American Institute of Mathematical Sciences

Advanced
doi: 10.3934/mcrf.2019041

Finite element error estimates for one-dimensional elliptic optimal control by BV-functions

Dominik Hafemeyer 1,, , Florian Mannel 2, , Ira Neitzel 3, and  Boris Vexler 1,

1. 

Department of Mathematics, Technische Universität München, Boltzmannstr. 3, 85748 Garching b. München, Germany
2. 

Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstr. 36, 8010 Graz, Austria
3. 

Institute for Numerical Simulation, Universität Bonn, Endenicher Allee 19b, 53115 Bonn, Germany

* Corresponding author: Dominik Hafemeyer

Received  February 2019 Revised  June 2019 Published  August 2019

We consider an optimal control problem governed by a one-dimensional elliptic equation that involves univariate functions of bounded variation as controls. For the discretization of the state equation we use linear finite elements and for the control discretization we analyze two strategies. First, we use variational discretization of the control and show that the $ L^2 $- and $ L^\infty $-error for the state and the adjoint state are of order $ {\mathcal O}(h^2) $ and that the $ L^1 $-error of the control behaves like $ {\mathcal O}(h^2) $, too. These results rely on a structural assumption that implies that the optimal control of the original problem is piecewise constant and that the adjoint state has nonvanishing first derivative at the jump points of the control. If, second, piecewise constant control discretization is used, we obtain $ L^2 $-error estimates of order $ \mathcal{O}(h) $ for the state and $ W^{1, \infty} $-error estimates of order $ \mathcal{O}(h) $ for the adjoint state. Under the same structural assumption as before we derive an $ L^1 $-error estimate of order $ \mathcal{O}(h) $ for the control. We discuss optimization algorithms and provide numerical results for both discretization schemes indicating that the error estimates are optimal.

Keywords: Optimal control, BV-functions, optimality conditions, numerical analysis, finite elements.
Mathematics Subject Classification: 26A45, 49J20, 49M25, 65N15, 65N30.
Citation: Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler. Finite element error estimates for one-dimensional elliptic optimal control by BV-functions. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2019041
References:
[1]

W. Alt, R. Baier, F. Lempio and M. Gerdts, Approximations of linear control problems with bang-bang solutions, Optimization, 62 (2013), 9-32.  doi: 10.1080/02331934.2011.568619.  Google Scholar

[2]

W. Alt, U. Felgenhauer and M. Seydenschwanz, Euler discretization for a class of nonlinear optimal control problems with control appearing linearly, Comput. Optim. Appl., 69 (2018), 825-856.  doi: 10.1007/s10589-017-9969-7.  Google Scholar

[3] 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
[4]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, Second edition. MOS-SIAM Series on Optimization, 17. SIAM, Philadelphia, PA, Mathematical Optimization Society, 2014. doi: 10.1137/1.9781611973488.  Google Scholar

[5]

S. Bartels, Total variation minimization with finite elements: Convergence and iterative solution, SIAM J. Numer. Anal., 50 (2012), 1162-1180.  doi: 10.1137/11083277X.  Google Scholar

[6]

S. Bartels and M. Milicevic, Iterative finite element solution of a constrained total variation regularized model problem, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1207-1232.  doi: 10.3934/dcdss.2017066.  Google Scholar

[7]

L. Bonifacius, K. Pieper and B Vexler, Error estimates for space-time discretization of parabolic time-optimal control problems with bang-bang controls, SIAM J. Control Optim., 57 (2019), 1730-1756.  doi: 10.1137/18M1213816.  Google Scholar

[8]

K. Bredies and D. Vicente, A perfect reconstruction property for pde-constrained total-variation minimization with application in quantitative susceptibility mapping, ESAIM Control Optim. Calc. Var., Accepted for Publication. Google Scholar

[9]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[10]

J. Casado-Díaz, C. Castro, M. Luna-Laynez and E. Zuazua, Numerical approximation of a one-dimensional elliptic optimal design problem, Multiscale Model. Simul., 9 (2011), 1181-1216.  doi: 10.1137/10081928X.  Google Scholar

[11]

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.  Google Scholar

[12]

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

[13]

E. Casas, P. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the $p$-Laplace equation. I. convergence result, SIAM J. Control Optim., 54 (2016), 1406-1422.  doi: 10.1137/15M1028108.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

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

[17]

E. Casas and K. Kunisch, Analysis of optimal control problems of semilinear elliptic equations by BV-functions, Set-Valued Var. Anal., 27 (2019), 355-379.  doi: 10.1007/s11228-018-0482-7.  Google Scholar

[18]

