# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021014

## A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system

 1 Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Germany 2 Department of Mathematics, Technical University of Munich, Boltzmannstraße 3, 85748 Garching, Germany

* Corresponding author: Marita Holtmannspötter

Received  April 2019 Revised  December 2019 Published  March 2021

In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with an ODE that has to hold true in almost all points in space. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the discretization of the control we employ the same discretization technique which turns out to be equivalent to a variational discretization approach. We provide error estimates of optimal order both for the discretization of the state equation as well as for the optimal control. Numerical experiments are added to illustrate the proven rates of convergence.

Citation: Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021014
##### References:
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Johnson, Adaptive finite element mothods for parabolic problems I: A linear model problem, SIAM Journal on Numerical Analysis, 28 (1991), 43-77.  doi: 10.1137/0728003.  Google Scholar [10] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems II: optimal error estimates in $L_\infty L_2$ and $L_\infty L_\infty$, SIAM Journal on Numerical Analysis, 32 (1995), 706-740.  doi: 10.1137/0732033.  Google Scholar [11] L. C. Evans, Partial Differential Equations Vol. 19, American Mathematical Society, Providence, Rhode Island, 1998. Google Scholar [12] M. Gerdts and S.-J. Kimmerle, Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin, Discrete Contin. Dyn. Syst. 2015, Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 515–524. doi: 10.3934/proc.2015.0515.  Google Scholar [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Boston, MA, 1985.  Google Scholar [14] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.  Google Scholar [15] D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition, Mathematical and Computer Modelling, 38 (2003), 1003-1028.  doi: 10.1016/S0895-7177(03)90102-6.  Google Scholar [16] S.-J. Kimmerle and M. Gerdts, Necessary optimality conditions and a semi-smooth Newton approach for an optimal control problem of a coupled system of Saint-Venant equations and ordinary differential equations, Pure Appl. Funct. Anal., 1 (2016), 231-256.   Google Scholar [17] S.-J. Kimmerle, M. Gerdts and R. Herzog, Optimal control of an elastic crane-trolley-load system - a case study for optimal control of coupled ODE-PDE systems, Mathematical and Computer Modelling of Dynamical Systems, 24 (2018), 182-206.  doi: 10.1080/13873954.2017.1405046.  Google Scholar [18] A. Logg, K. A. Mardal and G. N. Wells, Automated solution of Differential Equations by the Finite Element Method, Springer Verlag, 2012. Google Scholar [19] D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control Problems part I: Problems without control constraints, SIAM Journal on Control and Optimization, 47 (2008), 1150-1177.  doi: 10.1137/070694016.  Google Scholar [20] D. Meidner and B. Vexler, Optimal Error Estimates for Fully Discrete Galerkin Approximations of Semilinear Parabolic Equations, ESAIM: M2AN, 52 (2018), 2307-2325.  doi: 10.1051/m2an/2018040.  Google Scholar [21] C. Meyer, S. M. Schnepp and O. Thoma, Optimal control of the inhomogeneous relativistic Maxwell-Newton-Lorentz equations, SIAM J. Control Optim., 54 (2016), 2490-2525.  doi: 10.1137/14100083X.  Google Scholar [22] C. Meyer and L. M. Susu, Analysis of a viscous two-field gradient damage model, part I: Existence and uniqueness, Z. Anal. Anwend., 38 (2019), 249-286.  doi: 10.4171/ZAA/1637.  Google Scholar [23] C. Meyer and L. Susu, Analysis of a viscous two-field gradient damage model, part II: Penalization limit, Z. Anal. Anwend., 38 (2019), 439-474.  doi: 10.4171/ZAA/1645.  Google Scholar [24] I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems, Numerische Mathematik, 120 (2011), 345-386.  doi: 10.1007/s00211-011-0409-9.  Google Scholar [25] L. Susu, Analysis and optimal control of a damage model with penalty, PhD Thesis, TU Dortmund, 2017. Google Scholar [26] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen, vol. 2, Vieweg+Teubner, 2009. Google Scholar [27] G. Wachsmuth, Optimal Control of Quasistatic Plasticity, PhD Thesis, TU Chemnitz, 2011. Google Scholar

