September  2017, 7(3): 325-344. doi: 10.3934/naco.2017021

Analysis of optimal boundary control for a three-dimensional reaction-diffusion system

1. 

Zhuhai College of Jilin University, Zhuhai, China

2. 

School of Science, Curtin University, Australia

3. 

Department of Mathematics and Statistics, Curtin University, Australia

4. 

School of Business, National University of Singapore, Singapore

5. 

Business School, Nankai University, Tianjin, China

* Corresponding author

The reviewing process of the paper was handled by Shengjie Li as Guest Editor

Received  April 2016 Revised  May 2017 Published  July 2017

Fund Project: This work is partially supported by Australian Research Council Grant DP160102189 and by a grant from Curtin University, Australia.

This paper is concerned with optimal boundary control of a three dimensional reaction-diffusion system in a more general form than what has been presented in the literature. The state equations are analyzed and the optimal control problem is investigated. Necessary and sufficient optimality conditions are derived. The model is widely applicable due to its generality. Some examples in applications are discussed.

Citation: Wanli Yang, Jie Sun, Su Zhang. Analysis of optimal boundary control for a three-dimensional reaction-diffusion system. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 325-344. doi: 10.3934/naco.2017021
References:
[1]

W. BarthelC. John and F. Tröltzsch, Optimal boundary control of a system of reaction diffusion equations, Z. Angew. Math. Mech., 90 (2010), 966-982.  doi: 10.1002/zamm.200900359.

[2]

E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327.  doi: 10.1137/S0363012995283637.

[3]

E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control, Jahresbericht der Deutschen Mathematiker-Vereinigung, 117 (2015), 3-44.  doi: 10.1365/s13291-014-0109-3.

[4]

Z. Chen and K. H. Hoffmann, Numerical solutions of the optimal control problem governed by a phase field model, in Estimation and Control of Distributed Parameter Systems (eds W. Desch, F. Kappel and K. Kunisch), ISNM Vol. 100, Birkhäuser Verlag, Basel, (1991), 79–97. doi: 10.1007/978-3-0348-6418-3_5.

[5]

L. C. Evans, Partial Differential Equations, in: Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, Rhode Island, 1998.

[6]

R. Griesse, Parametric Sensitivity Analysis for Control-Constrained Optimal Control Problems Governed by Systems of Parabolic Partial Differential Equations, PhD thesis, University of Bayreuth, Bayreuth, 2003.

[7]

R. Griesse and S. Volkwein, A primal-dual active set strategy for optimal boundary control of a nonlinear reaction-diffusion system, SIAM J. Control Optim., 44 (2005), 467-494.  doi: 10.1137/S0363012903438696.

[8]

R. Griesse and S. Volkwein, Parametric sensitivity analysis for optimal boundary control of a 3D reaction-diffusion system, in: Large-Scale Nonlinear Optimization Nonconvex Optimization and its Applications, 83 (2006), 127–149. doi: 10.1007/0-387-30065-1_9.

[9]

M. Heinkenschloss, The numerical solution of a control problem governed by a phase field model, Optim. Methods Software, 7 (1997), 211-263.  doi: 10.1080/10556789708805656.

[10]

M. Heinkenschloss and E. W. Sachs, Numerical solution of a constrained control problem for a phase field model, in Control and Estimation of Distributed Parameter Systems (eds. W. Desch, F. Kappel and K. Kunisch), ISNM Vol. 118, Birkhäuser Verlag, Basel, (1994), 171–188.

[11]

M. Hintz, R. Pinnau and M. Ulbrich, Optimization with PDF Constraints, Springer, 2009.

[12]

Y. JiangY. He and J. Sun, Proximal analysis and the minimal time function of a class of semilinear control systems, J. Optim. Theory. Appl., 169 (2016), 784-800.  doi: 10.1007/s10957-015-0848-z.

[13]

C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 87 (1982), 165-198.  doi: 10.1016/0022-247X(82)90160-3.