E. Casas, K. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement, Appl. Math. Optim., 40 (1999), 229-257.  doi: 10.1007/s002459900124.  Google Scholar

[19]

E. Casas, M. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity, Comput. Optim. Appl., 70 (2018), 239-266.  doi: 10.1007/s10589-018-9979-0.  Google Scholar

[20]

E. Casas, D. Wachsmuth and G. Wachsmuth, Second-order analysis and numerical approximation for bang-bang bilinear control problems, SIAM J. Control Optim., 56 (2018), 4203-4227.  doi: 10.1137/17M1139953.  Google Scholar

[21]

I. Chryssoverghi, Approximate gradient/penalty methods with general discretization schemes for optimal control problems, in Large-Scale Scientific Computing, Lecture Notes in Comput. Sci., Springer, Berlin, 3743 (2006), 199–207. doi: 10.1007/11666806_21.  Google Scholar

[22]

C. Clason, F. Kruse and K. Kunisch, Total variation regularization of multi-material topology optimization, ESAIM, Math. Model. Numer. Anal., 52 (2018), 275-303.  doi: 10.1051/m2an/2017061.  Google Scholar

[23]

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.  Google Scholar

[24]

K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls, Comput. Optim. Appl., 51 (2012), 931-939.  doi: 10.1007/s10589-010-9365-z.  Google Scholar

[25]

A. L. Dontchev, An a priori estimate for discrete approximations in nonlinear optimal control, SIAM J. Control Optim., 34 (1996), 1315-1328.  doi: 10.1137/S036301299426948X.  Google Scholar

[26]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order runge-kutta approximations in control constrained optimal control, SIAM J. Numer. Anal., 38 (2000), 202-226.  doi: 10.1137/S0036142999351765.  Google Scholar

[27]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159. Springer, New York, 2004. doi: 10.1007/978-1-4757-4355-5.  Google Scholar

[28]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[29]

M. Hinze and T. Quyen, Iterated total variation regularization with finite element methods for reconstruction the source term in elliptic systems, preprint, arXiv: 1901.10278. Google Scholar

[30]

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.  Google Scholar

[31]

K. Kunisch, P. 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

[32]

J. Peypouquet, Convex Optimization in Normed Spaces: Theory, Methods and Examples, With a foreword by Hedy Attouch, SpringerBriefs in Optimization, Springer, Cham, 2015. doi: 10.1007/978-3-319-13710-0.  Google Scholar

[33]

K. Pieper, B. Quoc Tang, P. Trautmann and D. Walter, Inverse point source location with the helmholtz equation on a bounded domain, preprint, arXiv: 1805.03310. Google Scholar

[34]

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.  Google Scholar

[35]

P. Trautmann, B. Vexler and A. Zlotnik, Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients, Math. Control Relat. Fields, 8 (2018), 411-449.  doi: 10.3934/mcrf.2018017.  Google Scholar

[36]

M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, MOS-SIAM Series on Optimization, 11. SIAM, Philadelphia, PA, Mathematical Optimization Society, 2011. doi: 10.1137/1.9781611970692.  Google Scholar

[37]

V. Veliov, On the time-discretization of control systems, SIAM J. Control Optim., 35 (1997), 1470-1486.  doi: 10.1137/S0363012995288987.  Google Scholar

[38]

V. M. Veliov, Error analysis of discrete approximations to bang-bang optimal control problems: The linear case, Control Cybern., 34 (2005), 967-982.   Google Scholar

[39]

D. Walter, On Sparse Sensor Placement for Parameter Identifcation Problems with Partial Differential Equations, PhD thesis, Technische Universität München, 2018. Google Scholar

[40]

M. F. Wheeler, An optimal $L_{\infty }$ error estimate for Galerkin approximations to solutions of two-point boundary value problems, SIAM J. Numer. Anal., 10 (1973), 914-917.  doi: 10.1137/0710077.  Google Scholar

[41]

W. P. Ziemer, Weakly Differentiable Functions, Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, 120. Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

References:
[1]

W. Alt, R. Baier, F. Lempio and M. Gerdts, Approximations of linear control problems with bang-bang solutions, Optimization, 62 (2013), 9-32.  doi: 10.1080/02331934.2011.568619.  Google Scholar

[2]

W. Alt, U. Felgenhauer and M. Seydenschwanz, Euler discretization for a class of nonlinear optimal control problems with control appearing linearly, Comput. Optim. Appl., 69 (2018), 825-856.  doi: 10.1007/s10589-017-9969-7.  Google Scholar

[3] 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
[4]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization, Second edition. MOS-SIAM Series on Optimization, 17. SIAM, Philadelphia, PA, Mathematical Optimization Society, 2014. doi: 10.1137/1.9781611973488.  Google Scholar

