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

Hongwei Lou; Jiongmin Yong

doi:10.3934/mcrf.2018003

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2018, Volume 8, Issue 1: 57-88. Doi: 10.3934/mcrf.2018003

This issue Previous Article Optimal voltage control of non-stationary eddy current problems Next Article Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes

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

Received: February 28, 2017

Revised: September 30, 2017

Published: January 2018

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

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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.

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Mathematics Subject Classification: Primary: 49K20, 35J61; Secondary: 35Q93.

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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.
[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.