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September  2020, 10(3): 591-622. doi: 10.3934/mcrf.2020012

Optimal control of the linear wave equation by time-depending BV-controls: A semi-smooth Newton approach

1. 

Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität, Heinrichstr. 36, Graz, 8010, Austria

2. 

Radon Institute, Austrian Academy of Sciences, and Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität, Heinrichstr. 36, Graz, 8010, Austria

* Corresponding author: Sebastian Engel

Received  October 2018 Revised  September 2019 Published  December 2019

An optimal control problem for the linear wave equation with control cost chosen as the BV semi-norm in time is analyzed. This formulation enhances piecewise constant optimal controls and penalizes the number of jumps. Existence of optimal solutions and necessary optimality conditions are derived. With numerical realisation in mind, the regularization by $ H^1 $ functionals is investigated, and the asymptotic behavior as this regularization tends to zero is analyzed. For the $ H^1- $regularized problems the semi-smooth Newton algorithm can be used to solve the first order optimality conditions with super-linear convergence rate. Examples are constructed which show that the distributional derivative of an optimal control can be a mix of absolutely continuous measures with respect to the Lebesgue measure, a countable linear combination of Dirac measures, and Cantor measures. Numerical results illustrate and support the analytical results.

Citation: Sebastian Engel, Karl Kunisch. Optimal control of the linear wave equation by time-depending BV-controls: A semi-smooth Newton approach. Mathematical Control & Related Fields, 2020, 10 (3) : 591-622. doi: 10.3934/mcrf.2020012
References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.  Google Scholar

[2]

E. Casas, A review on sparse solutions in optimal control of partial differential equations, SeMA Journal, 74 (2017), 319-344.  doi: 10.1007/s40324-017-0121-5.  Google Scholar

[3]

E. Casas and K. Kunisch, Optimal control of the two-dimensional stationary Navier-Stokes equations with measure valued controls, SIAM Journal on Control and Optimization, 57 (2019), 1328-1354.  doi: 10.1137/18M1185582.  Google Scholar

[4]

E. Casas and K. Kunisch, Parabolic control problems in space-time measure space, ESAIM Control Optim. Calc. Var., 22 (2016), 355-370.  doi: 10.1051/cocv/2015008.  Google Scholar

[5]

E. Casas and E. Zuazua, Spike controls for elliptic and parabolic PDEs, Systems Control Lett., 62 (2013), 311-318.  doi: 10.1016/j.sysconle.2013.01.001.  Google Scholar

[6]

E. CasasC. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM J. Control Optimization, 50 (2012), 1735-1752.  doi: 10.1137/110843216.  Google Scholar

[7]

E. CasasC. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM Journal Control Optimization, 51 (2013), 28-63.  doi: 10.1137/120872395.  Google Scholar

[8]

E. CasasB. Vexler and E. Zuazua, Sparse initial data identification for parabolic PDE and its finite element approximations, Math. Control Relat. Fields, 5 (2015), 377-399.  doi: 10.3934/mcrf.2015.5.377.  Google Scholar

[9]

E. CasasF. Kruse and K. Kunisch, Optimal control of semilinear parabolic equations by BV-functions, SIAM Journal on Control and Optimization, 55 (2017), 1752-1788.  doi: 10.1137/16M1056511.  Google Scholar

[10]

A. ChambolleV. CasellesM. NovagaD. Cremers and T. Pock, An introduction to total variation for image analysis, Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Ser. Comput. Appl. Math., Walter de Gruyter, Berlin, 9 (2009), 263-340.  doi: 10.1515/9783110226157.263.  Google Scholar

[11]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESIAM Control Optim. Calc. Var., 17 (2011), 243-266.  doi: 10.1051/cocv/2010003.  Google Scholar

[12]

C. Clason and K. Kunisch, A measure space approach to optimal source placement, Comput. Optim. Appl., 53 (2012), 155-171.  doi: 10.1007/s10589-011-9444-9.  Google Scholar

[13]