[5]

S. Bartels, Total variation minimization with finite elements: Convergence and iterative solution, SIAM J. Numer. Anal., 50 (2012), 1162-1180.  doi: 10.1137/11083277X.  Google Scholar

[6]

S. Bartels and M. Milicevic, Iterative finite element solution of a constrained total variation regularized model problem, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1207-1232.  doi: 10.3934/dcdss.2017066.  Google Scholar

[7]

L. Bonifacius, K. Pieper and B Vexler, Error estimates for space-time discretization of parabolic time-optimal control problems with bang-bang controls, SIAM J. Control Optim., 57 (2019), 1730-1756.  doi: 10.1137/18M1213816.  Google Scholar

[8]

K. Bredies and D. Vicente, A perfect reconstruction property for pde-constrained total-variation minimization with application in quantitative susceptibility mapping, ESAIM Control Optim. Calc. Var., Accepted for Publication. Google Scholar

[9]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Third edition, Texts in Applied Mathematics, 15. Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[10]

J. Casado-Díaz, C. Castro, M. Luna-Laynez and E. Zuazua, Numerical approximation of a one-dimensional elliptic optimal design problem, Multiscale Model. Simul., 9 (2011), 1181-1216.  doi: 10.1137/10081928X.  Google Scholar

[11]

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.  Google Scholar

[12]

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

[13]

E. Casas, P. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the $p$-Laplace equation. I. convergence result, SIAM J. Control Optim., 54 (2016), 1406-1422.  doi: 10.1137/15M1028108.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

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

[17]

E. Casas and K. Kunisch, Analysis of optimal control problems of semilinear elliptic equations by BV-functions, Set-Valued Var. Anal., 27 (2019), 355-379.  doi: 10.1007/s11228-018-0482-7.  Google Scholar

[18]

E. Casas, K. Kunisch and C. Pola, Regularization by functions of bounded variation and applications to image enhancement, Appl. Math. Optim., 40 (1999), 229-257.  doi: 10.1007/s002459900124.  Google Scholar

[19]

E. Casas, M. Mateos and A. Rösch, Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity, Comput. Optim. Appl., 70 (2018), 239-266.  doi: 10.1007/s10589-018-9979-0.  Google Scholar

[20]

E. Casas, D. Wachsmuth and G. Wachsmuth, Second-order analysis and numerical approximation for bang-bang bilinear control problems, SIAM J. Control Optim., 56 (2018), 4203-4227.  doi: 10.1137/17M1139953.  Google Scholar

[21]

I. Chryssoverghi, Approximate gradient/penalty methods with general discretization schemes for optimal control problems, in Large-Scale Scientific Computing, Lecture Notes in Comput. Sci., Springer, Berlin, 3743 (2006), 199–207. doi: 10.1007/11666806_21.  Google Scholar

[22]

C. Clason, F. Kruse and K. Kunisch, Total variation regularization of multi-material topology optimization, ESAIM, Math. Model. Numer. Anal., 52 (2018), 275-303.  doi: 10.1051/m2an/2017061.  Google Scholar

[23]

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.  Google Scholar

[24]

K. Deckelnick and M. Hinze, A note on the approximation of elliptic control problems with bang-bang controls, Comput. Optim. Appl., 51 (2012), 931-939.  doi: 10.1007/s10589-010-9365-z.  Google Scholar

[25]

A. L. Dontchev, An a priori estimate for discrete approximations in nonlinear optimal control, SIAM J. Control Optim., 34 (1996), 1315-1328.  doi: 10.1137/S036301299426948X.  Google Scholar

[26]

A. L. Dontchev, W. W. Hager and V. M. Veliov, Second-order runge-kutta approximations in control constrained optimal control, SIAM J. Numer. Anal., 38 (2000), 202-226.  doi: 10.1137/S0036142999351765.  Google Scholar

[27]

A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Sciences, 159. Springer, New York, 2004. doi: 10.1007/978-1-4757-4355-5.  Google Scholar

[28]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80. Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.  Google Scholar

[29]

M. Hinze and T. Quyen, Iterated total variation regularization with finite element methods for reconstruction the source term in elliptic systems, preprint, arXiv: 1901.10278. Google Scholar

[30]

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.  Google Scholar

[31]

K. Kunisch, P. 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

[32]

J. Peypouquet, Convex Optimization in Normed Spaces: Theory, Methods and Examples, With a foreword by Hedy Attouch, SpringerBriefs in Optimization, Springer, Cham, 2015. doi: 10.1007/978-3-319-13710-0.  Google Scholar

[33]

