# American Institute of Mathematical Sciences

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

 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.

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. [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.

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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.
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