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  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, 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

[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]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[2]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[3]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[4]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[5]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[6]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020271

[7]

Reza Lotfi, Zahra Yadegari, Seyed Hossein Hosseini, Amir Hossein Khameneh, Erfan Babaee Tirkolaee, Gerhard-Wilhelm Weber. A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020158

[8]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[9]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[10]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

[11]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[12]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[13]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044

[14]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[15]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[16]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[17]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (146)
  • HTML views (352)
  • Cited by (0)

Other articles
by authors

[Back to Top]