K. Pieper, B. Quoc Tang, P. Trautmann and D. Walter, Inverse point source location with the helmholtz equation on a bounded domain, preprint, arXiv: 1805.03310. Google Scholar

[34]

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.  Google Scholar

[35]

P. Trautmann, B. Vexler and A. Zlotnik, Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients, Math. Control Relat. Fields, 8 (2018), 411-449.  doi: 10.3934/mcrf.2018017.  Google Scholar

[36]

M. Ulbrich, Semismooth Newton Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces, MOS-SIAM Series on Optimization, 11. SIAM, Philadelphia, PA, Mathematical Optimization Society, 2011. doi: 10.1137/1.9781611970692.  Google Scholar

[37]

V. Veliov, On the time-discretization of control systems, SIAM J. Control Optim., 35 (1997), 1470-1486.  doi: 10.1137/S0363012995288987.  Google Scholar

[38]

V. M. Veliov, Error analysis of discrete approximations to bang-bang optimal control problems: The linear case, Control Cybern., 34 (2005), 967-982.   Google Scholar

[39]

D. Walter, On Sparse Sensor Placement for Parameter Identifcation Problems with Partial Differential Equations, PhD thesis, Technische Universität München, 2018. Google Scholar

[40]

M. F. Wheeler, An optimal $L_{\infty }$ error estimate for Galerkin approximations to solutions of two-point boundary value problems, SIAM J. Numer. Anal., 10 (1973), 914-917.  doi: 10.1137/0710077.  Google Scholar

[41]

W. P. Ziemer, Weakly Differentiable Functions, Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics, 120. Springer, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

Figure 1.  Example 1: A semi-discrete solution to the data from Section 5.3. The discretization parameter $ h $ is roughly $ 3.8\cdot 10^{-6} $. The inclusions provided in Corollary 2 are clearly visible
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Example 1: Convergence plots of the errors of the solutions to the semi-discrete problem (Pvd) compared to the exact solution. The exact solution is known
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Example 1: Convergence plots of the errors of the solutions to the fully discrete problem (Pcd) compared to the exact solution. The exact solution is known
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Example 2: A variationally discrete solution to the data from Section 5.4. The discretization parameter $ h $ is roughly $ 3.8\cdot 10^{-6} $. The inclusions provided in Corollary 2 are clearly visible
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Example 2: Convergence plots of the errors of the solutions to the semi-discrete problem (Pvd) compared to an approximation of the exact solution. The reference solution is computed as solution to (Pvd) with $ h_{\text{ref}}\approx 3.8\cdot 10^{-6} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Example 2: Convergence plots of the errors of the solutions to the fully discrete problem (Pcd) compared to an approximation of the exact solution. The reference solution is computed as solution to (Pcd) with $ h_{\text{ref}}\approx 2.4\cdot 10^{-7} $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
[2]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[3]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559
[4]

Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101
[5]

Heinz Schättler, Urszula Ledzewicz, Helmut Maurer. Sufficient conditions for strong local optimality in optimal control problems with $L_{2}$-type objectives and control constraints. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2657-2679. doi: 10.3934/dcdsb.2014.19.2657
[6]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[7]

Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027
[8]

Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235
[9]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47
[10]

Luong V. Nguyen. A note on optimality conditions for optimal exit time problems. Mathematical Control & Related Fields, 2015, 5 (2) : 291-303. doi: 10.3934/mcrf.2015.5.291
[11]

Stefano Bianchini, Daniela Tonon. A decomposition theorem for $BV$ functions. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1549-1566. doi: 10.3934/cpaa.2011.10.1549
[12]

Fredrik Hellman, Patrick Henning, Axel Målqvist. Multiscale mixed finite elements. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1269-1298. doi: 10.3934/dcdss.2016051
[13]

Miniak-Górecka Alicja, Nowakowski Andrzej. Sufficient optimality conditions for a class of epidemic problems with control on the boundary. Mathematical Biosciences & Engineering, 2017, 14 (1) : 263-275. doi: 10.3934/mbe.2017017
[14]

Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial & Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329
[15]

Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096
[16]

Radoslaw Pytlak. Numerical procedure for optimal control of higher index DAEs. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 647-670. doi: 10.3934/dcds.2011.29.647
[17]

Emmanuel Trélat. Optimal control of a space shuttle, and numerical simulations. Conference Publications, 2003, 2003 (Special) : 842-851. doi: 10.3934/proc.2003.2003.842
[18]

Philip Trautmann, Boris Vexler, Alexander Zlotnik. Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients. Mathematical Control & Related Fields, 2018, 8 (2) : 411-449. doi: 10.3934/mcrf.2018017
[19]

Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107
[20]

Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (31)
  • HTML views (157)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences