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September  2015, 5(3): 377-399. doi: 10.3934/mcrf.2015.5.377

Sparse initial data identification for parabolic PDE and its finite element approximations

Eduardo Casas 1, , Boris Vexler 2, and  Enrique Zuazua 3,

1. 

Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain
2. 

Centre for Mathematical Sciences, Technische Universität München, Bolzmannstrasse 3, D-85747 Garching b. München, Germany
3. 

BCAM - Basque Center for Applied Mathematics, Mazarredo, 14, E-48009 Bilbao-Basque Country

Received  August 2014 Revised  November 2014 Published  July 2015

We address the problem of inverse source identification for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a finite number of pointwise locations. Moreover, we discuss the finite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.
Keywords: approximate controllability, Parabolic equations, Borel measures., sparse controls.
Mathematics Subject Classification: 35K15, 49K20, 93B05, 93C2.
Citation: Eduardo Casas, Boris Vexler, Enrique Zuazua. Sparse initial data identification for parabolic PDE and its finite element approximations. Mathematical Control and Related Fields, 2015, 5 (3) : 377-399. doi: 10.3934/mcrf.2015.5.377
References:
[1]

E. Di Benedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 487-535.

[2]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, Berlin, Heidelberg, 2008, Third edition. doi: 10.1007/978-0-387-75934-0.

[3]

E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327. doi: 10.1137/S0363012995283637.

[4]

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

[5]

________, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28-63. doi: 10.1137/120872395.

[6]

E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820. doi: 10.1137/110834366.

[7]

E. Casas and K. Kunisch, Optimal control of semilinear elliptic equations in measure spaces, SIAM J. Control Optim., 52 (2014), 339-364. doi: 10.1137/13092188X.

[8]

________, Parabolic control problems in space-time measure spaces, To appear in ESAIM Control Optim. Calc. Var.

[9]

E. Casas and F. Tröltzsch, Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls, SIAM J. Control Optim., 52 (2014), 1010-1033. doi: 10.1137/130917314.

[10]

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

[11]

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

[12]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland-Elsevier, New York, 1976.

[13]

C. Fabre, J. P. Puel and E. Zuazua, On the density of the range of the semigroup for semilinear heat equations, Control and optimal design of distributed parameter systems (Minneapolis, MN, 1992), IMA Vol. Math. Appl., Springer, New York, 70 (1995), 73-91. doi: 10.1007/978-1-4613-8460-1_4.

[14]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Advances in Differential Equations, 5 (2000), 465-514.

[15]

W. Gong, M. Hinze and Z. Zhou, A Priori Error Analysis for Finite Element Approximation of Parabolic Optimal Control Problems with Pointwise Control, SIAM J. Control Optim., 52 (2014), 97-119. doi: 10.1137/110840133.

[16]

J. A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces, Adv. Differential Equations, 12 (2007), 1031-1078.

[17]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London-Melbourne, 1985.

[18]

A. Hansbo, Strong stability and non-smooth data error estimates for discretizations of linear parabolic problems, BIT, 42 (2002), 351-379. doi: 10.1023/A:1021903109720.

[19]

R. Herzog, G. 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.

[20]

J. Kovats, Real analytic solutions of parabolic equations with time-measurable coefficients, Proc. Amer. Math. Soc., 130 (2002), 1055-1064. doi: 10.1090/S0002-9939-01-06163-9.

[21]

K. Kunisch, K. 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.

[22]

D. Leykekhman and B. Vexler, Optimal a priori error estimates of parabolic optimal control problems with pointwise control, SIAM J. Numer. Anal., 51 (2013), 2797-2821. doi: 10.1137/120885772.

[23]

Y. Li, S. Osher and R. Tsai, Heat source identification based on $l_1$ constrained minimization, Inverse Probl. and Imaging, 8 (2014), 199-221. doi: 10.3934/ipi.2014.8.199.

[24]

J.-L. Lions and B. Malgrange, Sur l'unicité rétrograde des équations paraboliques, Math. Scand., 8 (1960), 277-286.

[25]

D. Meidner, R. Rannacher and B. Vexler, A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time, SIAM J. Control Optim., 49 (2011), 1961-1997. doi: 10.1137/100793888.

[26]

K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM J. Control Optim., 51 (2013), 2788-2808. doi: 10.1137/120889137.

[27]

J. P. Raymond and H. Zidani, Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations, App. Math. Optim., 39 (1999), 143-177. doi: 10.1007/s002459900102.

[28]

W. Rudin, Real and Complex Analysis, McGraw-Hill, London, 1970.

[29]

A. H. Schatz, V. C. Thomée and L. B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl. Math., 33 (1980), 265-304. doi: 10.1002/cpa.3160330305.

