
-
Previous Article
Optimal investment problem with delay under partial information
- MCRF Home
- This Issue
-
Next Article
Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme
Finite element error estimates for one-dimensional elliptic optimal control by BV-functions
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 |
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.
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. |
[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. |
[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.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[37] |
V. Veliov,
On the time-discretization of control systems, SIAM J. Control Optim., 35 (1997), 1470-1486.
doi: 10.1137/S0363012995288987. |
[38] |
V. M. Veliov,
Error analysis of discrete approximations to bang-bang optimal control problems: The linear case, Control Cybern., 34 (2005), 967-982.
|
[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. |
[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. |
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. |
[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. |
[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.
![]() |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[37] |
V. Veliov,
On the time-discretization of control systems, SIAM J. Control Optim., 35 (1997), 1470-1486.
doi: 10.1137/S0363012995288987. |
[38] |
V. M. Veliov,
Error analysis of discrete approximations to bang-bang optimal control problems: The linear case, Control Cybern., 34 (2005), 967-982.
|
[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. |
[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. |






[1] |
Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025 |
[2] |
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, 2021 doi: 10.3934/mcrf.2021014 |
[3] |
Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021012 |
[4] |
Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205 |
[5] |
Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021031 |
[6] |
Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 |
[7] |
Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021007 |
[8] |
Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215 |
[9] |
Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183 |
[10] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[11] |
Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021040 |
[12] |
John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026 |
[13] |
Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021 doi: 10.3934/mcrf.2021009 |
[14] |
Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 |
[15] |
Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021012 |
[16] |
Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190 |
[17] |
Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020136 |
[18] |
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209 |
[19] |
Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91 |
[20] |
Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]