S. Engel, Optimal Control and Bayesian Inversion for Linear Second-Order Hyperbolic Equations by BV-Functions in Time, PhD thesis, Karl-Franzens-Universität Graz, 2018. Google Scholar

[14]

M. GugatA. Keimer and G. Leugering, Optimal distributed control of the wave equation subject to state constraints, Z. angew. Math. Mech., 89 (2009), 420-444.  doi: 10.1002/zamm.200800196.  Google Scholar

[15]

D. Hafemeyer, Optimal Control of Differential Equations Using BV-Functions, thesis, Technical University of Munich, 2015. Google Scholar

[16]

R. HerzogG. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations, SIAM J. Control Optim., 50 (2012), 943-963.  doi: 10.1137/100815037.  Google Scholar

[17]

M. HintermüllerK. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Control Optim., 13 (2002), 865-888.  doi: 10.1137/S1052623401383558.  Google Scholar

[18]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23. Springer, New York, 2009.  Google Scholar

[19]

K. KunischK. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.  doi: 10.1137/140959055.  Google Scholar

[20]

K. KunischP. Trautmann and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., 54 (2016), 1212-1244.  doi: 10.1137/141001366.  Google Scholar

[21]

O. A. Ladyzhenskaya, Boundary value problems of mathematical physics, Nauka, Moscow, (1973), 407 pp.  Google Scholar

[22]

I. Lasiecka, Control theory for partial differential equations volume 2: Abstract hyperbolic-like systems over a finite time horizon, Cambridge Univ. Press, (2000), 645–1067. Google Scholar

[23]

I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions. Ⅱ. General boundary data, Journal of Differential Equations, 94 (1991), 112-164.  doi: 10.1016/0022-0396(91)90106-J.  Google Scholar

[24]

I. LasieckaJ.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, Journal de Mathématiques Pures et Appliquées, 65 (1986), 149-192.   Google Scholar

[25]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971.  Google Scholar

[26]

K. Pieper, Finite Element Discretization and Efficient Numerical Solution of Elliptic and Parabolic Sparse Control Problems, PhD thesis, Technical University Munich, 2015. Google Scholar

[27]

M. Reed and S. Barry, Functional Analysis. I, Academic Press, Inc., 1980. Google Scholar

[28]

A. A. Zlotnik, Convergence Rate Estimates of Finite-Element Methods for Second-Order Hyperbolic Equations. Numerical Methods and Applications, Guri I. Marchuk, CRC Press, 1994.  Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000.  Google Scholar

[2]

E. Casas, A review on sparse solutions in optimal control of partial differential equations, SeMA Journal, 74 (2017), 319-344.  doi: 10.1007/s40324-017-0121-5.  Google Scholar

[3]

E. Casas and K. Kunisch, Optimal control of the two-dimensional stationary Navier-Stokes equations with measure valued controls, SIAM Journal on Control and Optimization, 57 (2019), 1328-1354.  doi: 10.1137/18M1185582.  Google Scholar

[4]

E. Casas and K. Kunisch, Parabolic control problems in space-time measure space, ESAIM Control Optim. Calc. Var., 22 (2016), 355-370.  doi: 10.1051/cocv/2015008.  Google Scholar

[5]

E. Casas and E. Zuazua, Spike controls for elliptic and parabolic PDEs, Systems Control Lett., 62 (2013), 311-318.  doi: 10.1016/j.sysconle.2013.01.001.  Google Scholar

[6]

E. CasasC. Clason and K. Kunisch, Approximation of elliptic control problems in measure spaces with sparse solutions, SIAM J. Control Optimization, 50 (2012), 1735-1752.  doi: 10.1137/110843216.  Google Scholar

[7]

E. CasasC. Clason and K. Kunisch, Parabolic control problems in measure spaces with sparse solutions, SIAM Journal Control Optimization, 51 (2013), 28-63.  doi: 10.1137/120872395.  Google Scholar

[8]

