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December  2020, 9(4): 1027-1040. doi: 10.3934/eect.2020050

Relaxation of optimal control problems driven by nonlinear evolution equations

Leszek Gasiński 1,, and  Nikolaos S. Papageorgiou 2,

1. 

Pedagogical University of Krakow, Department of Mathematics, 2, 30-084 Kraków, Poland
2. 

National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

* Corresponding author

Dedicated to Professor Meir Shillor on the occasion of his 70th birthday

Received  October 2019 Published  March 2020

We consider a nonlinear optimal control problem with dynamics described by a nonlinear evolution equation defined on an evolution triple of spaces. Both the dynamics and the cost functional are not convex and so an optimal pair need not exist. For this reason using tools from multivalued analysis and from convex analysis, we introduce a relaxed version of the problem. No Young measures are involved in our relaxation method. We show that the relaxed problem is admissible.

Keywords: Evolution triple, admissible relaxation, weak norm, optimal pair, cost functional.
Mathematics Subject Classification: 49J20, 49J35, 34G20.
Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. Relaxation of optimal control problems driven by nonlinear evolution equations. Evolution Equations & Control Theory, 2020, 9 (4) : 1027-1040. doi: 10.3934/eect.2020050
References:
[1]

N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, North-Holland Publishing Co., New York-Amsterdam, 1981.  Google Scholar

[2]

G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics Series, 207. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar

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G. Buttazzo and G. Dal Maso, $\Gamma $-convergence and optimal control problems, J. Optim. Theory Appl., 38 (1982), 385-407.  doi: 10.1007/BF00935345.  Google Scholar

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L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics (New York), 17. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

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H. O. Fattorini, Optimal control problems for distributed parameter systems in Banach spaces, Appl. Math. Optim., 28 (1993), 225-257.  doi: 10.1007/BF01200380.  Google Scholar

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H. O. Fattorini, Existence theory and the maximum principle for relaxed infinite-dimensional optimal control problems, SIAM J. Control Optim., 32 (1994), 311-331.  doi: 10.1137/S0363012991220244.  Google Scholar

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H. O. Fattorini, Relaxed controls, differential inclusions, existence theorems, and the maximum principle in nonlinear infinite-dimensional control theory, Evolution Equations, Control Theory, and Biomathematics, Lecture Notes in Pure and Appl. Math., Dekker, New York, 155 (1994), 185-204.   Google Scholar

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H. O. Fattorini, Relaxation theorems, differential inclusions, and Filippov's theorem for relaxed controls in semilinear infinite-dimensional systems, J. Differential Equations, 112 (1994), 131-153.  doi: 10.1006/jdeq.1994.1097.  Google Scholar

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L.Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Series in Mathematical Analysis and Applications, 8. Chapman & Hall/CRC, Boca Raton, FL, 2005.  Google Scholar

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L.Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 1. Problem Books in Mathematics, Springer, Cham, 2014.  Google Scholar

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L.Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2. Nonlinear Analysis, Springer, Cham, 2016. doi: 10.1007/978-3-319-27817-9.  Google Scholar

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S. H. Hou, Existence theorems of optimal control problems in Banach spaces, Nonlinear Anal., 7 (1983), 239-257.  doi: 10.1016/0362-546X(83)90069-X.  Google Scholar

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S. C. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. I. Theory, Mathematics and its Applications, 419. Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.  Google Scholar

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J. Kreulich, Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings, Adv. Nonlinear Anal., 8 (2019), 1-28.  doi: 10.1515/anona-2016-0075.  Google Scholar

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K. L. Kuttler, J. Li and M. Shillor, A general product measurability theorem with applications to variational inequalities, Electron. J. Differential Equations, 2016 (2016), 12 pp.  Google Scholar

[21]

X. Y. Liu, Z. H. Liu and X. Fu, Relaxation in nonconvex optimal control problems described by fractional differential equations, J. Math. Anal. Appl., 409 (2014), 446-458.  doi: 10.1016/j.jmaa.2013.07.032.  Google Scholar

[22]

N. S. Papageorgiou, Optimal control of nonlinear evolution inclusions, J. Optim. Theory Appl., 67 (1990), 321-354.  doi: 10.1007/BF00940479.  Google Scholar

[23]

N. S. Papageorgiou, Existence theory for nonlinear distributed parameter optimal control problems, Japan J. Indust. Appl. Math., 12 (1995), 457-485.  doi: 10.1007/BF03167239.  Google Scholar

[24]

