# American Institute of Mathematical Sciences

March  2018, 8(1): 195-215. doi: 10.3934/mcrf.2018009

## Error analysis for global minima of semilinear optimal control problems

 1 Schwerpunkt Optimierung und Approximation, Universität Hamburg, Bundesstraße 55,20146 Hamburg, Germany 2 Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2,39106 Magdeburg, Germany

* Corresponding author: Michael Hinze

Received  April 2017 Revised  September 2017 Published  January 2018

In [2] we consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization, where we provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. With the present work we complement our investigations of [2] in that we prove an error estimate for those discrete global solutions. Numerical experiments confirm our analytical findings.

Citation: Ahmad Ahmad Ali, Klaus Deckelnick, Michael Hinze. Error analysis for global minima of semilinear optimal control problems. Mathematical Control and Related Fields, 2018, 8 (1) : 195-215. doi: 10.3934/mcrf.2018009
##### References:
 [1] A. Ahmad Ali, Optimal Control of Semilinear Elliptic PDEs with State Constraints -Numerical Analysis and Implementation, PhD thesis, Dissertation, Hamburg, Universität Hamburg, 2017. [2] A. Ahmad Ali, K. Deckelnick and M. Hinze, Global minima for semilinear optimal control problems, Computational Optimization and Applications, 65 (2016), 261-288. [3] N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, 23 (2002), 201-229. [4] E. Casas, L2 Estimates for the finite element method for the Dirichlet problem with singular data, Numerische Mathematik, 47 (1985), 627-632. [5] E. Casas, Control of an elliptic problem with pointwise state constraints, SIAM Journal on Control and Optimization, 24 (1986), 1309-1318. [6] E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM Journal on Control and Optimization, 31 (1993), 993-1006. [7] E. Casas, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 345-374. [8] E. Casas, Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 575-589. [9] E. Casas, J. C. De Los Reyes and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM Journal on Optimization, 19 (2008), 616-643. [10] E. Casas and L. A. Fernández, Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state, Applied Mathematics and Optimization, 27 (1993), 35-56. [11] E. Casas and M. Mateos, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM Journal on Control and Optimization, 40 (2002), 1431-1454. [12] E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems, Comput. Appl. Math., 21 (2002), 67-100, Special issue in memory of Jacques-Louis Lions. [13] E. Casas, M. Mateos and B. Vexler, New regularity results and improved error estimates for optimal control problems with state constraints, ESAIM. Control, Optimisation and Calculus of Variations, 20 (2014), 803-822. [14] E. Casas and F. Tröltzsch, Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 581-600. [15] E. Casas and F. Tröltzsch, Second order optimality conditions and their role in pde control, Jahresbericht der Deutschen Mathematiker-Vereinigung, 117 (2015), 3-44. [16] K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM Journal on Numerical Analysis, 45 (2007), 1937-1953. [17] K. Deckelnick and M. Hinze, A finite element approximation to elliptic control problems in the presence of control and state constraints, Hamburger Beiträge zur Angewandten Mathematik, (2007). [18] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 69, SIAM, 2011. [19] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, vol. 23 of Mathematical Modelling: Theory and Applications, Springer, New York, 2009. [20] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-61. [21] M. Hinze and C. Meyer, Stability of semilinear elliptic optimal control problems with pointwise state constraints, Computational Optimization and Applications, 52 (2012), 87-114. [22] M. Hinze and A. Rösch, Discretization of optimal control problems, in Constrained Optimization and Optimal Control for Partial Differential Equations, Springer, 160 (2012), 391-430. [23] M. Hinze and F. Tröltzsch, Discrete concepts versus error analysis in pde-constrained optimization, GAMM-Mitteilungen, 33 (2010), 148-162. [24] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. [25] P. Merino, F. Tröltzsch and B. Vexler, Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 167-188. [26] C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints, Control and Cybernetics, 37 (2008), 51-83. [27] I. Neitzel, J. Pfefferer and A. Rösch, Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM Journal on Control and Optimization, 53 (2015), 874-904. [28] I. Neitzel and W. Wollner, A priori L2-discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints, Numer. Math., online first (2017).

