American Institute of Mathematical Sciences

Advanced
March  2018, 8(1): 57-88. doi: 10.3934/mcrf.2018003

Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls

Hongwei Lou 1, and  Jiongmin Yong 2,

1. 

School of Mathematical Sciences, and LMNS, Fudan University, Shanghai 200433, China,
2. 

Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

Dedicated to Professor Eduardo Casas at his 60th birthdate

Received  March 2017 Revised  October 2017 Published  January 2018

Fund Project: The first author was supported in part by NSFC grant 11371104, the second author was supported in part by NSF grant DMS-1406776.

An optimal control problem for a semilinear elliptic equation of divergenceform is considered. Both the leading term and the semilinear term of the state equationcontain the control. The well-known Pontryagin type maximum principle for the optimal controls is the first-order necessary condition. When such a first-order necessary condition is singular in some sense, certain type of the second-order necessary condition will come in naturally. The aim of this paper is to explore such kind of conditions for our optimal control problem.

Keywords: Optimal control, semilinear elliptic equation, control in leading term, second-order necessary conditions.
Mathematics Subject Classification: Primary: 49K20, 35J61; Secondary: 35Q93.
Citation: Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. Google Scholar

[2]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. Google Scholar

[3]

J. F. Bonnans and A. Hermant, No-gap second-order optimality conditions for optimal control problems with a single state constraint and control, Math. Program., Ser. B, 117 (2009), 21-50. Google Scholar

[4]

J. F. Bonnans and A. Hermant, Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints, Ann. I. H. Poincare, 26 (2009), 561-598. Google Scholar

[5]

E. Casas, Optimal control in coefficients of elliptic equations with state constraints, Appl. Math. Optim., 26 (1992), 21-37. Google Scholar

[6]

E. Casas, Second order analysis for bang-bang control problems of PDEs, SIAM J. Control Optim., 50 (2012), 2355-2372. Google Scholar

[7]

E. Casas, J. C. de Los Reye and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616-643. Google Scholar

[8]

E. Casas and M. Mateos, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM J. Control Optim., 40 (2002), 1431-1454. Google Scholar

[9]

E. Casas and F. Tröltzsch, Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations, Appl. Math. Optim., 39 (1999), 211-227. Google Scholar

[10]

E. Casas and F. Tröltzsch, Second-order necessary and sufficient optimality conditions for optimality conditions for optimization problems and applications to control theory, SIAM J. Optim., 13 (2002), 406-431. Google Scholar

[11]

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 Optim., 48 (2009), 688-718. Google Scholar

[12]

E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: Improving results expected from abstract theory, SIAM J. Optim., 22 (2012), 261-279. Google Scholar

[13]

E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control, Jahresber Dtsch Math-Ver, 117 (2015), 3-44. Google Scholar

[14]

E. Casas, F. Tröltzsch and A. Unger, Second-order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations, SIAM J. Control Optim., 38 (2000), 1369-1391. Google Scholar

[15]

H. O. Fattorini, Relaxed controls in infinite dimensional systems, International Series of Numerical Mathematics, 100 (1991), 115-128. Google Scholar

[16]

R. Gabasov and F. M. Kirillova, High order necessary conditions for optimality, SIAM J. Control Optim., 10 (1972), 127-168. Google Scholar

[17]

R. Gamkrelidze, Principle of Optimal Control Theory, Plenum Press, New York, 1978. Google Scholar

[18]

H. J. Kelly, A second variation test for singular extremals, AIAA J., 2 (1964), 1380-1382. Google Scholar

[19]

H. W. Knobloch, Higher Order Necessary Conditions in Optimal Control Theory, Lecture Notes in Control & Inform. Sci., Springer-Verlag, New York, 1981. doi: 10.1007/978-1-4612-0873-0. Google Scholar

[20]

R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals, AIAA J., 3 (1965), 1439-1444. Google Scholar

[21]

A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. Control Optim., 15 (1977), 256-293. Google Scholar

