Averaged control Enrique Zuazua 1 Ikerbasque & BCAM & CIMI - Toulouse enrique.zuazua@gmail.com enzuazua.net CIMI-Toulouse, April 2014 April 29, 2014 1 Funded by the ERC Advanced Grant NUMERIWAVES E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 1 / 50 jueves, 17 de julio de 14
Outline Motivation 1 Finite-dimensional systems 2 Averaged control Averaged observation Comparison with simultaneous controllability Limitations Averaged control of PDEs : The finite case 3 Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities Averaged control of PDEs: Continuous averages 4 Continuous averages of heat equations An abstract setting Continuous averages of wave equations Perspectives and open problems 5 E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 2 / 50 jueves, 17 de julio de 14
Motivation Motivation Often parameters of the control system under consideration are not fully known. It is then natural to look for robust control strategies, independent of the unknown parameters, and performing, overall, optimally. We introduce the notion of “averaged control” that consists, simply, on controlling the average of solutions with respect to the unknown parameters. As we shall see, this leads to interesting new problems in the frame of observability of parameter dependent systems. The problems are linked, but di ff erent, to those arising in the context of simultaneous observation and control. E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 3 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Averaged control Outline Motivation 1 Finite-dimensional systems 2 Averaged control Averaged observation Comparison with simultaneous controllability Limitations Averaged control of PDEs : The finite case 3 Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities Averaged control of PDEs: Continuous averages 4 Continuous averages of heat equations An abstract setting Continuous averages of wave equations Perspectives and open problems 5 E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 4 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Averaged control Consider the finite dimensional linear control system ⇢ x 0 ( t ) = A ( ν ) x ( t ) + Bu ( t ) , 0 < t < T , (1) x (0) = x 0 . In (1) the (column) vector valued function ∈ R N is the state of the system, A ( ν ) is � � x ( t , ν ) = x 1 ( t , ν ) , . . . , x N ( t , ν ) a N × N − matrix and u = u ( t ) is a M -component control vector in R M , M ≤ N . The matrix A is assumed to depend on a parameter ν in a continuous manner. To fix ideas we will assume that the parameter ν ranges within the interval (0 , 1). Note however that, to simplify the presentation, the control operator B has been taken to be independent of ν , the same as the initial datum x 0 ∈ R N to be controlled. But the same analysis applies when both of them depend on ν . E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 5 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Averaged control Given a control time T > 0 and a final target x 1 ∈ R N we look for a control u such that the solution of (1) satisfies Z 1 x ( T , ν ) d ν = x 1 . (2) 0 This concept of averaged controllability di ff ers from that of simultaneous controllability in which one is interested on controlling all states simultaneously and not only its average. When A is independent of the parameter ν , controllable systems can be fully characterized in algebraic terms by the rank condition h i B , AB , . . . , A N − 1 B = N . (3) rank E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 6 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Averaged control The following holds: Theorem Averaged controllability holds if and only the following rank condition is satisfied: Z 1 Z 1 h i [ A ( ν )] 2 d ν B , . . . [ A ( ν )] d ν B , = N . (4) rank B , 0 0 E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 7 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Averaged observation Outline Motivation 1 Finite-dimensional systems 2 Averaged control Averaged observation Comparison with simultaneous controllability Limitations Averaged control of PDEs : The finite case 3 Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities Averaged control of PDEs: Continuous averages 4 Continuous averages of heat equations An abstract setting Continuous averages of wave equations Perspectives and open problems 5 E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 8 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Averaged observation Let us now characterize the property of averaged control in terms of the observability of the adjoint system. The adjoint system depends also on the parameter ν : ⇢ − ϕ 0 ( t ) = A ⇤ ( ν ) ϕ ( t ) , t ∈ (0 , T ) (5) ϕ ( T ) = ϕ 0 . Note that, for all values of the parameter ν , we take the same datum for ϕ at t = T . This is so because our analysis is limited to the problem of averaged controllability. The corresponding averaged observability property reads: Z T Z 1 2 � � | ϕ 0 | 2 ≤ C � � � B ⇤ ϕ ( t , ν ) d ν (6) dt . � � � 0 0 E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 9 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Averaged observation The reason is the following duality identity: Z T Z Z Z x ( T , ν ) d ν , ϕ 0 > − < x 0 , ϕ (0 , ν ) d ν > = < u ( t ) , B ∗ ϕ ( t , ν ) d ν < 0 In fact, once the averaged observability inequality above is satisfied, the control of minimal L 2 (0 , T ) -norm can be built by minimizing the quadratic functional below within the class of solutions of the parameter-dependent adjoint system, i.e. minimizing 2 R T R 1 � � = 1 � ϕ 0 � 0 B ∗ ( ν ) ϕ ( t , ν ) d ν J dt � � (7) 2 0 � � R 1 − < x 1 , ϕ 0 > + < x 0 , 0 ϕ (0 , ν ) d ν > in R N . The functional is continuous and convex, and its coercivity is guaranteed by the averaged observability inequality. The control is then Z u ( t ) = B ∗ ϕ ( t , ν ) d ν , ˜ where ˜ ϕ is the solution of the adjoint system associated to the minimizer ϕ 0 of the functional J . ˜ E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 10 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Averaged observation Since we are working in the finite-dimensional context, the observability inequality (6) is equivalent to the following uniqueness property: Z 1 ∀ t ∈ [0 , T ] ⇒ ϕ 0 ≡ 0 . B ∗ ϕ ( t , ν ) d ν = 0 (8) 0 To analyze this inequality we use the following representation of the adjoint state: ϕ ( t , ν ) = exp[ A ∗ ( ν )( T − t )] ϕ 0 . Then, the fact that Z 1 B ∗ ϕ ( t , ν ) d ν = 0 ∀ t ∈ [0 , T ] 0 is equivalent to Z 1 exp[ A ∗ ( ν )( t − T )] d ν ϕ 0 = 0 B ∗ ∀ t ∈ [0 , T ] . 0 The result follows using the time analyticity of the matrix exponentials, and the classical argument consisting in taking consecutive derivatives at time t = T . E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 11 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Averaged observation Z 1 Z 1 [ A ∗ ( ν )] k d ν ϕ 0 = 0 , ∀ k ≥ 1 . exp[ A ∗ ( ν )( t − T )] d ν ϕ 0 ≡ 0 ∼ B ∗ B ∗ 0 0 Contrarily to the classical rank condition for the controllability of a given system, in the present context of averaged control, all moments, of an arbitrarily high order, need to be taken into account in the characterization. For instance, if the dependence of A ( ν ) with respect to ν is odd, we see that all the terms involving an odd power vanish, and only the even ones remain... E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 12 / 50 jueves, 17 de julio de 14
Finite-dimensional systems Comparison with simultaneous controllability Outline Motivation 1 Finite-dimensional systems 2 Averaged control Averaged observation Comparison with simultaneous controllability Limitations Averaged control of PDEs : The finite case 3 Finite averages of wave equations Additive perturbations of PDE Systems of heat equations Ingham like inequalities Averaged control of PDEs: Continuous averages 4 Continuous averages of heat equations An abstract setting Continuous averages of wave equations Perspectives and open problems 5 E. Zuazua (Ikerbasque-BCAM-CIMI) Averaged control-CIMI2014 April 29, 2014 13 / 50 jueves, 17 de julio de 14
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