
-
Previous Article
On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations
- DCDS-S Home
- This Issue
-
Next Article
Weak solutions of stochastic reaction diffusion equations and their optimal control
Hybrid optimal control problems for a class of semilinear parabolic equations
1. | Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010 Graz, Austria |
2. | Radon Institute, Austrian Academy of Sciences, Altenberger Strasse 69, 4040 Linz, Austria |
A class of optimal control problems of hybrid nature governed by semilinear parabolic equations is considered. These problems involve the optimization of switching times at which the dynamics, the integral cost, and the bounds on the control may change. First- and second-order optimality conditions are derived. The analysis is based on a reformulation involving a judiciously chosen transformation of the time domains. For autonomous systems and a time-independent integral cost, we prove that the Hamiltonian is constant in time when evaluated along the optimal controls and trajectories. A numerical example is provided.
References:
[1] |
S. Aniţa, V. Arnăutu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011. From mathematical models to numerical simulation with MATLAB®.
doi: 10.1007/978-0-8176-8098-5. |
[2] |
W. Barthel, C. John and F. Tröltzsch,
Optimal boundary control of a system of reaction diffusion equations, ZAMM Z. Angew. Math. Mech., 90 (2010), 966-982.
doi: 10.1002/zamm.200900359. |
[3] |
T. Bayen and F. J. Silva,
Second order analysis for strong solutions in the optimal control of parabolic equations, SIAM Journal on Control and Optimization, 54 (2016), 819-844.
doi: 10.1137/141000415. |
[4] |
L. Bourdin and E. Trélat,
Optimal sampled-data control, and generalizations on time scales, Mathematical Control and Related Fields, 6 (2016), 53-94.
doi: 10.3934/mcrf.2016.6.53. |
[5] |
E. Casas, J. C. de los Reyes and F. Tröltzsch,
Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616-643.
doi: 10.1137/07068240X. |
[6] |
E. Casas and K. Kunisch,
Stabilization by sparse controls for a class of semilinear parabolic equations, SIAM J. Control Optim., 55 (2017), 512-532.
doi: 10.1137/16M1084298. |
[7] |
E. Casas and F. Tröltzsch,
Second-order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM Journal on Optimization, 13 (2002), 406-431.
doi: 10.1137/S1052623400367698. |
[8] |
E. Casas and F. Tröltzsch,
Second order optimality conditions and their role in PDE control, Jahresber. Dtsch. Math.-Ver., 117 (2015), 3-44.
doi: 10.1365/s13291-014-0109-3. |
[9] |
C. Clason, A. Rund and K. Kunisch,
Nonconvex penalization of switching control of partial differential equations, Systems Control Lett., 106 (2017), 1-8.
doi: 10.1016/j.sysconle.2017.05.006. |
[10] |
C. Clason, A. Rund, K. Kunisch and R. C. Barnard,
A convex penalty for switching control of partial differential equations, Systems & Control Letters, 89 (2016), 66-73.
doi: 10.1016/j.sysconle.2015.12.013. |
[11] |
S. Court, K. Kunisch and L. Pfeiffer,
Optimal control for a class of infinite dimensional systems involving an $L^∞$-term in the cost functional, Z. Angew. Math. Mech., 98 (2018), 569-588.
doi: 10.1002/zamm.201600199. |
[12] |
J. C. Dunn, On second order sufficient conditions for structured nonlinear programs in infinitedimensional function spaces, In Mathematical programming with data perturbations, volume 195 of Lecture Notes in Pure and Appl. Math., pages 83–107. Dekker, New York, 1998. |
[13] |
H. O. Fattorini, Invariance of the hamiltonian in control problems for semilinear parabolic distributed parameter systems, Control and estimation of distributed parameter systems: nonlinear phenomena (Vorau, 1993), 115–130, Internat. Ser. Numer. Math., 118, Birkhäuser, Basel, 1994. |
[14] |
M. Garavello and B. Piccoli,
Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887.
doi: 10.1137/S0363012903416219. |
[15] |
M. Heinkenschloss,
The numerical solution of a control problem governed by a phase field model, Optim. Methods Softw., 7 (1997), 211-263.
doi: 10.1080/10556789708805656. |
[16] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, volume 23 of Mathematical Modelling: Theory and Applications, Springer, New York, 2009. |
[17] |
B. Hu and J. Yong,
Pontryagin maximum principle for semilinear and quasilinear parabolic equations with pointwise state constraints, SIAM Journal on Control and Optimization, 33 (1995), 1857-1880.
doi: 10.1137/S0363012993250074. |
[18] |
K. Ito and K. Kunisch,
Semismooth Newton methods for time-optimal control for a class of ODEs, SIAM J. Control Optim., 48 (2010), 3997-4013.