N. S. Papageorgiou, On parametric evolution inclusions of the subdifferential type with applications to optimal control problems, Trans. Amer. Math. Soc., 347 (1995), 203-231.  doi: 10.1090/S0002-9947-1995-1282896-X.  Google Scholar

[25]

N. S. Papageorgiou, Optimal control and admissible relaxation of uncertain nonlinear elliptic systems, J. Math. Anal. Appl., 197 (1996), 27-41.  doi: 10.1006/jmaa.1996.0004.  Google Scholar

[26]

N. S. Papageorgiou and F. Papalini, Existence and relaxation for finite-dimensional optimal control problems driven by maximal monotone operators, Z. Anal. Anwendungen, 22 (2003), 863-898.  doi: 10.4171/ZAA/1177.  Google Scholar

[27]

N. S. Papageorgiou and V. D. Rǎdulescu, Periodic solutions for time-dependent subdifferential evolution inclusions, Evol. Equ. Control Theory, 6 (2017), 277-297.  doi: 10.3934/eect.2017015.  Google Scholar

[28]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions, Adv. Nonlinear Anal., 6 (2017), 199-235.  doi: 10.1515/anona-2016-0096.  Google Scholar

[29]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear second order evolution inclusions with noncoercive viscosity term, J. Differential Equations, 264 (2018), 4749-4763.  doi: 10.1016/j.jde.2017.12.022.  Google Scholar

[30]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Periodic solutions for a class of evolution inclusions, Comput. Math. Appl., 75 (2018), 3047-3065.  doi: 10.1016/j.camwa.2018.01.031.  Google Scholar

[31]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Periodic solutions for implicit evolution inclusions, Evol. Equ. Control Theory, 8 (2019), 621-631.  doi: 10.3934/eect.2019029.  Google Scholar

[32]

N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovš, Nonlinear Analysis - Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar

[33]

N. S. Papageorgiou, C. Vetro and F. Vetro, Relaxation for a class of control systems with unilateral constraints, Acta Appl. Math., publisher online, http://dx.doi.org/10.1007/s10440-019-00270-4. Google Scholar

[34]

T. Roubíček, Relaxation in Optimization Theory and Variational Calculus, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 4, Walter de Gruyter & Co., Berlin, 1997.  Google Scholar

show all references

References:
[1]

N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Systems, North-Holland Publishing Co., New York-Amsterdam, 1981.  Google Scholar

[2]

G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Mathematics Series, 207. Longman Scientific & Technical, Harlow, copublished in the United States with John Wiley & Sons, Inc., New York, 1989.  Google Scholar

[3]

G. Buttazzo and G. Dal Maso, $\Gamma $-convergence and optimal control problems, J. Optim. Theory Appl., 38 (1982), 385-407.  doi: 10.1007/BF00935345.  Google Scholar

[4]

L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics (New York), 17. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[5] L. Egghe, Stopping Time Techniques for Analysts and Probabilists, London Mathematical Society Lecture Note Series, 100. Cambridge University Press, Cambridge, 1984.  doi: 10.1017/CBO9780511526176.  Google Scholar
[6]

I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999. doi: 10.1137/1.9781611971088.  Google Scholar

[7]

H. O. Fattorini, Optimal control problems for distributed parameter systems in Banach spaces, Appl. Math. Optim., 28 (1993), 225-257.  doi: 10.1007/BF01200380.  Google Scholar

[8]

H. O. Fattorini, Existence theory and the maximum principle for relaxed infinite-dimensional optimal control problems, SIAM J. Control Optim., 32 (1994), 311-331.  doi: 10.1137/S0363012991220244.  Google Scholar

[9]

H. O. Fattorini, Relaxed controls, differential inclusions, existence theorems, and the maximum principle in nonlinear infinite-dimensional control theory, Evolution Equations, Control Theory, and Biomathematics, Lecture Notes in Pure and Appl. Math., Dekker, New York, 155 (1994), 185-204.   Google Scholar

[10]

H. O. Fattorini, Relaxation theorems, differential inclusions, and Filippov's theorem for relaxed controls in semilinear infinite-dimensional systems, J. Differential Equations, 112 (1994), 131-153.  doi: 10.1006/jdeq.1994.1097.  Google Scholar

[11] H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications, 62. Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511574795.  Google Scholar
[12]

L.Gasiński and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Series in Mathematical Analysis and Applications, 8. Chapman & Hall/CRC, Boca Raton, FL, 2005.  Google Scholar

[13]

L.Gasiński and N. S. Papageorgiou, Nonlinear Analysis, Series in Mathematical Analysis and Applications, 9. Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

[14]

