x , t ) = κ ( y ( x , t − τ ) − y ( x , t )) ”Pyragas type” �� ∞ � u ( x , t ) = κ g ( τ ) y ( x , t − τ ) d τ − y ( x , t ) ”Nonlocal time-delayed”. 0 We concentrate on a finite interval of time and the nonlocal version : �� T � u ( x , t ) = κ g ( τ ) y ( x , t − τ ) d τ − y ( x , t ) . 0 We will often suppress the dependence on x . Notice, however, that g = g ( t ) does not depend on the spatial variable. F. Tröltzsch (TU Berlin) 7 / 35

Feedback system Related feedback system �� T � ∂ t y ( x , t ) − ∂ xx y ( x , t ) + R ( y ( x , t )) = κ g ( τ ) y ( x , t − τ ) d τ − y ( x , t ) 0 y ( x , s ) = y 0 ( x , s ) , s ≤ 0 , x ∈ Ω , ∂ x y ( a , t ) = ∂ x y ( b , t ) = 0 , t ∈ ( 0 , T ) . F. Tröltzsch (TU Berlin) 8 / 35

Some references J. Löber, R. Coles, J. Siebert, H. Engel, E. Schöll, Control of chemical wave propagation in Engineering of Chemical Complexity II. A. S. Mikhailov, G. Ertl (Eds.), World Scientific, Singapore, 2014. J. Siebert, S. Alonso, M. Bär, E. Schöll, Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as limiting case of differential advection. Phys. Rev. E 89, 052909 (2014) . J. Siebert, E. Schöll, Front and Turing patterns induced by Mexican-hat-like nonlocal feedback. arXiv 1411.6561 (2014) . F. Tröltzsch (TU Berlin) 9 / 35

Forward problem: g �→ y Depending on the chosen feedback kernel g , different solutions y are generated. We numerically confirmed some results by Löber et al. (2014). F. Tröltzsch (TU Berlin) 10 / 35

Forward problem: g �→ y Depending on the chosen feedback kernel g , different solutions y are generated. We numerically confirmed some results by Löber et al. (2014). Ω = ( 0 , 200 ) , T = 400, y 0 : Natural wave front starting from a step function. ”Weak gamma delay kernel” g ( t ) = e − t , y 1 = 0 , y 3 = 1 F. Tröltzsch (TU Berlin) 10 / 35

Forward problem: g �→ y Depending on the chosen feedback kernel g , different solutions y are generated. We numerically confirmed some results by Löber et al. (2014). Ω = ( 0 , 200 ) , T = 400, y 0 : Natural wave front starting from a step function. ”Weak gamma delay kernel” g ( t ) = e − t , y 1 = 0 , y 3 = 1 y 2 = 0 . 25 , κ = − 1 . 65 y 2 = 0 . 5 , κ = − 1 . 4 F. Tröltzsch (TU Berlin) 10 / 35

Ω = ( 0 , 200 ) , T = 200, y 0 : Natural wave starting from a step function. ”Strong gamma delay kernel” g ( t ) = t e − t y 2 = 0 , κ = 2 y 2 = 0 , κ = − 2 F. Tröltzsch (TU Berlin) 11 / 35

� Design problem: y �→ g In the forward problem , we computed the function y associated with a given kernel g . In the design problem , this is reversed: Find a kernel g such that the solution y g associated with g is as close as possible to a given desired function � y . F. Tröltzsch (TU Berlin) 12 / 35