doi: 10.1137/090753905. |
[19] |
K. Kunisch, K. Pieper and A. Rund,
Time optimal control for a reaction diffusion system arising in cardiac electrophysiology -- a monolithic approach, ESAIM: M2AN, 50 (2016), 381-414.
doi: 10.1051/m2an/2015048. |
[20] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R. I., 1968. |
[21] |
L. Li, Y. Gao and H. Wang,
Second order sufficient optimality conditions for hybrid control problems with state jump, J. Ind. Manag. Optim., 11 (2015), 329-343.
doi: 10.3934/jimo.2015.11.329. |
[22] |
J. Merger, A. Borzí and R. Herzog,
Optimal control of a system of reaction-diffusion equations modeling the wine fermentation process, Optimal Control Applications and Methods, 38 (2017), 112-132.
doi: 10.1002/oca.2246. |
[23] |
M. Raydan,
The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim., 7 (1997), 26-33.
doi: 10.1137/S1052623494266365. |
[24] |
J.-P. Raymond and F. Tröltzsch,
Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints, Discrete Contin. Dynam. Systems, 6 (2000), 431-450.
doi: 10.3934/dcds.2000.6.431. |
[25] |
J. P. Raymond and H. Zidani,
Pontryagin's principle for time-optimal problems, J. Optim. Theory Appl., 101 (1999), 375-402.
doi: 10.1023/A:1021793611520. |
[26] |
J. P. Raymond and H. Zidani,
Time optimal problems with boundary controls, Differential Integral Equations, 13 (2000), 1039-1072.
|
[27] |
F. Rüffler and F. M. Hante,
Optimal switching for hybrid semilinear evolutions, Nonlinear Analysis: Hybrid Systems, 22 (2016), 215-227.
doi: 10.1016/j.nahs.2016.05.001. |
[28] |
F. J. Silva,
Second order analysis for the optimal control of parabolic equations under control and final state constraints, Set-Valued and Variational Analysis, 24 (2016), 57-81.
doi: 10.1007/s11228-015-0337-4. |
[29] |
F. Tröltzsch, Optimal Control of Partial Differential Equations, volume 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels.
doi: 10.1090/gsm/112. |
show all references
References:
[1] |
S. Aniţa, V. Arnăutu and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser/Springer, New York, 2011. From mathematical models to numerical simulation with MATLAB®.
doi: 10.1007/978-0-8176-8098-5. |
[2] |
W. Barthel, C. John and F. Tröltzsch,
Optimal boundary control of a system of reaction diffusion equations, ZAMM Z. Angew. Math. Mech., 90 (2010), 966-982.
doi: 10.1002/zamm.200900359. |
[3] |
T. Bayen and F. J. Silva,
Second order analysis for strong solutions in the optimal control of parabolic equations, SIAM Journal on Control and Optimization, 54 (2016), 819-844.
doi: 10.1137/141000415. |
[4] |
L. Bourdin and E. Trélat,
Optimal sampled-data control, and generalizations on time scales, Mathematical Control and Related Fields, 6 (2016), 53-94.
doi: 10.3934/mcrf.2016.6.53. |
[5] |
E. Casas, J. C. de los Reyes and F. Tröltzsch,
Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616-643.
doi: 10.1137/07068240X. |
[6] |
E. Casas and K. Kunisch,
Stabilization by sparse controls for a class of semilinear parabolic equations, SIAM J. Control Optim., 55 (2017), 512-532.
doi: 10.1137/16M1084298. |
[7] |
E. Casas and F. Tröltzsch,
Second-order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM Journal on Optimization, 13 (2002), 406-431.
doi: 10.1137/S1052623400367698. |
[8] |
E. Casas and F. Tröltzsch,
Second order optimality conditions and their role in PDE control, Jahresber. Dtsch. Math.-Ver., 117 (2015), 3-44.
doi: 10.1365/s13291-014-0109-3. |
[9] |
C. Clason, A. Rund and K. Kunisch,
Nonconvex penalization of switching control of partial differential equations, Systems Control Lett., 106 (2017), 1-8.
doi: 10.1016/j.sysconle.2017.05.006. |
[10] |
C. Clason, A. Rund, K. Kunisch and R. C. Barnard,
A convex penalty for switching control of partial differential equations, Systems & Control Letters, 89 (2016), 66-73.
doi: 10.1016/j.sysconle.2015.12.013. |
[11] |
S. Court, K. Kunisch and L. Pfeiffer,
Optimal control for a class of infinite dimensional systems involving an $L^∞$-term in the cost functional, Z. Angew. Math. Mech., 98 (2018), 569-588.
