-
Previous Article
Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes
- MCRF Home
- This Issue
-
Next Article
Optimal voltage control of non-stationary eddy current problems
Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls
1. | School of Mathematical Sciences, and LMNS, Fudan University, Shanghai 200433, China, |
2. | Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA |
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.
References:
[1] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
|
[2] |
G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. |
[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.
|
[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.
|
[5] |
E. Casas,
Optimal control in coefficients of elliptic equations with state constraints, Appl. Math. Optim., 26 (1992), 21-37.
|
[6] |
E. Casas,
Second order analysis for bang-bang control problems of PDEs, SIAM J. Control Optim., 50 (2012), 2355-2372.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[13] |
E. Casas and F. Tröltzsch,
Second order optimality conditions and their role in PDE control, Jahresber Dtsch Math-Ver, 117 (2015), 3-44.
|
[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.
|
[15] |
H. O. Fattorini,
Relaxed controls in infinite dimensional systems, International Series of Numerical Mathematics, 100 (1991), 115-128.
|
[16] |
R. Gabasov and F. M. Kirillova,
High order necessary conditions for optimality, SIAM J. Control Optim., 10 (1972), 127-168.
|
[17] |
R. Gamkrelidze, Principle of Optimal Control Theory, Plenum Press, New York, 1978. |
[18] |
H. J. Kelly,
A second variation test for singular extremals, AIAA J., 2 (1964), 1380-1382.
|
[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. |
[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.
|
[22] |
B. Li and H. Lou,
Optimality Conditions for semilinear hyperbolic equations with controls in coefficients, Appl. Math. Optim., 65 (2012), 371-402.
|
[23] |
X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. |
[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. |
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[32] |
J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. |
[33] |
A. Zygmund, Trigonometric Series, 3$^{rd}$ edition, Cambridge University Press, Cambridge, 2002. |
show all references
References:
[1] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
|
[2] |
G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002. |
[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.
|
[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.
|
[5] |
E. Casas,
Optimal control in coefficients of elliptic equations with state constraints, Appl. Math. Optim., 26 (1992), 21-37.
|
[6] |
E. Casas,
Second order analysis for bang-bang control problems of PDEs, SIAM J. Control Optim., 50 (2012), 2355-2372.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[13] |
E. Casas and F. Tröltzsch,
Second order optimality conditions and their role in PDE control, Jahresber Dtsch Math-Ver, 117 (2015), 3-44.
|
[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.
|
[15] |
H. O. Fattorini,
Relaxed controls in infinite dimensional systems, International Series of Numerical Mathematics, 100 (1991), 115-128.
|
[16] |
R. Gabasov and F. M. Kirillova,
High order necessary conditions for optimality, SIAM J. Control Optim., 10 (1972), 127-168.
|
[17] |
R. Gamkrelidze, Principle of Optimal Control Theory, Plenum Press, New York, 1978. |
[18] |
H. J. Kelly,
A second variation test for singular extremals, AIAA J., 2 (1964), 1380-1382.
|
[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. |
[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.
|
[22] |
B. Li and H. Lou,
Optimality Conditions for semilinear hyperbolic equations with controls in coefficients, Appl. Math. Optim., 65 (2012), 371-402.
|
[23] |
X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. |
[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. |
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[32] |
J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. |
[33] |
A. Zygmund, Trigonometric Series, 3$^{rd}$ edition, Cambridge University Press, Cambridge, 2002. |
[1] |
Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020052 |
[2] |
Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 |
[3] |
Hongbo Guan, Yong Yang, Huiqing Zhu. A nonuniform anisotropic FEM for elliptic boundary layer optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1711-1722. doi: 10.3934/dcdsb.2020179 |
[4] |
Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020045 |
[5] |
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 |
[6] |
Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2020164 |
[7] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[8] |
Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176 |
[9] |
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 |
[10] |
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 |
[11] |
Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020107 |
[12] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 |
[13] |
Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011 |
[14] |
Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439 |
[15] |
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 |
[16] |
Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213 |
[17] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
[18] |
Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control & Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032 |
[19] |
Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026 |
[20] |
Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020053 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]