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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

 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.

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] M. Garavello and B. Piccoli, Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887. doi: 10.1137/S0363012903416219. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [26] J. P. Raymond and H. Zidani, Time optimal problems with boundary controls, Differential Integral Equations, 13 (2000), 1039-1072. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [14] M. Garavello and B. Piccoli, Hybrid necessary principle, SIAM J. Control Optim., 43 (2005), 1867-1887. doi: 10.1137/S0363012903416219. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [26] J. P. Raymond and H. Zidani, Time optimal problems with boundary controls, Differential Integral Equations, 13 (2000), 1039-1072. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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