[30]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.

[31]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Second edition, Spinger-Verlag, Berlin, 2006.

[32]

G. Wachsmuth and D. Wachsmuth, Convergence and regularisation results for optimal control problems with sparsity functional, ESAIM Control Optim. Calc. Var., 17 (2011), 858-886. doi: 10.1051/cocv/2010027.

show all references

References:
[1]

E. Di Benedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 487-535.

[2]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, Berlin, Heidelberg, 2008, Third edition. doi: 10.1007/978-0-387-75934-0.

[3]

E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327. doi: 10.1137/S0363012995283637.

[4]

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

[5]

________, Parabolic control problems in measure spaces with sparse solutions, SIAM J. Control Optim., 51 (2013), 28-63. doi: 10.1137/120872395.

[6]

E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820. doi: 10.1137/110834366.

[7]

E. Casas and K. Kunisch, Optimal control of semilinear elliptic equations in measure spaces, SIAM J. Control Optim., 52 (2014), 339-364. doi: 10.1137/13092188X.

[8]

________, Parabolic control problems in space-time measure spaces, To appear in ESAIM Control Optim. Calc. Var.

[9]

E. Casas and F. Tröltzsch, Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls, SIAM J. Control Optim., 52 (2014), 1010-1033. doi: 10.1137/130917314.

[10]

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

[11]

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

[12]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland-Elsevier, New York, 1976.

[13]

C. Fabre, J. P. Puel and E. Zuazua, On the density of the range of the semigroup for semilinear heat equations, Control and optimal design of distributed parameter systems (Minneapolis, MN, 1992), IMA Vol. Math. Appl., Springer, New York, 70 (1995), 73-91. doi: 10.1007/978-1-4613-8460-1_4.

[14]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Advances in Differential Equations, 5 (2000), 465-514.

[15]

W. Gong, M. Hinze and Z. Zhou, A Priori Error Analysis for Finite Element Approximation of Parabolic Optimal Control Problems with Pointwise Control, SIAM J. Control Optim., 52 (2014), 97-119. doi: 10.1137/110840133.

[16]

J. A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces, Adv. Differential Equations, 12 (2007), 1031-1078.

[17]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London-Melbourne, 1985.

[18]

A. Hansbo, Strong stability and non-smooth data error estimates for discretizations of linear parabolic problems, BIT, 42 (2002), 351-379. doi: 10.1023/A:1021903109720.

[19]

R. Herzog, G. 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.

[20]

J. Kovats, Real analytic solutions of parabolic equations with time-measurable coefficients, Proc. Amer. Math. Soc., 130 (2002), 1055-1064. doi: 10.1090/S0002-9939-01-06163-9.

[21]

K. Kunisch, K. 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.

[22]

D. Leykekhman and B. Vexler, Optimal a priori error estimates of parabolic optimal control problems with pointwise control, SIAM J. Numer. Anal., 51 (2013), 2797-2821. doi: 10.1137/120885772.

[23]

Y. Li, S. Osher and R. Tsai, Heat source identification based on $l_1$ constrained minimization, Inverse Probl. and Imaging, 8 (2014), 199-221. doi: 10.3934/ipi.2014.8.199.

[24]

J.-L. Lions and B. Malgrange, Sur l'unicité rétrograde des équations paraboliques, Math. Scand., 8 (1960), 277-286.

[25]

D. Meidner, R. Rannacher and B. Vexler, A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time, SIAM J. Control Optim., 49 (2011), 1961-1997. doi: 10.1137/100793888.

[26]

K. Pieper and B. Vexler, A priori error analysis for discretization of sparse elliptic optimal control problems in measure space, SIAM J. Control Optim., 51 (2013), 2788-2808. doi: 10.1137/120889137.

[27]

J. P. Raymond and H. Zidani, Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations, App. Math. Optim., 39 (1999), 143-177. doi: 10.1007/s002459900102.

[28]

W. Rudin, Real and Complex Analysis, McGraw-Hill, London, 1970.

[29]

A. H. Schatz, V. C. Thomée and L. B. Wahlbin, Maximum norm stability and error estimates in parabolic finite element equations, Comm. Pure Appl. Math., 33 (1980), 265-304. doi: 10.1002/cpa.3160330305.

[30]

G. Stadler, Elliptic optimal control problems with $L^1$-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), 159-181. doi: 10.1007/s10589-007-9150-9.

[31]

V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Second edition, Spinger-Verlag, Berlin, 2006.

[32]

G. Wachsmuth and D. Wachsmuth, Convergence and regularisation results for optimal control problems with sparsity functional, ESAIM Control Optim. Calc. Var., 17 (2011), 858-886. doi: 10.1051/cocv/2010027.

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