doi: 10.1002/zamm.201600199. |
[12] |
J. C. Dunn, On second order sufficient conditions for structured nonlinear programs in infinitedimensional function spaces, In Mathematical programming with data perturbations, volume 195 of Lecture Notes in Pure and Appl. Math., pages 83–107. Dekker, New York, 1998. |
[13] |
H. O. Fattorini, Invariance of the hamiltonian in control problems for semilinear parabolic distributed parameter systems, Control and estimation of distributed parameter systems: nonlinear phenomena (Vorau, 1993), 115–130, Internat. Ser. Numer. Math., 118, Birkhäuser, Basel, 1994. |
[14] |
M. Garavello and B. Piccoli,
Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887.
doi: 10.1137/S0363012903416219. |
[15] |
M. Heinkenschloss,
The numerical solution of a control problem governed by a phase field model, Optim. Methods Softw., 7 (1997), 211-263.
doi: 10.1080/10556789708805656. |
[16] |
M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, volume 23 of Mathematical Modelling: Theory and Applications, Springer, New York, 2009. |
[17] |
B. Hu and J. Yong,
Pontryagin maximum principle for semilinear and quasilinear parabolic equations with pointwise state constraints, SIAM Journal on Control and Optimization, 33 (1995), 1857-1880.
doi: 10.1137/S0363012993250074. |
[18] |
K. Ito and K. Kunisch,
Semismooth Newton methods for time-optimal control for a class of ODEs, SIAM J. Control Optim., 48 (2010), 3997-4013.
doi: 10.1137/090753905. |
[19] |
K. Kunisch, K. Pieper and A. Rund,
Time optimal control for a reaction diffusion system arising in cardiac electrophysiology -- a monolithic approach, ESAIM: M2AN, 50 (2016), 381-414.
doi: 10.1051/m2an/2015048. |
[20] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R. I., 1968. |
[21] |
L. Li, Y. Gao and H. Wang,
Second order sufficient optimality conditions for hybrid control problems with state jump, J. Ind. Manag. Optim., 11 (2015), 329-343.
doi: 10.3934/jimo.2015.11.329. |
[22] |
J. Merger, A. Borzí and R. Herzog,
Optimal control of a system of reaction-diffusion equations modeling the wine fermentation process, Optimal Control Applications and Methods, 38 (2017), 112-132.
doi: 10.1002/oca.2246. |
[23] |
M. Raydan,
The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim., 7 (1997), 26-33.
doi: 10.1137/S1052623494266365. |
[24] |
J.-P. Raymond and F. Tröltzsch,
Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints, Discrete Contin. Dynam. Systems, 6 (2000), 431-450.
doi: 10.3934/dcds.2000.6.431. |
[25] |
J. P. Raymond and H. Zidani,
Pontryagin's principle for time-optimal problems, J. Optim. Theory Appl., 101 (1999), 375-402.
doi: 10.1023/A:1021793611520. |
[26] |
J. P. Raymond and H. Zidani,
Time optimal problems with boundary controls, Differential Integral Equations, 13 (2000), 1039-1072.
|
[27] |
F. Rüffler and F. M. Hante,
Optimal switching for hybrid semilinear evolutions, Nonlinear Analysis: Hybrid Systems, 22 (2016), 215-227.
doi: 10.1016/j.nahs.2016.05.001. |
[28] |
F. J. Silva,
Second order analysis for the optimal control of parabolic equations under control and final state constraints, Set-Valued and Variational Analysis, 24 (2016), 57-81.
doi: 10.1007/s11228-015-0337-4. |
[29] |
F. Tröltzsch, Optimal Control of Partial Differential Equations, volume 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. Theory, methods and applications, Translated from the 2005 German original by Jürgen Sprekels.
doi: 10.1090/gsm/112. |
[1] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 |
[2] |
Nguyen Anh Tuan, Donal O'Regan, Dumitru Baleanu, Nguyen H. Tuan. On time fractional pseudo-parabolic equations with nonlocal integral conditions. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020109 |
[3] |
Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020045 |
[4] |
Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052 |
[5] |
Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 |
[6] |
Nguyen Huy Tuan, Vo Van Au, Runzhang Xu. Semilinear Caputo time-fractional pseudo-parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020282 |
[7] |
Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061 |
[8] |
Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020046 |
[9] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[10] |
Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179 |
[11] |
Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020383 |
[12] |
Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267 |
[13] |
Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020398 |
[14] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[15] |
Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032 |
[16] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
[17] |
Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020053 |
[18] |
Juliana Fernandes, Liliane Maia. Blow-up and bounded solutions for a semilinear parabolic problem in a saturable medium. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1297-1318. doi: 10.3934/dcds.2020318 |
[19] |
Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020055 |
[20] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]