[14]

J. P. Raymond and H. Zidani, Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations, Appl. Math. Optim., 39 (1999), 143-177.  doi: 10.1007/s002459900102.

[15]

E. Sachs, A parabolic control problem with a boundary condition of the Stefan-Boltzman type, Z. Angew. Math. Mech., 58 (1978), 443-449.  doi: 10.1002/zamm.19780581005.

[16]

F. Tröltzsch, Optimal Control of Partial Differential Equations - Theory, Methods and Applications, in: Graduate Studies in Mathematics, Vol. 112, American Mathematical Society, Providence, 2010. doi: 10.1090/gsm/112.

show all references

References:
[1]

W. BarthelC. John and F. Tröltzsch, Optimal boundary control of a system of reaction diffusion equations, Z. Angew. Math. Mech., 90 (2010), 966-982.  doi: 10.1002/zamm.200900359.

[2]

E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327.  doi: 10.1137/S0363012995283637.

[3]

E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control, Jahresbericht der Deutschen Mathematiker-Vereinigung, 117 (2015), 3-44.  doi: 10.1365/s13291-014-0109-3.

[4]

Z. Chen and K. H. Hoffmann, Numerical solutions of the optimal control problem governed by a phase field model, in Estimation and Control of Distributed Parameter Systems (eds W. Desch, F. Kappel and K. Kunisch), ISNM Vol. 100, Birkhäuser Verlag, Basel, (1991), 79–97. doi: 10.1007/978-3-0348-6418-3_5.

[5]

L. C. Evans, Partial Differential Equations, in: Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, Rhode Island, 1998.

[6]

R. Griesse, Parametric Sensitivity Analysis for Control-Constrained Optimal Control Problems Governed by Systems of Parabolic Partial Differential Equations, PhD thesis, University of Bayreuth, Bayreuth, 2003.

[7]

R. Griesse and S. Volkwein, A primal-dual active set strategy for optimal boundary control of a nonlinear reaction-diffusion system, SIAM J. Control Optim., 44 (2005), 467-494.  doi: 10.1137/S0363012903438696.

[8]

R. Griesse and S. Volkwein, Parametric sensitivity analysis for optimal boundary control of a 3D reaction-diffusion system, in: Large-Scale Nonlinear Optimization Nonconvex Optimization and its Applications, 83 (2006), 127–149. doi: 10.1007/0-387-30065-1_9.

[9]

M. Heinkenschloss, The numerical solution of a control problem governed by a phase field model, Optim. Methods Software, 7 (1997), 211-263.  doi: 10.1080/10556789708805656.

[10]

M. Heinkenschloss and E. W. Sachs, Numerical solution of a constrained control problem for a phase field model, in Control and Estimation of Distributed Parameter Systems (eds. W. Desch, F. Kappel and K. Kunisch), ISNM Vol. 118, Birkhäuser Verlag, Basel, (1994), 171–188.

[11]

M. Hintz, R. Pinnau and M. Ulbrich, Optimization with PDF Constraints, Springer, 2009.

[12]

Y. JiangY. He and J. Sun, Proximal analysis and the minimal time function of a class of semilinear control systems, J. Optim. Theory. Appl., 169 (2016), 784-800.  doi: 10.1007/s10957-015-0848-z.

[13]

C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl., 87 (1982), 165-198.  doi: 10.1016/0022-247X(82)90160-3.

[14]

J. P. Raymond and H. Zidani, Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations, Appl. Math. Optim., 39 (1999), 143-177.  doi: 10.1007/s002459900102.

[15]

E. Sachs, A parabolic control problem with a boundary condition of the Stefan-Boltzman type, Z. Angew. Math. Mech., 58 (1978), 443-449.  doi: 10.1002/zamm.19780581005.

[16]

F. Tröltzsch, Optimal Control of Partial Differential Equations - Theory, Methods and Applications, in: Graduate Studies in Mathematics, Vol. 112, American Mathematical Society, Providence, 2010. doi: 10.1090/gsm/112.

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