×

Optimal control of convective FitzHugh-Nagumo equation. (English) Zbl 1373.65045

Summary: We investigate smooth and sparse optimal control problems for convective FitzHugh-Nagumo equation with traveling wave solutions in moving excitable media. The cost function includes distributed space-time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several numerical results are presented for the control of the traveling waves. We also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing Tikhonov regularization parameter. Further, we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced Hessian.

MSC:

65K10 Numerical optimization and variational techniques
49J20 Existence theories for optimal control problems involving partial differential equations
49M25 Discrete approximations in optimal control

Software:

CG_DESCENT
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Murray, J. D., Mathematical biology: I. an introduction, (2002), Springer-Verlag New York
[2] Karasözen, B.; Küçükseyhan, T.; Uzunca, M., Structure preserving integration and model order reduction of skew-gradient reaction-diffusion systems, Ann. Oper. Res., 1-28, (2015)
[3] Ermakova, E. A.; Panteleev, M. A.; Shnol, E. E., Blood coagulation and propagation of autowaves in flow, Pathophysiol. Haemost. Thromb., 34, 135-142, (2005)
[4] Ermakova, E. A.; Shnol, E. E.; Panteleev, M. A.; Butylin, A. A.; Volpert, V.; Ataullakhanov, F. I., On propagation of excitation waves in moving media: the Fitzhugh-Nagumo model, PLoS One, 4, 2, E4454, (2009)
[5] Lobanov, A.; Starozhilova, T., The effect of convective flows on blood coagulation processes, Pathophysiol. Haemost. Thromb., 34, 121-134, (2005)
[6] Buchholz, R.; Engel, H.; Kammann, E.; Tröltzsch, F., On the optimal control of the Schlögl-model, Comput. Optim. Appl., 56, 1, 153-185, (2013) · Zbl 1273.49006
[7] Casas, E.; Ryll, C.; Tröltzsch, F., Sparse optimal control of the Schlögl and Fitzhugh-Nagumo systems, Comput. Methods Appl. Math., 13, 4, 415-442, (2013) · Zbl 1393.49019
[8] Casas, E.; Ryll, C.; Tröltzsch, F., Second order and stability analysis for optimal sparse control of the Fitzhugh-Nagumo equation, SIAM J. Control Optim., 53, 4, 2168-2202, (2015) · Zbl 1326.49032
[9] Ryll, C.; Löber, J.; Martens, S.; Engel, H.; Tröltzsch, F., Analytical, optimal, and sparse optimal control of traveling wave solutions to reaction-diffusion systems, (Schöll, E.; Klapp, L. S.H.; Hövel, P., Control of Self-Organizing Nonlinear Systems, (2016), Springer International Publishing), 189-210, (Chapter 10)
[10] Stoll, M.; Pearson, J. W.; Maini, P. K., Fast solvers for optimal control problems from pattern fotrmation, J. Comput. Phys., 304, 27-45, (2016) · Zbl 1349.92035
[11] Breiten, T.; Kunisch, K., Riccati-based feedback control of the monodomain equations with the Fitzhugh-Nagumo model, SIAM J. Control Optim., 52, 6, 4057-4081, (2014) · Zbl 1316.93047
[12] Stadler, G., Elliptic optimal control problems with \(L^1\)-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44, 1, 159-181, (2009) · Zbl 1185.49031
[13] Casas, E.; Herzog, R.; Wachsmuth, G., Optimality conditions and error analysis of semilinear elliptic control problems with \(L^1\) cost functional, SIAM J. Optim., 22, 34, 795-820, (2012) · Zbl 1278.49026
[14] Wachsmuth, G.; Wachsmuth, D., Convergence and regularization results for optimal control problems with sparsity function, ESAIM Control Optim. Calc. Var., 17, 3, 858-886, (2010) · Zbl 1228.49032
[15] Leykekhman, D.; Heinkenschloss, M., Local error analysis of discontinuous Galerkin methods for advection-dominated elliptic linear-quadratic optimal control problems, SIAM J. Numer. Anal., 50, 4, 2012-2038, (2012) · Zbl 1253.49018
[16] Yücel, H.; Benner, P., Distributed optimal control problems governed by coupled convection dominated pdes with control constraints, (Abdulle, A.; Deparis, S.; Kressner, D.; Nobile, F.; Picasso, M., Numerical Mathematics and Advanced Applications, ENUMATH 2013, Lecture Notes in Computational Science and Engineering, vol. 103, (2015), Springer International Publishing), 469-478 · Zbl 1328.65145
[17] Yücel, H.; Karasözen, B., Adaptive symmetric interior penalty Galerkin (SIPG) method for optimal control of convection diffusion equations with control constraints, Optimization, 63, 145-166, (2014) · Zbl 1302.65258
[18] Akman, T.; Karasözen, B., Variational time discretization methods for optimal control problems governed by diffusion-convection-reaction equations, J. Comput. Appl. Math., 272, 41-56, (2014) · Zbl 1293.49065
[19] Akman, T.; Yücel, H.; Karasözen, B., A priori error analysis of the upwind symmetric interior penalty Galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equations, Comput. Optim. Appl., 57, 703-729, (2014) · Zbl 1301.49072
[20] Yücel, H.; Benner, P., Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations, Comput. Optim. Appl., 62, 291-321, (2015) · Zbl 1333.49047
[21] Yücel, H.; Stoll, M.; Benner, P., A discontinous Galerkin method for optimal control problems governed by a system of convection-diffusion PDEs with nonlinear reaction terms, Comput. Math. Appl., 70, 2414-2431, (2015)
[22] Hager, W. W.; Zhang, H., Algorithm 851: CG_DESCENT, a conjugate gradient method with guaranteed descent, ACM Trans. Math. Software, 32, 1, 113-137, (2006) · Zbl 1346.90816
[23] Casas, E.; Tröltzsch, F., Second order optimality conditions and their role in PDE control, Jahresber. Deutsch. Math.-Verein., 117, 1, 3-44, (2015) · Zbl 1311.49002
[24] Rösch, A.; Wachsmuth, D., Numerical verification of optimality conditions, SIAM J. Control Optim., 47, 5, 2557-2581, (2008) · Zbl 1171.49018
[25] Rösch, A.; Wachsmuth, D., A-posteriori error estimates for optimal control problems with state and control constraints, Numer. Math., 120, 733-762, (2012) · Zbl 1247.65087
[26] Kammann, E.; Tröltzsch, F.; Volkwein, S., A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD, ESAIM: M2AN, 47, 2, 555-581, (2013) · Zbl 1282.49021
[27] Lass, O.; Trenz, S.; Volkwein, S., Optimality conditions and POD a-posteriori error estimates for a semilinear parabolic optimal control, Konstanzer Schr. Math., 345, (2015)
[28] Pao, C., Nonlinear parabolic and elliptic equations, (1992), Plenum Press New York
[29] Tröltzsch, F., (Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics, vol. 112, (2010), American Mathematical Society Providence, RI) · Zbl 1195.49001
[30] Barthel, W.; John, C.; Tröltzsch, F., Optimal boundary control of a system of reaction diffusion equations, ZAMM Z. Angew. Math. Mech., 90, 12, 966-982, (2010) · Zbl 1375.49030
[31] Griesse, R., Parametric sensitivity analysis for control-constrained optimal control problems governed by parabolic partial differential equations, (2003), Institut für Mathematik, Universität Bayreuth Bayreuth, Germany, (Ph.D. thesis)
[32] Jackson, D. E., Existence and regularity for the Fitzhugh-Nagumo equations with inhomogeneous boundary conditions, Nonlinear Anal.-Theor., 14, 3, 201-216, (1990) · Zbl 0712.35048
[33] Gudi, T.; Pani, A. K., Discontinuous Galerkin methods for quasi-linear elliptic problems of nonmonotone type, SIAM J. Numer. Anal., 45, 1, 163-192, (2007) · Zbl 1140.65082
[34] Griesse, R.; Volkwein, S., A primal-dual active set strategy for optimal boundary control of a nonlinear reaction-diffusion system, SIAM J. Control Optim., 44, 2, 467-494, (2005) · Zbl 1090.49024
[35] Park, J. Y.; Park, S. H., Optimal control problems for anti-periodic quasi-linear hemivariational inequalities, Optim. Control Appl. Methods, 28, 4, 275-287, (2007)
[36] Mittelmann, H. D., Verification of second-order sufficient optimality conditions for semilinear elliptic and parabolic control problems, Comput. Optim. Appl., 20, 1, 93-110, (2001) · Zbl 0986.49011
[37] Arada, N.; Casas, E.; Tröltzsch, F., Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl., 23, 2, 201-229, (2002) · Zbl 1033.65044
[38] Casas, E.; Tröltzsch, F., Second-order optimality conditions for weak and strong local solutions of parabolic optimal control problems, Vietnam J. Math., 44, 181-202, (2016) · Zbl 1337.49038
[39] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779, (2002) · Zbl 1008.65080
[40] Rivière, B., (Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Theory and Implementation, Frontiers in Applied Mathematics, vol. 35, (2008), Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA) · Zbl 1153.65112
[41] Ciarlet, P. G., (The Finite Element Method for Elliptic Problems, Classics Appl. Math., vol. 40, (2002), SIAM Philadelphia, PA)
[42] Cangiani, A.; Chapman, J.; Georgoulis, E. H.; Jensen, M., On local super-penalization of interior penalty discontinuous Galerkin methods, Int. J. Numer. Anal. Model., 11, 3, 478-495, (2014)
[43] Yücel, H.; Heinkenschloss, M.; Karasözen, B., Distributed optimal control of diffusion-convection-reaction equations using discontinuous Galerkin methods, (Numerical Mathematics and Advanced Applications 2011, (2013), Springer-Verlag Berlin, Heidelberg), 389-397 · Zbl 1267.65076
[44] Hager, W. W.; Zhang, H., A survey of nonlinear conjugate gradient methods, Pac. J. Optim., 2, 1, 35-58, (2006) · Zbl 1117.90048
[45] Nocedal, J.; Wright, S. J., Numerical optimization, (2006), Springer Verlag Berlin, Heidelberg, New York · Zbl 1104.65059
[46] Herzog, R.; Kunisch, K., Algorithms for PDE-constrained optimization, GAMM-Mitt., 33, 2, 163-176, (2010) · Zbl 1207.49034
[47] Peruzzi, N. J.; Chavarette, F. R.; Balthazar, J. M.; Tusset, A. M.; Perticarrari, A. L.P. M.; Brasil, R. M.L. R.F., The dynamic behavior of a parametrically excited time-periodic MEMS taking into account parametric errors, J. Vib. Control, 22, 4101-4110, (2016) · Zbl 1373.93132
[48] Tusset, A. M.; Piccirillo, V.; Bueno, A.; Balthazar, J. M.; Sado, D.; Felix, J. L.P.; Brasil, R. M.L. R.F., Chaos control and sensitivity analysis of a double pendulum arm excited by an RLC circuit based nonlinear shaker, J. Vib. Control, 17, 3621-3637, (2016)
[49] Ryll, C.; Tröltzsch, F., Proper orthogonal decomposition in sparse optimal control of some reaction diffusion equations using model predictive control, Proc. Appl. Math. Mech., 14, 1, 883-884, (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.