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##### References:
 [1] T. Breiten, K. Kunisch and S. S. Rodrigues, Feedback stabilization to nonstationary solutions of a class of reaction diffusion equations of FitzHugh-Nagumo type, SIAM J. Control Optim., 55 (2017), 2684-2713.  doi: 10.1137/15M1038165.  Google Scholar [2] E. Casas and K. Chrysafinos, A discontinuous Galerkin time-stepping scheme for the velocity tracking problem, SIAM Journal on Numerical Analysis, 50 (2012), 2281-2306.  doi: 10.1137/110829404.  Google Scholar [3] E. Casas and K. Chrysafinos, Analysis of the velocity tracking control problem for the 3D evolutionary Navier-Stokes equations, SIAM Journal on Control and Optimization, 54 (2016), 99-128.  doi: 10.1137/140978107.  Google Scholar [4] K. Chudej, H. J. Pesch, M. Wächter, G. Sachs and F. Le Bras, Instationary heat-constrained trajectory optimization of a hypersonic space vehicle by ODE-PDE-constrained optimal control, Variational Analysis and Aerospace Engineering, Springer Optim. Appl., 33, Springer, New York, 2009,127–144. doi: 10.1007/978-0-387-95857-6_8.  Google Scholar [5] B. J. Dimitrijevic and K. Hackl, A method for gradient enhancement of continuum damage models, Technische Mechanik, Ruhr-Universität Bochum, 28 (2008), 43-52.   Google Scholar [6] B. J. Dimitrijevic and K. Hackl, A regularization framework for damage-plasticity models via gradient enhancement of the free energy, International Journal for Numerical Methods in Biomedical Engineering, 27 (2011), 1199-1210.  doi: 10.1002/cnm.1350.  Google Scholar [7] E. Emmrich, Gewöhnliche und Operator-differentialgleichungen, Vieweg, 2004. doi: 10.1007/978-3-322-80240-8.  Google Scholar [8] K. Eriksson, C. Johnson and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér, 19 (1985), 611-643.  doi: 10.1051/m2an/1985190406111.  Google Scholar [9] K. Eriksson and C. Johnson, Adaptive finite element mothods for parabolic problems I: A linear model problem, SIAM Journal on Numerical Analysis, 28 (1991), 43-77.  doi: 10.1137/0728003.  Google Scholar [10] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems II: optimal error estimates in $L_\infty L_2$ and $L_\infty L_\infty$, SIAM Journal on Numerical Analysis, 32 (1995), 706-740.  doi: 10.1137/0732033.  Google Scholar [11] L. C. Evans, Partial Differential Equations Vol. 19, American Mathematical Society, Providence, Rhode Island, 1998. Google Scholar [12] M. Gerdts and S.-J. Kimmerle, Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin, Discrete Contin. Dyn. Syst. 2015, Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 515–524. doi: 10.3934/proc.2015.0515.  Google Scholar [13] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Boston, MA, 1985.  Google Scholar [14] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-61.  doi: 10.1007/s10589-005-4559-5.  Google Scholar [15] D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using proper orthogonal decomposition, Mathematical and Computer Modelling, 38 (2003), 1003-1028.  doi: 10.1016/S0895-7177(03)90102-6.  Google Scholar [16] S.-J. Kimmerle and M. Gerdts, Necessary optimality conditions and a semi-smooth Newton approach for an optimal control problem of a coupled system of Saint-Venant equations and ordinary differential equations, Pure Appl. Funct. Anal., 1 (2016), 231-256.   Google Scholar [17] S.-J. Kimmerle, M. Gerdts and R. Herzog, Optimal control of an elastic crane-trolley-load system - a case study for optimal control of coupled ODE-PDE systems, Mathematical and Computer Modelling of Dynamical Systems, 24 (2018), 182-206.  doi: 10.1080/13873954.2017.1405046.  Google Scholar [18] A. Logg, K. A. Mardal and G. N. Wells, Automated solution of Differential Equations by the Finite Element Method, Springer Verlag, 2012. Google Scholar [19] D. Meidner and B. Vexler, A priori error estimates for space-time finite element discretization of parabolic optimal control Problems part I: Problems without control constraints, SIAM Journal on Control and Optimization, 47 (2008), 1150-1177.  doi: 10.1137/070694016.  Google Scholar [20] D. Meidner and B. Vexler, Optimal Error Estimates for Fully Discrete Galerkin Approximations of Semilinear Parabolic Equations, ESAIM: M2AN, 52 (2018), 2307-2325.  doi: 10.1051/m2an/2018040.  Google Scholar [21] C. Meyer, S. M. Schnepp and O. Thoma, Optimal control of the inhomogeneous relativistic Maxwell-Newton-Lorentz equations, SIAM J. Control Optim., 54 (2016), 2490-2525.  doi: 10.1137/14100083X.  Google Scholar [22] C. Meyer and L. M. Susu, Analysis of a viscous two-field gradient damage model, part I: Existence and uniqueness, Z. Anal. Anwend., 38 (2019), 249-286.  doi: 10.4171/ZAA/1637.  Google Scholar [23] C. Meyer and L. Susu, Analysis of a viscous two-field gradient damage model, part II: Penalization limit, Z. Anal. Anwend., 38 (2019), 439-474.  doi: 10.4171/ZAA/1645.  Google Scholar [24] I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems, Numerische Mathematik, 120 (2011), 345-386.  doi: 10.1007/s00211-011-0409-9.  Google Scholar [25] L. Susu, Analysis and optimal control of a damage model with penalty, PhD Thesis, TU Dortmund, 2017. Google Scholar [26] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen, vol. 2, Vieweg+Teubner, 2009. Google Scholar [27] G. Wachsmuth, Optimal Control of Quasistatic Plasticity, PhD Thesis, TU Chemnitz, 2011. Google Scholar
Refinement of the temporal discretization for $N = 66049$(left) and $N = 263169$(right) nodes
Refinement of the spatial discretization for $M = 512$(left) and $M = 4096$(right) time steps
Errors $\Vert l-l_\sigma\Vert_{I\times\Omega}$ for different spatial and temporal mesh refinements
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