• Previous Article
    Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix
  • MCRF Home
  • This Issue
  • Next Article
    A convergent hierarchy of non-linear eigenproblems to compute the joint spectral radius of nonnegative matrices
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  September 2020 Early access  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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[21]

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

[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.

[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.

[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. 

[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.

[26]

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

[27]

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

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[18]

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

[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.

[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.

[21]

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

[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.

[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.

[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. 

[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.

[26]

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

[27]

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

[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.

Figure 4.  In this figure we see one possible shape for $ \overline{u}_i $
[1]

Hongwei Lou, Junjie Wen, Yashan Xu. Time optimal control problems for some non-smooth systems. Mathematical Control and Related Fields, 2014, 4 (3) : 289-314. doi: 10.3934/mcrf.2014.4.289

[2]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control and Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011

[3]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control and Related Fields, 2021, 11 (3) : 521-554. doi: 10.3934/mcrf.2020052

[4]

Mikhail I. Belishev, Aleksei F. Vakulenko. Non-smooth unobservable states in control problem for the wave equation in $\mathbb{R}^3$. Evolution Equations and Control Theory, 2014, 3 (2) : 247-256. doi: 10.3934/eect.2014.3.247

[5]

Ying Zhang, Changjun Yu, Yingtao Xu, Yanqin Bai. Minimizing almost smooth control variation in nonlinear optimal control problems. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1663-1683. doi: 10.3934/jimo.2019023

[6]

Giuseppe Tomassetti. Smooth and non-smooth regularizations of the nonlinear diffusion equation. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1519-1537. doi: 10.3934/dcdss.2017078

[7]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 903-920. doi: 10.3934/dcdsb.2021073

[8]

Paul Glendinning. Non-smooth pitchfork bifurcations. Discrete and Continuous Dynamical Systems - B, 2004, 4 (2) : 457-464. doi: 10.3934/dcdsb.2004.4.457

[9]

Deepak Singh, Bilal Ahmad Dar, Do Sang Kim. Sufficiency and duality in non-smooth interval valued programming problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 647-665. doi: 10.3934/jimo.2018063

[10]

R.M. Brown, L.D. Gauthier. Inverse boundary value problems for polyharmonic operators with non-smooth coefficients. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022006

[11]

Matthias Gerdts, Martin Kunkel. Convergence analysis of Euler discretization of control-state constrained optimal control problems with controls of bounded variation. Journal of Industrial and Management Optimization, 2014, 10 (1) : 311-336. doi: 10.3934/jimo.2014.10.311

[12]

Nurullah Yilmaz, Ahmet Sahiner. Generalization of hyperbolic smoothing approach for non-smooth and non-Lipschitz functions. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021170

[13]

Luis Bayón, Jose Maria Grau, Maria del Mar Ruiz, Pedro Maria Suárez. A hydrothermal problem with non-smooth Lagrangian. Journal of Industrial and Management Optimization, 2014, 10 (3) : 761-776. doi: 10.3934/jimo.2014.10.761

[14]

Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure and Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161

[15]

Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064

[16]

Xiaoshan Chen, Xun Li, Fahuai Yi. Optimal stopping investment with non-smooth utility over an infinite time horizon. Journal of Industrial and Management Optimization, 2019, 15 (1) : 81-96. doi: 10.3934/jimo.2018033

[17]

Roberto Triggiani. Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space. Evolution Equations and Control Theory, 2016, 5 (4) : 489-514. doi: 10.3934/eect.2016016

[18]

Laetitia Paoli. A proximal-like algorithm for vibro-impact problems with a non-smooth set of constraints. Conference Publications, 2011, 2011 (Special) : 1186-1195. doi: 10.3934/proc.2011.2011.1186

[19]

Chao Zhang, Lihe Wang, Shulin Zhou, Yun-Ho Kim. Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2559-2587. doi: 10.3934/cpaa.2014.13.2559

[20]

Oleksiy V. Kapustyan, Pavlo O. Kasyanov, José Valero. Structure and regularity of the global attractor of a reaction-diffusion equation with non-smooth nonlinear term. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4155-4182. doi: 10.3934/dcds.2014.34.4155

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (290)
  • HTML views (425)
  • Cited by (1)

Other articles
by authors

[Back to Top]