An optimal osmotic control problem for a concrete dam system

Renzhao Chen; Xuezhang Hou

doi:10.3934/cpaa.2021082

Article Contents

2021, Volume 20, Issue 6: 2341-2359. Doi: 10.3934/cpaa.2021082

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An optimal osmotic control problem for a concrete dam system

Renzhao Chen^1, and
Xuezhang Hou^2,

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

^* Corresponding author
^* Corresponding author

Received: October 31, 2020

Revised: February 28, 2021

Early access: May 2021

Published: June 2021

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Abstract

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.

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Mathematics Subject Classification: Primary: 93C20, 93D15, 35B35, 35P10.

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Figure 1. Drainage set

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Figure 2. Drainage pore in coordinate system

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Figure 3. Divide of Ω

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Figure 4. Sketch vertical section

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References

[1]	R. A. Adams, Sobolev Space, New york, Academic Press, 1975.
[2]	J. Bear, Dynamics of Fluids in Porous Media, American Elsevier Publishing Company, INC. 1972.
[3]	M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.
[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.
[5]	R. Z. Chen, Optimal boundary control of prabolic system on doubly connected rogin in new space, Sci. China, 38 (1995), 933-944.
[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.
[7]	O. A. Lady$\check{z}$enskaja and N. N. Ural'ceva, Linear and Quasilinear Equation of Elliptic Type, Academic Press, New York, 1968.
[8]	J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
[9]	J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I. Springer-Verlag, Berlin, 1972.
[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.
[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.
[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.
[13]	K. Yosida, Functional Analysis, Springer-Verlag, 1978.