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  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 & Control Theory, 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

[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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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

[1]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115

[2]

Kerem Uǧurlu. Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D. Communications on Pure & Applied Analysis, 2017, 16 (1) : 189-208. doi: 10.3934/cpaa.2017009

[3]

Jingzhi Tie, Qing Zhang. Switching between a pair of stocks: An optimal trading rule. Mathematical Control & Related Fields, 2018, 8 (3&4) : 965-999. doi: 10.3934/mcrf.2018042

[4]

Giuseppe Buttazzo, Serena Guarino Lo Bianco, Fabrizio Oliviero. Optimal location problems with routing cost. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1301-1317. doi: 10.3934/dcds.2014.34.1301

[5]

Tomáš Roubíček. On certain convex compactifications for relaxation in evolution problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 467-482. doi: 10.3934/dcdss.2011.4.467

[6]

Nguyen Thieu Huy, Pham Van Bang. Invariant stable manifolds for partial neutral functional differential equations in admissible spaces on a half-line. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2993-3011. doi: 10.3934/dcdsb.2015.20.2993

[7]

Wei Xu, Liying Yu, Gui-Hua Lin, Zhi Guo Feng. Optimal switching signal design with a cost on switching action. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019068

[8]

Liman Dai, Xingfu Zou. Effects of superinfection and cost of immunity on host-parasite co-evolution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 809-829. doi: 10.3934/dcdsb.2017040

[9]

Antonio Azzollini. On a functional satisfying a weak Palais-Smale condition. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1829-1840. doi: 10.3934/dcds.2014.34.1829

[10]

Donglei Du, Xiaoyue Jiang, Guochuan Zhang. Optimal preemptive online scheduling to minimize lp norm on two processors. Journal of Industrial & Management Optimization, 2005, 1 (3) : 345-351. doi: 10.3934/jimo.2005.1.345

[11]

Donghui Yang, Jie Zhong. Optimal actuator location of the minimum norm controls for stochastic heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1081-1095. doi: 10.3934/mcrf.2018046

[12]

Julius Fergy T. Rabago, Hideyuki Azegami. A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem. Evolution Equations & Control Theory, 2019, 8 (4) : 785-824. doi: 10.3934/eect.2019038

[13]

Yujing Wang, Changjun Yu, Kok Lay Teo. A new computational strategy for optimal control problem with a cost on changing control. Numerical Algebra, Control & Optimization, 2016, 6 (3) : 339-364. doi: 10.3934/naco.2016016

[14]

Shuren Liu, Qiying Hu, Yifan Xu. Optimal inventory control with fixed ordering cost for selling by internet auctions. Journal of Industrial & Management Optimization, 2012, 8 (1) : 19-40. doi: 10.3934/jimo.2012.8.19

[15]

Sebastian Albrecht, Marion Leibold, Michael Ulbrich. A bilevel optimization approach to obtain optimal cost functions for human arm movements. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 105-127. doi: 10.3934/naco.2012.2.105

[16]

José C. Bellido, Pablo Pedregal. Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 967-982. doi: 10.3934/dcds.2002.8.967

[17]

Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605

[18]

Nobuyuki Kato. Linearized stability and asymptotic properties for abstract boundary value functional evolution problems. Conference Publications, 1998, 1998 (Special) : 371-387. doi: 10.3934/proc.1998.1998.371

[19]

Jordi-Lluís Figueras, Àlex Haro. Triple collisions of invariant bundles. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2069-2082. doi: 10.3934/dcdsb.2013.18.2069

[20]

Wenyan Zhang, Shu Xu, Shengji Li, Xuexiang Huang. Generalized weak sharp minima of variational inequality problems with functional constraints. Journal of Industrial & Management Optimization, 2013, 9 (3) : 621-630. doi: 10.3934/jimo.2013.9.621

2019 Impact Factor: 0.953

Article outline

[Back to Top]