[22]

B. Li and H. Lou, Optimality Conditions for semilinear hyperbolic equations with controls in coefficients, Appl. Math. Optim., 65 (2012), 371-402. Google Scholar

[23]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. Google Scholar

[24]

H. Lou, Second-order necessary/sufficient optimality conditions for optimal control problems in the absence of linear structure, Discrete and Continuous Dynamical Systems-Series B, 14 (2010), Special Issue, 1445-1464. Google Scholar

[25]

H. Lou, Optimality conditions for semilinear parabolic equations with controls in leading term, ESAIM: Control, Optimization and Calculus of Variations, 17 (2011), 975-994. Google Scholar

[26]

H. Lou and J. Yong, Optimality conditions for semilinear elliptic equations with leading term containing controls, SIAM J. Control Optim., 48 (2009), 2366-2387. Google Scholar

[27]

H. D. Mittelmann, Verification of second-order sufficient optimality conditions for semilinear elliptic and parabolic control problems, Comp. Optim. Appl., 20 (2001), 93-110. Google Scholar

[28]

J. P. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints, Discrete Contin. Dynam. Systems, 6 (2000), 431-450. Google Scholar

[29]

A. Rösch and F. Tröltzsch, Sufficient second-order optimality conditions for a parabolic optimal control problem with pointwise control-state constraints, SIAM J. Control Optim., 42 (2003), 138-154. Google Scholar

[30]

A. Rösch and F. Tröltzsch, Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints, SIAM J. Optim., 17 (2006), 776-794. Google Scholar

[31]

L. Wang and P. He, Second-order optimality conditions for optimal control problems governed by 3-dimensional Nevier-Stokes equations, Acta Math. Scientia, 26 (2006), 729-734. Google Scholar

[32]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. Google Scholar

[33]

A. Zygmund, Trigonometric Series, 3$^{rd}$ edition, Cambridge University Press, Cambridge, 2002. Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. Google Scholar

[2]

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. Google Scholar

[3]

J. F. Bonnans and A. Hermant, No-gap second-order optimality conditions for optimal control problems with a single state constraint and control, Math. Program., Ser. B, 117 (2009), 21-50. Google Scholar

[4]

J. F. Bonnans and A. Hermant, Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints, Ann. I. H. Poincare, 26 (2009), 561-598. Google Scholar

[5]

E. Casas, Optimal control in coefficients of elliptic equations with state constraints, Appl. Math. Optim., 26 (1992), 21-37. Google Scholar

[6]

E. Casas, Second order analysis for bang-bang control problems of PDEs, SIAM J. Control Optim., 50 (2012), 2355-2372. Google Scholar

[7]

E. Casas, J. C. de Los Reye and F. Tröltzsch, Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints, SIAM J. Optim., 19 (2008), 616-643. Google Scholar

[8]

E. Casas and M. Mateos, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM J. Control Optim., 40 (2002), 1431-1454. Google Scholar

[9]

E. Casas and F. Tröltzsch, Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations, Appl. Math. Optim., 39 (1999), 211-227. Google Scholar

[10]

E. Casas and F. Tröltzsch, Second-order necessary and sufficient optimality conditions for optimality conditions for optimization problems and applications to control theory, SIAM J. Optim., 13 (2002), 406-431. Google Scholar

[11]

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 Optim., 48 (2009), 688-718. Google Scholar

[12]

E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: Improving results expected from abstract theory, SIAM J. Optim., 22 (2012), 261-279. Google Scholar

[13]

E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control, Jahresber Dtsch Math-Ver, 117 (2015), 3-44. Google Scholar

[14]

E. Casas, F. Tröltzsch and A. Unger, Second-order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations, SIAM J. Control Optim., 38 (2000), 1369-1391. Google Scholar

[15]

H. O. Fattorini, Relaxed controls in infinite dimensional systems, International Series of Numerical Mathematics, 100 (1991), 115-128. Google Scholar

[16]

R. Gabasov and F. M. Kirillova, High order necessary conditions for optimality, SIAM J. Control Optim., 10 (1972), 127-168. Google Scholar

[17]

R. Gamkrelidze, Principle of Optimal Control Theory, Plenum Press, New York, 1978. Google Scholar

[18]

H. J. Kelly, A second variation test for singular extremals, AIAA J., 2 (1964), 1380-1382. Google Scholar

[19]

H. W. Knobloch, Higher Order Necessary Conditions in Optimal Control Theory, Lecture Notes in Control & Inform. Sci., Springer-Verlag, New York, 1981. doi: 10.1007/978-1-4612-0873-0. Google Scholar

[20]

R. E. Kopp and H. G. Moyer, Necessary conditions for singular extremals, AIAA J., 3 (1965), 1439-1444. Google Scholar

[21]

A. J. Krener, The high order maximal principle and its application to singular extremals, SIAM J. Control Optim., 15 (1977), 256-293. Google Scholar

[22]

B. Li and H. Lou, Optimality Conditions for semilinear hyperbolic equations with controls in coefficients, Appl. Math. Optim., 65 (2012), 371-402. Google Scholar

[23]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. Google Scholar

[24]

H. Lou, Second-order necessary/sufficient optimality conditions for optimal control problems in the absence of linear structure, Discrete and Continuous Dynamical Systems-Series B, 14 (2010), Special Issue, 1445-1464. Google Scholar

[25]

H. Lou, Optimality conditions for semilinear parabolic equations with controls in leading term, ESAIM: Control, Optimization and Calculus of Variations, 17 (2011), 975-994. Google Scholar

[26]

H. Lou and J. Yong, Optimality conditions for semilinear elliptic equations with leading term containing controls, SIAM J. Control Optim., 48 (2009), 2366-2387. Google Scholar

[27]

H. D. Mittelmann, Verification of second-order sufficient optimality conditions for semilinear elliptic and parabolic control problems, Comp. Optim. Appl., 20 (2001), 93-110. Google Scholar

[28]

J. P. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints, Discrete Contin. Dynam. Systems, 6 (2000), 431-450. Google Scholar

[29]

A. Rösch and F. Tröltzsch, Sufficient second-order optimality conditions for a parabolic optimal control problem with pointwise control-state constraints, SIAM J. Control Optim., 42 (2003), 138-154. Google Scholar

[30]

A. Rösch and F. Tröltzsch, Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints, SIAM J. Optim., 17 (2006), 776-794. Google Scholar

[31]

L. Wang and P. He, Second-order optimality conditions for optimal control problems governed by 3-dimensional Nevier-Stokes equations, Acta Math. Scientia, 26 (2006), 729-734. Google Scholar

[32]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. Google Scholar

[33]

A. Zygmund, Trigonometric Series, 3$^{rd}$ edition, Cambridge University Press, Cambridge, 2002. Google Scholar

[1]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445
[2]

Bo Li, Hongwei Lou. Cesari-type conditions for semilinear elliptic equation with leading term containing controls. Mathematical Control & Related Fields, 2011, 1 (1) : 41-59. doi: 10.3934/mcrf.2011.1.41
[3]

Leonardo Colombo, David Martín de Diego. Second-order variational problems on Lie groupoids and optimal control applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6023-6064. doi: 10.3934/dcds.2016064
[4]

M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070
[5]

Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485
[6]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[7]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018
[8]

Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929
[9]

Leonardo Colombo. Second-order constrained variational problems on Lie algebroids: Applications to Optimal Control. Journal of Geometric Mechanics, 2017, 9 (1) : 1-45. doi: 10.3934/jgm.2017001
[10]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011
[11]

Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559
[12]

Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022
[13]

Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1399-1419. doi: 10.3934/jimo.2018101
[14]

Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial & Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329
[15]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47
[16]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018
[17]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[18]

Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323
[19]

Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control & Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019
[20]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (61)
  • HTML views (203)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences