September  2017, 7(3): 493-506. doi: 10.3934/mcrf.2017018

On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions

School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China

* Corresponding author:Shu Luan

Received  May 2016 Revised  January 2017 Published  July 2017

Fund Project: This work was partially supported by the National Natural Science Foundation of China under grants 11301472 and 11601213, the Natural Science Foundation of Guangdong Province under grant 2014A030307011, the Training Program Project for Outstanding Young Teachers of Colleges and Universities in Guangdong Province under grant Yq2014116 and the Characteristic Innovation Project of Common Colleges and Universities in Guangdong Province under grant 2014KTSCX158.

An optimal control problem governed by a class of semilinear elliptic equations with nonlinear Neumann boundary conditions is studied in this paper. It is pointed out that the cost functional considered may not be convex. Using a relaxation method, some existence results of an optimal control are obtained.

Citation: Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control and Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018
References:
[1]

H. Amann and M. G. Crandall, On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J., 27 (1977), 779-790.  doi: 10.1512/iumj.1978.27.27050.

[2]

Z. Artstein, On a variational problem, J. Math. Anal. Appl., 45 (1974), 405-415.  doi: 10.1016/0022-247X(74)90081-X.

[3]

E. J. Balder, New existence results for optimal controls in the absence of convexity: The importance of extremality, SIAM J. Control Optim., 32 (1994), 890-916.  doi: 10.1137/S0363012990193099.

[4]

A. Cellinaand and G. Colombo, On a classical problem of the calculus of variations without convexity assumptions, Ann. Inst. H. Poincaré Anal. Non Linéaire., 7 (1990), 97-106.  doi: 10.1016/S0294-1449(16)30306-7.

[5]

L. Cesari, Optimization Theory and Applications, Problems with Ordinary Differential Equations, Spring, New York, 1983. doi: 10.1007/978-1-4613-8165-5.

[6]

P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter & Co. , Berlin, 1997. doi: 10.1515/9783110804775.

[7]

F. Flores-Bazán and S. Perrotta, Nonconvex variational problems related to a hyperbolic equation, SIAM J. Control Optim, 37 (1999), 1751-1766.  doi: 10.1137/S0363012998332299.

[8]

G. Giuseppina and M. Federica, On the existence of optimal controls for SPDEs with boundary noise and boundary control, SIAM J. Control Optim., 51 (2013), 1909-1939.  doi: 10.1137/110855855.

[9]

V. O. Kapustyan and O. P. Kogut, On the existence of optimal coefficient controls for a nonlinear Neumann boundary value problem, Diff. Eqs., 46 (2010), 923-938.  doi: 10.1134/S0012266110070013.

[10]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser Boston, Cambridge, MA, 1995. doi: 10.1007/978-1-4612-4260-4.

[11]

P. Lin and G. S. Wang, Some properties for blowup parabolic equations and their application, J. Math. Pures Appl., 101 (2014), 223-255.  doi: 10.1016/j.matpur.2013.06.001.

[12]

H. W. Lou, Existence and nonexistence results of an optimal control problem by using relaxed control, SIAM J. Control Optim., 46 (2007), 1923-1941.  doi: 10.1137/050628386.

[13]

H. W. Lou, Analysis of the optimal relaxed control to an optimal control problem, Appl. Math. Optim., 59 (2009), 75-97.  doi: 10.1007/s00245-008-9045-x.

[14]

H. W. LouJ. J. Wen and Y. S. Xu, Time optimal control problems for some non-smooth systems, Math. Control Relat. F., 4 (2014), 289-314.  doi: 10.3934/mcrf.2014.4.289.

[15]

Q. Lü and G. S. Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations, SIAM J. Control Optim., 49 (2011), 1124-1149.  doi: 10.1137/10081277X.

[16]

S. Luan, Nonexistence and existence of an optimal control problem governed by a class of semilinear elliptic equations, J. Optim. Theory Appl., 158 (2013), 1-10.  doi: 10.1007/s10957-012-0244-x.

[17]

S. Luan, Nonexistence and existence results of an optimal control problem governed by a class of multisolution semilinear elliptic equations, Nonlinear Anal., 128 (2015), 380-390.  doi: 10.1016/j.na.2015.08.015.

[18]

E. J. McShane, Generalized curves, Duke Math. J., 6 (1940), 513-536.  doi: 10.1215/S0012-7094-40-00642-1.

[19]

L. W. Neustadt, The existence of optimal controls in the absence of convexity conditions, J. Math. Anal. Appl., 7 (1963), 110-117.  doi: 10.1016/0022-247X(63)90081-7.

[20]

K. D. PhungG. S. Wang and X. Zhang, On the existence of time optimal controls for linear evolution equations, Discrete Contin. Dyn. Syst. Ser. B., 8 (2007), 925-941.  doi: 10.3934/dcdsb.2007.8.925.

[21]

J. P. Raymond, Existence theorems in optimal control theory without convexity assumptions, J. Optim. Theory Appl., 67 (1990), 109-132.  doi: 10.1007/BF00939738.

