-
Previous Article
Weak solutions of stochastic reaction diffusion equations and their optimal control
- DCDS-S Home
- This Issue
-
Next Article
On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations
December
2018, 11(6): 1031-1060.
doi: 10.3934/dcdss.2018060
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.
Keywords:
Hybrid optimal control problems,
semilinear parabolic equations,
PDE-constrained optimization,
optimality conditions,
transversality conditions.
Mathematics Subject Classification:
Primary: 49K20, 93C30; Secondary: 35K58.
Citation:
Sébastien Court, Karl Kunisch, Laurent Pfeiffer. Hybrid optimal control problems for a class of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - S,
2018, 11
(6)
: 1031-1060.
doi: 10.3934/dcdss.2018060
![]()
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] |
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 |
| [2] |
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 |
| [3] |
Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791 |
| [4] |
Savin Treanţă. Characterization of efficient solutions for a class of PDE-constrained vector control problems. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019035 |
| [5] |
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 |
| [6] |
Yong Xia. New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problems. Journal of Industrial & Management Optimization, 2009, 5 (4) : 881-892. doi: 10.3934/jimo.2009.5.881 |
| [7] |
Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483 |
| [8] |
Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014 |
| [9] |
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 |
| [10] |
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 |
| [11] |
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 |
| [12] |
Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018 |
| [13] |
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 |
| [14] |
J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431 |
| [15] |
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 |
| [16] |
Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial & Management Optimization, 2018, 14 (3) : 1041-1054. doi: 10.3934/jimo.2017089 |
| [17] |
Juan Carlos De los Reyes, Carola-Bibiane Schönlieb. Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization. Inverse Problems & Imaging, 2013, 7 (4) : 1183-1214. doi: 10.3934/ipi.2013.7.1183 |
| [18] |
Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003 |
| [19] |
Ziye Shi, Qingwei Jin. Second order optimality conditions and reformulations for nonconvex quadratically constrained quadratic programming problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 871-882. doi: 10.3934/jimo.2014.10.871 |
| [20] |
Tan Bui-Thanh, Quoc P. Nguyen. FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems. Inverse Problems & Imaging, 2016, 10 (4) : 943-975. doi: 10.3934/ipi.2016028 |
2017 Impact Factor: 0.561
Tools
Metrics
Other articles
by authors
[Back to Top]