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##### References:
 [1] A. Ahmad Ali, Optimal Control of Semilinear Elliptic PDEs with State Constraints -Numerical Analysis and Implementation, PhD thesis, Dissertation, Hamburg, Universität Hamburg, 2017. [2] A. Ahmad Ali, K. Deckelnick and M. Hinze, Global minima for semilinear optimal control problems, Computational Optimization and Applications, 65 (2016), 261-288. [3] N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, 23 (2002), 201-229. [4] E. Casas, L2 Estimates for the finite element method for the Dirichlet problem with singular data, Numerische Mathematik, 47 (1985), 627-632. [5] E. Casas, Control of an elliptic problem with pointwise state constraints, SIAM Journal on Control and Optimization, 24 (1986), 1309-1318. [6] E. Casas, Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM Journal on Control and Optimization, 31 (1993), 993-1006. [7] E. Casas, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 345-374. [8] E. Casas, Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 575-589. [9] E. Casas, J. C. De Los Reyes and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM Journal on Optimization, 19 (2008), 616-643. [10] E. Casas and L. A. Fernández, Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state, Applied Mathematics and Optimization, 27 (1993), 35-56. [11] E. Casas and M. Mateos, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM Journal on Control and Optimization, 40 (2002), 1431-1454. [12] E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems, Comput. Appl. Math., 21 (2002), 67-100, Special issue in memory of Jacques-Louis Lions. [13] E. Casas, M. Mateos and B. Vexler, New regularity results and improved error estimates for optimal control problems with state constraints, ESAIM. Control, Optimisation and Calculus of Variations, 20 (2014), 803-822. [14] E. Casas and F. Tröltzsch, Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 581-600. [15] E. Casas and F. Tröltzsch, Second order optimality conditions and their role in pde control, Jahresbericht der Deutschen Mathematiker-Vereinigung, 117 (2015), 3-44. [16] K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state-constrained elliptic control problem, SIAM Journal on Numerical Analysis, 45 (2007), 1937-1953. [17] K. Deckelnick and M. Hinze, A finite element approximation to elliptic control problems in the presence of control and state constraints, Hamburger Beiträge zur Angewandten Mathematik, (2007). [18] P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 69, SIAM, 2011. [19] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, vol. 23 of Mathematical Modelling: Theory and Applications, Springer, New York, 2009. [20] M. Hinze, A variational discretization concept in control constrained optimization: The linear-quadratic case, Computational Optimization and Applications, 30 (2005), 45-61. [21] M. Hinze and C. Meyer, Stability of semilinear elliptic optimal control problems with pointwise state constraints, Computational Optimization and Applications, 52 (2012), 87-114. [22] M. Hinze and A. Rösch, Discretization of optimal control problems, in Constrained Optimization and Optimal Control for Partial Differential Equations, Springer, 160 (2012), 391-430. [23] M. Hinze and F. Tröltzsch, Discrete concepts versus error analysis in pde-constrained optimization, GAMM-Mitteilungen, 33 (2010), 148-162. [24] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics, 31. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000. [25] P. Merino, F. Tröltzsch and B. Vexler, Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 167-188. [26] C. Meyer, Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints, Control and Cybernetics, 37 (2008), 51-83. [27] I. Neitzel, J. Pfefferer and A. Rösch, Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation, SIAM Journal on Control and Optimization, 53 (2015), 874-904. [28] I. Neitzel and W. Wollner, A priori L2-discretization error estimates for the state in elliptic optimization problems with pointwise inequality state constraints, Numer. Math., online first (2017).
Example 1: $\|\bar p_h\|_{L^4}$ and $\eta(\alpha)$ vs. $\alpha$.
Example 1: The unique global minimum together with its state and the associated multipliers.
Example 1: Errors for the optimal control and its state versus the mesh size.
Example 2: The unique global minimum together with its state and the associated multipliers.
Example 2: Errors for the optimal control and its state versus the mesh size.
Example 1: EOC for the optimal control and its state.
 Levels EOC$_{u_{L2}}$ EOC$_{y_{H1}}$ EOC$_{y_{L2}}$ EOC$_{y_{L\infty}}$ 2-3 1.187645 0.833842 1.464788 1.058334 3-4 1.078183 0.948273 1.708822 1.758387 4-5 1.027290 0.985352 1.794456 1.657899 5-6 1.016702 0.997996 1.831198 1.514376
 Levels EOC$_{u_{L2}}$ EOC$_{y_{H1}}$ EOC$_{y_{L2}}$ EOC$_{y_{L\infty}}$ 2-3 1.187645 0.833842 1.464788 1.058334 3-4 1.078183 0.948273 1.708822 1.758387 4-5 1.027290 0.985352 1.794456 1.657899 5-6 1.016702 0.997996 1.831198 1.514376
Example 2: EOC for the optimal control and its state.
 Levels EOC$_{u_{L2}}$ EOC$_{y_{H1}}$ EOC$_{y_{L2}}$ EOC$_{y_{L\infty}}$ 2-3 1.910131 1.015420 1.686197 1.399920 3-4 1.983722 1.017850 1.934978 1.876141 4-5 1.982064 1.005320 1.996849 1.993434 5-6 1.944839 1.003374 2.035042 2.027269
 Levels EOC$_{u_{L2}}$ EOC$_{y_{H1}}$ EOC$_{y_{L2}}$ EOC$_{y_{L\infty}}$ 2-3 1.910131 1.015420 1.686197 1.399920 3-4 1.983722 1.017850 1.934978 1.876141 4-5 1.982064 1.005320 1.996849 1.993434 5-6 1.944839 1.003374 2.035042 2.027269
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