# American Institute of Mathematical Sciences

June  2021, 20(6): 2341-2359. doi: 10.3934/cpaa.2021082

## An optimal osmotic control problem for a concrete dam system

 1 Department of Mathematics, Northeast Normal University, Changchun 130024, China 2 Mathematics Department, Towson University, Towson, Maryland 21252, USA

* Corresponding author

Received  November 2020 Revised  March 2021 Published  May 2021

In this paper, an optimal control problem for a concrete dam system is considered. First, a mathematical model on the optimal osmotic control for basis of concrete dams is built up, and an optimal line-wise control of the system governed by the hybrid problem for elliptic partial differential equations is investigated. Then, the regularity of the generalized solution to the adjoint state equations, and the existence and uniqueness of the ${\rm L^2}$-solution for state equations are discussed and examined. Subsequently, the existence and uniqueness of the optimal control for the system, and a necessary and sufficient conditions for a control to be optimal and the optimality system are claimed and derived. Finally, the applications of the penalty shifting method with calculation of the optimal control of the system are studied, and the convergence of the method on an appropriate Hilbert space is claimed and proved.

Citation: Renzhao Chen, Xuezhang Hou. An optimal osmotic control problem for a concrete dam system. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2341-2359. doi: 10.3934/cpaa.2021082
##### References:
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##### References:
 [1] R. A. Adams, Sobolev Space, New york, Academic Press, 1975.   Google Scholar [2] J. Bear, Dynamics of Fluids in Porous Media, American Elsevier Publishing Company, INC. 1972. Google Scholar [3] M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.   Google Scholar [4] E. Casas and J. P. Raymond, Error estimates for the numerical approximation of dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611.  doi: 10.1137/050626600.  Google Scholar [5] R. Z. Chen, Optimal boundary control of prabolic system on doubly connected rogin in new space, Sci. China, 38 (1995), 933-944.   Google Scholar [6] G. Dipillo and L. Grippo, On the method multipliers for a class of distributed parameter systems, inProceedeings of 2nd IFAC Symposium on Control of Distributed Parameter Systems, Warmick, England, 1977.  Google Scholar [7] O. A. Lady$\check{z}$enskaja and N. N. Ural'ceva, Linear and Quasilinear Equation of Elliptic Type, Academic Press, New York, 1968.   Google Scholar [8] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.  Google Scholar [9] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I. Springer-Verlag, Berlin, 1972.  Google Scholar [10] Y. S. Luo, Necessary optimality conditions for some control problems of elliptic equations with venttsel boundary conditions, Appl. Math. Optim., 61 (2010), 337-351.  doi: 10.1007/s00245-009-9078-8.  Google Scholar [11] D. Tonon, M. Aronna and D. Kalise, Optimal Control: Novel Directional and Applications, Lecture Notes in Mathematics, 2017. doi: 10.1007/978-3-319-60771-9.  Google Scholar [12] S. Y. Weng, H. Y. Gao, R. Z. Chen and X. Z. Hou, Penalty shifting method on calaulation of optimal linewise control of dam-detouring osmtic system for concrete dams with heterogemous, Appl. Math. Comput., 169 (2005), 1129-1141.  doi: 10.1016/j.amc.2004.10.089.  Google Scholar [13] K. Yosida, Functional Analysis, Springer-Verlag, 1978.  Google Scholar
Drainage set
Drainage pore in coordinate system
Divide of Ω
Sketch vertical section
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