E. CasasB. Vexler and E. Zuazua, Sparse initial data identification for parabolic PDE and its finite element approximations, Math. Control Relat. Fields, 5 (2015), 377-399.  doi: 10.3934/mcrf.2015.5.377.  Google Scholar

[9]

E. CasasF. Kruse and K. Kunisch, Optimal control of semilinear parabolic equations by BV-functions, SIAM Journal on Control and Optimization, 55 (2017), 1752-1788.  doi: 10.1137/16M1056511.  Google Scholar

[10]

A. ChambolleV. CasellesM. NovagaD. Cremers and T. Pock, An introduction to total variation for image analysis, Theoretical Foundations and Numerical Methods for Sparse Recovery, Radon Ser. Comput. Appl. Math., Walter de Gruyter, Berlin, 9 (2009), 263-340.  doi: 10.1515/9783110226157.263.  Google Scholar

[11]

C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces, ESIAM Control Optim. Calc. Var., 17 (2011), 243-266.  doi: 10.1051/cocv/2010003.  Google Scholar

[12]

C. Clason and K. Kunisch, A measure space approach to optimal source placement, Comput. Optim. Appl., 53 (2012), 155-171.  doi: 10.1007/s10589-011-9444-9.  Google Scholar

[13]

S. Engel, Optimal Control and Bayesian Inversion for Linear Second-Order Hyperbolic Equations by BV-Functions in Time, PhD thesis, Karl-Franzens-Universität Graz, 2018. Google Scholar

[14]

M. GugatA. Keimer and G. Leugering, Optimal distributed control of the wave equation subject to state constraints, Z. angew. Math. Mech., 89 (2009), 420-444.  doi: 10.1002/zamm.200800196.  Google Scholar

[15]

D. Hafemeyer, Optimal Control of Differential Equations Using BV-Functions, thesis, Technical University of Munich, 2015. Google Scholar

[16]

R. HerzogG. Stadler and G. Wachsmuth, Directional sparsity in optimal control of partial differential equations, SIAM J. Control Optim., 50 (2012), 943-963.  doi: 10.1137/100815037.  Google Scholar

[17]

M. HintermüllerK. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Control Optim., 13 (2002), 865-888.  doi: 10.1137/S1052623401383558.  Google Scholar

[18]

M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23. Springer, New York, 2009.  Google Scholar

[19]

K. KunischK. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.  doi: 10.1137/140959055.  Google Scholar

[20]

K. KunischP. Trautmann and B. Vexler, Optimal control of the undamped linear wave equation with measure valued controls, SIAM J. Control Optim., 54 (2016), 1212-1244.  doi: 10.1137/141001366.  Google Scholar

[21]

O. A. Ladyzhenskaya, Boundary value problems of mathematical physics, Nauka, Moscow, (1973), 407 pp.  Google Scholar

[22]

I. Lasiecka, Control theory for partial differential equations volume 2: Abstract hyperbolic-like systems over a finite time horizon, Cambridge Univ. Press, (2000), 645–1067. Google Scholar

[23]

I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions. Ⅱ. General boundary data, Journal of Differential Equations, 94 (1991), 112-164.  doi: 10.1016/0022-0396(91)90106-J.  Google Scholar

[24]

I. LasieckaJ.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, Journal de Mathématiques Pures et Appliquées, 65 (1986), 149-192.   Google Scholar

[25]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971.  Google Scholar

[26]

K. Pieper, Finite Element Discretization and Efficient Numerical Solution of Elliptic and Parabolic Sparse Control Problems, PhD thesis, Technical University Munich, 2015. Google Scholar

[27]

M. Reed and S. Barry, Functional Analysis. I, Academic Press, Inc., 1980. Google Scholar

[28]

A. A. Zlotnik, Convergence Rate Estimates of Finite-Element Methods for Second-Order Hyperbolic Equations. Numerical Methods and Applications, Guri I. Marchuk, CRC Press, 1994.  Google Scholar

Figure 4.  In this figure we see one possible shape for $ \overline{u}_i $
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