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

Relaxation of optimal control problems driven by nonlinear evolution equations

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  December 2020 Early access  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.

Citation: Leszek Gasiński, Nikolaos S. Papageorgiou. Relaxation of optimal control problems driven by nonlinear evolution equations. Evolution Equations and 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.

[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.

[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.

[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.

[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.
[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.

[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.

[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.

[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. 

[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.

[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.
[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

X. Y. LiuZ. 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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[28]

N. S. PapageorgiouV. 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.

[29]

N. S. PapageorgiouV. 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.

[30]

N. S. PapageorgiouV. 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.

[31]

N. S. PapageorgiouV. 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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.
[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.

[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.

[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.

[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. 

[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.

[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.
[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.

[13]

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

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[21]

X. Y. LiuZ. 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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[28]

N. S. PapageorgiouV. 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.

[29]

N. S. PapageorgiouV. 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.

[30]

N. S. PapageorgiouV. 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.

[31]

N. S. PapageorgiouV. 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.

[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.

[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.

[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.

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