[22]

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

[23]

P. Winkert, $L^∞$ estimates for nonlinear elliptic Neumann boundary value problems, Nonlinear Differ. Equ. Appl., 17 (2010), 289-302.  doi: 10.1007/s00030-009-0054-5.

[24]

L. C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, C. R. Sci. Lettres Varsovie, C. Ⅲ., 30 (1937), 212-234. 

show all references

References:
[1]

H. Amann and M. G. Crandall, On some existence theorems for semi-linear elliptic equations, Indiana Univ. Math. J., 27 (1977), 779-790.  doi: 10.1512/iumj.1978.27.27050.

[2]

Z. Artstein, On a variational problem, J. Math. Anal. Appl., 45 (1974), 405-415.  doi: 10.1016/0022-247X(74)90081-X.

[3]

E. J. Balder, New existence results for optimal controls in the absence of convexity: The importance of extremality, SIAM J. Control Optim., 32 (1994), 890-916.  doi: 10.1137/S0363012990193099.

[4]

A. Cellinaand and G. Colombo, On a classical problem of the calculus of variations without convexity assumptions, Ann. Inst. H. Poincaré Anal. Non Linéaire., 7 (1990), 97-106.  doi: 10.1016/S0294-1449(16)30306-7.

[5]

L. Cesari, Optimization Theory and Applications, Problems with Ordinary Differential Equations, Spring, New York, 1983. doi: 10.1007/978-1-4613-8165-5.

[6]

P. Drábek, A. Kufner and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities, Walter de Gruyter & Co. , Berlin, 1997. doi: 10.1515/9783110804775.

[7]

F. Flores-Bazán and S. Perrotta, Nonconvex variational problems related to a hyperbolic equation, SIAM J. Control Optim, 37 (1999), 1751-1766.  doi: 10.1137/S0363012998332299.

[8]

G. Giuseppina and M. Federica, On the existence of optimal controls for SPDEs with boundary noise and boundary control, SIAM J. Control Optim., 51 (2013), 1909-1939.  doi: 10.1137/110855855.

[9]

V. O. Kapustyan and O. P. Kogut, On the existence of optimal coefficient controls for a nonlinear Neumann boundary value problem, Diff. Eqs., 46 (2010), 923-938.  doi: 10.1134/S0012266110070013.

[10]

X. J. Li and J. M. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser Boston, Cambridge, MA, 1995. doi: 10.1007/978-1-4612-4260-4.

[11]

P. Lin and G. S. Wang, Some properties for blowup parabolic equations and their application, J. Math. Pures Appl., 101 (2014), 223-255.  doi: 10.1016/j.matpur.2013.06.001.

[12]

H. W. Lou, Existence and nonexistence results of an optimal control problem by using relaxed control, SIAM J. Control Optim., 46 (2007), 1923-1941.  doi: 10.1137/050628386.

[13]

H. W. Lou, Analysis of the optimal relaxed control to an optimal control problem, Appl. Math. Optim., 59 (2009), 75-97.  doi: 10.1007/s00245-008-9045-x.

[14]

H. W. LouJ. J. Wen and Y. S. Xu, Time optimal control problems for some non-smooth systems, Math. Control Relat. F., 4 (2014), 289-314.  doi: 10.3934/mcrf.2014.4.289.

[15]

Q. Lü and G. S. Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations, SIAM J. Control Optim., 49 (2011), 1124-1149.  doi: 10.1137/10081277X.

[16]

S. Luan, Nonexistence and existence of an optimal control problem governed by a class of semilinear elliptic equations, J. Optim. Theory Appl., 158 (2013), 1-10.  doi: 10.1007/s10957-012-0244-x.

[17]

S. Luan, Nonexistence and existence results of an optimal control problem governed by a class of multisolution semilinear elliptic equations, Nonlinear Anal., 128 (2015), 380-390.  doi: 10.1016/j.na.2015.08.015.

[18]

E. J. McShane, Generalized curves, Duke Math. J., 6 (1940), 513-536.  doi: 10.1215/S0012-7094-40-00642-1.

[19]

L. W. Neustadt, The existence of optimal controls in the absence of convexity conditions, J. Math. Anal. Appl., 7 (1963), 110-117.  doi: 10.1016/0022-247X(63)90081-7.

[20]

K. D. PhungG. S. Wang and X. Zhang, On the existence of time optimal controls for linear evolution equations, Discrete Contin. Dyn. Syst. Ser. B., 8 (2007), 925-941.  doi: 10.3934/dcdsb.2007.8.925.

[21]

J. P. Raymond, Existence theorems in optimal control theory without convexity assumptions, J. Optim. Theory Appl., 67 (1990), 109-132.  doi: 10.1007/BF00939738.

[22]

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

[23]

P. Winkert, $L^∞$ estimates for nonlinear elliptic Neumann boundary value problems, Nonlinear Differ. Equ. Appl., 17 (2010), 289-302.  doi: 10.1007/s00030-009-0054-5.

[24]

L. C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, C. R. Sci. Lettres Varsovie, C. Ⅲ., 30 (1937), 212-234. 

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