� Design problem: y �→ g In the forward problem , we computed the function y associated with a given kernel g . In the design problem , this is reversed: Find a kernel g such that the solution y g associated with g is as close as possible to a given desired function � y . Related optimal control problem: ”Optimize the Controller” �� 1 y ) 2 dxdt ( y g − � min 2 g ∈ C Q  � T    ∂ t y ( t ) − ∂ xx y ( t ) + R ( y ( t )) = κ g ( τ ) y ( t − τ ) d τ − κ y ( t )  0 y ( x , s ) = y 0 ( x , s ) , s ≤ 0 , x ∈ Ω     ∂ x y ( a , t ) = ∂ x y ( b , t ) = 0 , � T where C = { g ∈ L ∞ ( 0 , T ) : 0 ≤ g ( τ ) ≤ β, g ( τ ) d τ = 1 } . 0 F. Tröltzsch (TU Berlin) 12 / 35

� Design problem: y �→ g We must select realistic patterns � y . . . F. Tröltzsch (TU Berlin) 13 / 35

� Design problem: y �→ g We must select realistic patterns � y . . . Realistic Unrealistic F. Tröltzsch (TU Berlin) 13 / 35

Well-posedness of the problem Theorem (Control-to-state mapping) For each g ∈ L ∞ ( 0 , T ) and each y 0 ∈ C (¯ Ω × [ − T , 0 ]) , the feedback equation has a unique weak solution y g ∈ C ( Q ) . The mapping g �→ y g is of class C ∞ . Idea of the proof: � T ∂ t y + ∂ xx u + R ( y ) + κ y = κ g ( τ ) y ( x , t − τ ) d τ 0 � t � T = κ g ( τ ) y ( x , t − τ ) d τ + κ g ( τ ) y 0 ( x , t − τ ) d τ � �� � 0 t � �� � s Y g ( x , t ) � t = κ g ( t − s ) y ( x , s ) ds + Y g ( x , t ) 0 = K ( g ) y + Y g . F. Tröltzsch (TU Berlin) 14 / 35

Control-to-state mapping Next, we substitute y = e λ t v and get ∂ t v + ∂ xx v + e − λ t R ( e λ t v ) + ( λ + κ ) v − e − λ t K ( g )( e λ · v ) = e − λ t Y g . If λ is sufficiently large, the mapping in blue behaves like a monotone mapping. Now we proceed as in E. Casas, C. Ryll, F. Tröltzsch, Sparse optimal control of the Schlögl and FitzHugh-Nagumo systems , CMAM, 2013. We get a unique v g and the differentiability of the mapping g �→ v g . � F. Tröltzsch (TU Berlin) 15 / 35

Existence of an optimal kernel Corollary (Existence) The problem of optimal feedback design is solvable, i.e. there exists at least one optimal kernel ¯ g ∈ C. F. Tröltzsch (TU Berlin) 16 / 35

Existence of an optimal kernel Corollary (Existence) The problem of optimal feedback design is solvable, i.e. there exists at least one optimal kernel ¯ g ∈ C. We have associated necessary conditions. However, from now on we concentrate on a special choice of the kernel g . We optimize with respect to a particular class of step functions g . J. Löber, R. Coles, J. Siebert, H. Engel, E. Schöll, Control of chemical wave propagation in Engineering of Chemical Complexity II, A. S. Mikhailov, G. Ertl (Eds.), World Scientific, Singapore, 2014. F. Tröltzsch (TU Berlin) 16 / 35

Class of step functions Class of kernels g : Select 0 ≤ t 1 < t 2 ≤ T ;  1  , t 1 ≤ τ ≤ t 2 , t 2 − t 1 g ( τ ) =  0 , else. F. Tröltzsch (TU Berlin) 17 / 35

Class of step functions Class of kernels g : Select 0 ≤ t 1 < t 2 ≤ T ;  1  , t 1 ≤ τ ≤ t 2 , t 2 − t 1 g ( τ ) =  0 , else. Nonlocal feedback � t 2 κ ∂ t y ( x , t ) − ∂ xx y ( x , t ) + R ( y ( x , t )) = y ( x , t − τ ) d τ − κ y ( x , t ) t 2 − t 1 t 1 Here, κ, t 1 , t 2 are our control parameters to be optimized. F. Tröltzsch (TU Berlin) 17 / 35