L.Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 1. Problem Books in Mathematics, Springer, Cham, 2014.  Google Scholar

[15]

L.Gasiński and N. S. Papageorgiou, Exercises in Analysis. Part 2. Nonlinear Analysis, Springer, Cham, 2016. doi: 10.1007/978-3-319-27817-9.  Google Scholar

[16]

S. H. Hou, Existence theorems of optimal control problems in Banach spaces, Nonlinear Anal., 7 (1983), 239-257.  doi: 10.1016/0362-546X(83)90069-X.  Google Scholar

[17]

S. C. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. I. Theory, Mathematics and its Applications, 419. Kluwer Academic Publishers, Dordrecht, 1997. doi: 10.1007/978-1-4615-6359-4.  Google Scholar

[18]

S. C. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. Ⅱ. Applications, Mathematics and its Applications, 500. Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4615-4665-8_17.  Google Scholar

[19]

J. Kreulich, Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings, Adv. Nonlinear Anal., 8 (2019), 1-28.  doi: 10.1515/anona-2016-0075.  Google Scholar

[20]

K. L. Kuttler, J. Li and M. Shillor, A general product measurability theorem with applications to variational inequalities, Electron. J. Differential Equations, 2016 (2016), 12 pp.  Google Scholar

[21]

X. Y. Liu, Z. H. Liu and X. Fu, Relaxation in nonconvex optimal control problems described by fractional differential equations, J. Math. Anal. Appl., 409 (2014), 446-458.  doi: 10.1016/j.jmaa.2013.07.032.  Google Scholar

[22]

N. S. Papageorgiou, Optimal control of nonlinear evolution inclusions, J. Optim. Theory Appl., 67 (1990), 321-354.  doi: 10.1007/BF00940479.  Google Scholar

[23]

N. S. Papageorgiou, Existence theory for nonlinear distributed parameter optimal control problems, Japan J. Indust. Appl. Math., 12 (1995), 457-485.  doi: 10.1007/BF03167239.  Google Scholar

[24]

N. S. Papageorgiou, On parametric evolution inclusions of the subdifferential type with applications to optimal control problems, Trans. Amer. Math. Soc., 347 (1995), 203-231.  doi: 10.1090/S0002-9947-1995-1282896-X.  Google Scholar

[25]

N. S. Papageorgiou, Optimal control and admissible relaxation of uncertain nonlinear elliptic systems, J. Math. Anal. Appl., 197 (1996), 27-41.  doi: 10.1006/jmaa.1996.0004.  Google Scholar

[26]

N. S. Papageorgiou and F. Papalini, Existence and relaxation for finite-dimensional optimal control problems driven by maximal monotone operators, Z. Anal. Anwendungen, 22 (2003), 863-898.  doi: 10.4171/ZAA/1177.  Google Scholar

[27]

N. S. Papageorgiou and V. D. Rǎdulescu, Periodic solutions for time-dependent subdifferential evolution inclusions, Evol. Equ. Control Theory, 6 (2017), 277-297.  doi: 10.3934/eect.2017015.  Google Scholar

[28]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Sensitivity analysis for optimal control problems governed by nonlinear evolution inclusions, Adv. Nonlinear Anal., 6 (2017), 199-235.  doi: 10.1515/anona-2016-0096.  Google Scholar

[29]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear second order evolution inclusions with noncoercive viscosity term, J. Differential Equations, 264 (2018), 4749-4763.  doi: 10.1016/j.jde.2017.12.022.  Google Scholar

[30]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Periodic solutions for a class of evolution inclusions, Comput. Math. Appl., 75 (2018), 3047-3065.  doi: 10.1016/j.camwa.2018.01.031.  Google Scholar

[31]

N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Periodic solutions for implicit evolution inclusions, Evol. Equ. Control Theory, 8 (2019), 621-631.  doi: 10.3934/eect.2019029.  Google Scholar

[32]

N. S. Papageorgiou, V. D. Rǎdulescu and D. Repovš, Nonlinear Analysis - Theory and Methods, Springer Monographs in Mathematics, Springer, Cham, 2019. doi: 10.1007/978-3-030-03430-6.  Google Scholar

[33]

N. S. Papageorgiou, C. Vetro and F. Vetro, Relaxation for a class of control systems with unilateral constraints, Acta Appl. Math., publisher online, http://dx.doi.org/10.1007/s10440-019-00270-4. Google Scholar

[34]

T. Roubíček, Relaxation in Optimization Theory and Variational Calculus, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 4, Walter de Gruyter & Co., Berlin, 1997.  Google Scholar

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