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August  2022, 11(4): 1357-1372. doi: 10.3934/eect.2021047

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 Published  August 2022 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 and Control Theory, 2022, 11 (4) : 1357-1372. 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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[12]

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

[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.

[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.

[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.

[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.

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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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. 

[12]

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

[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.

[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.

[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.

[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.

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