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doi: 10.3934/eect.2021047
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Solution stability to parametric distributed optimal control problems with finite unilateral constraints

School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No.1 Dai Co Viet, Hanoi, Vietnam

* Corresponding author: Nguyen Hai Son

Received  March 2021 Revised  June 2021 Early access August 2021

Fund Project: The research is funded by Vietnam National Foundation for Scienceand Technology Development (NAFOSTED) under Grant Number 101.01-2019.308

This paper deals with stability of solution map to a parametric control problem governed by semilinear elliptic equations with finite unilateral constraints, where the objective functional is not convex. By using the first-order necessary optimality conditions, we derive some sufficient conditions under which the solution map is upper semicontinuous with respect to parameters.

Citation: Nguyen Hai Son. Solution stability to parametric distributed optimal control problems with finite unilateral constraints. Evolution Equations & Control Theory, doi: 10.3934/eect.2021047
References:
[1]

W. AltR. GriesseN. Metla and A. Rösch, Lipschitz stability for elliptic optimal control problems with mixed control-state contraints, Optimization, 59 (2010), 833-849.  doi: 10.1080/02331930902863749.  Google Scholar

[2]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, 1990.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010.  Google Scholar

[4]

E. Casas and F. Tröltzsch, First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations, SIAM J. Control and Optim., 48 (2009), 688-718.  doi: 10.1137/080720048.  Google Scholar

[5]

B. Dacorogna, Direct Methods in Calculus of Variations, Springer-Verlag Berlin Heidelberg, 1989. doi: 10.1007/978-3-642-51440-1.  Google Scholar

[6]

R. Griesse, Lipschitz stability of solutions to some state-constrained elliptic optimal control problems, Z. Anal. Anwend., 25 (2006), 435-455.  doi: 10.4171/ZAA/1300.  Google Scholar

[7]

B. T. Kien and V. H. Nhu, Second-order necessary optimality conditions for a class of semilinear elliptic optimal control problems with mixed pointwise constraints, SIAM J. Control Optim., 52 (2014), 1166-1202.  doi: 10.1137/130917570.  Google Scholar

[8]

B. T. KienV. H. Nhu and A. Rösch, Lower semicontinuous of the solution map to a parametric elliptic optimal control problem with mixed pointwise constraints, Optim., 64 (2015), 1219-1238.  doi: 10.1080/02331934.2013.853060.  Google Scholar

[9]

B. T. KienV. H. Nhu and N. H. Son, Second-order optimality conditions for a semilinear elliptic optimal control problem with mixed pointwise constraints, Set-Valued Var. Anal., 25 (2017), 177-210.  doi: 10.1007/s11228-016-0373-8.  Google Scholar

[10]

B. T. KienN. Q. TuanC.-F Wen and J.-C. Yao, $L^\infty $-Stability of a parametric optimal control problem governed by semilinear elliptic equations, Appl. Math. and Optim., 84 (2021), 849-876.  doi: 10.1007/s00245-020-09664-5.  Google Scholar

[11]

B. T. Kien and J.-C. Yao, Semicontinuity of the solution map to a parametric optimal control problem, Appl. Anal. and Optim., 2 (2018), 93-116.   Google Scholar

[12]

K. Malanowski and F. Tröltzsch, Lipschitz stability of solutions to parametric optimal control for elliptic equations, Control Cybern., 29 (2000), 237-256.   Google Scholar

[13]

V. H. NhuN. H. Anh and B. T. Kien, Hölder continuity of the solution map to an elliptic optimal control problem with mixed control-state constraints,, Taiwanese J. Math., 17 (2013), 1245-1266.  doi: 10.11650/tjm.17.2013.2617.  Google Scholar

[14]

N. H. Son, On the semicontinuity of the solution map to a parametric boundary control problem, Optim., 66 (2017), 311-329.  doi: 10.1080/02331934.2016.1274990.  Google Scholar

[15]

