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December  2018, 11(6): 1031-1060. doi: 10.3934/dcdss.2018060

Hybrid optimal control problems for a class of semilinear parabolic equations

Sébastien Court 1, , Karl Kunisch 1,2, and  Laurent Pfeiffer 1,,

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

* Corresponding author: Laurent Pfeiffer

Received  November 2016 Revised  May 2017 Published  June 2018

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 and 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.

Figure 1.  Values of the state $y_1$ and the control $u_1$, for different values of the time.
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