N. H. Son and T. A. Dao, Upper semicontinuity of the solution map to a parametric boundary optimal control problem with unbounded constraint sets, Submitted, http://arXiv.org/abs/2012.15196. Google Scholar

[16]

N. H. Son and N. B. Giang, Upper semicontinuity of the solution map to a parametric elliptic optimal control problem, Set-Valued Var. Anal., 29 (2021), 257-282.  doi: 10.1007/s11228-020-00546-0.  Google Scholar

show all references

References:
[1]

W. AltR. GriesseN. Metla and A. Rösch, Lipschitz stability for elliptic optimal control problems with mixed control-state contraints, Optimization, 59 (2010), 833-849.  doi: 10.1080/02331930902863749.  Google Scholar

[2]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, 1990.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010.  Google Scholar

[4]

E. Casas and F. Tröltzsch, First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations, SIAM J. Control and Optim., 48 (2009), 688-718.  doi: 10.1137/080720048.  Google Scholar

[5]

B. Dacorogna, Direct Methods in Calculus of Variations, Springer-Verlag Berlin Heidelberg, 1989. doi: 10.1007/978-3-642-51440-1.  Google Scholar

[6]

R. Griesse, Lipschitz stability of solutions to some state-constrained elliptic optimal control problems, Z. Anal. Anwend., 25 (2006), 435-455.  doi: 10.4171/ZAA/1300.  Google Scholar

[7]

B. T. Kien and V. H. Nhu, Second-order necessary optimality conditions for a class of semilinear elliptic optimal control problems with mixed pointwise constraints, SIAM J. Control Optim., 52 (2014), 1166-1202.  doi: 10.1137/130917570.  Google Scholar

[8]

B. T. KienV. H. Nhu and A. Rösch, Lower semicontinuous of the solution map to a parametric elliptic optimal control problem with mixed pointwise constraints, Optim., 64 (2015), 1219-1238.  doi: 10.1080/02331934.2013.853060.  Google Scholar

[9]

B. T. KienV. H. Nhu and N. H. Son, Second-order optimality conditions for a semilinear elliptic optimal control problem with mixed pointwise constraints, Set-Valued Var. Anal., 25 (2017), 177-210.  doi: 10.1007/s11228-016-0373-8.  Google Scholar

[10]

B. T. KienN. Q. TuanC.-F Wen and J.-C. Yao, $L^\infty $-Stability of a parametric optimal control problem governed by semilinear elliptic equations, Appl. Math. and Optim., 84 (2021), 849-876.  doi: 10.1007/s00245-020-09664-5.  Google Scholar

[11]

B. T. Kien and J.-C. Yao, Semicontinuity of the solution map to a parametric optimal control problem, Appl. Anal. and Optim., 2 (2018), 93-116.   Google Scholar

[12]

K. Malanowski and F. Tröltzsch, Lipschitz stability of solutions to parametric optimal control for elliptic equations, Control Cybern., 29 (2000), 237-256.   Google Scholar

[13]

V. H. NhuN. H. Anh and B. T. Kien, Hölder continuity of the solution map to an elliptic optimal control problem with mixed control-state constraints,, Taiwanese J. Math., 17 (2013), 1245-1266.  doi: 10.11650/tjm.17.2013.2617.  Google Scholar

[14]

N. H. Son, On the semicontinuity of the solution map to a parametric boundary control problem, Optim., 66 (2017), 311-329.  doi: 10.1080/02331934.2016.1274990.  Google Scholar

[15]

N. H. Son and T. A. Dao, Upper semicontinuity of the solution map to a parametric boundary optimal control problem with unbounded constraint sets, Submitted, http://arXiv.org/abs/2012.15196. Google Scholar

[16]

N. H. Son and N. B. Giang, Upper semicontinuity of the solution map to a parametric elliptic optimal control problem, Set-Valued Var. Anal., 29 (2021), 257-282.  doi: 10.1007/s11228-020-00546-0.  Google Scholar

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