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Research Article

A One Dimensional Elliptic Distributed Optimal Control Problem with Pointwise Derivative Constraints

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Pages 1549-1563 | Received 16 Mar 2020, Accepted 17 Jun 2020, Published online: 30 Jun 2020

References

  • Casas, E., Bonnans, J. F. (1988). Contrôle de systèmes elliptiques semilinéares comportant des contraintes sur l’état. In Brezzis, H., Lions, J. L., eds., Nonlinear Partial Differential Equations and Their Applications, vol. 8. New York: Longman, pp. 69–86.
  • Casas, E., Fernandez, L. A. (1993). Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state. Appl. Math. Optim. 27(1):35–56. DOI: 10.1007/BF01182597.
  • Deckelnick, K., Günther, A., Hinze, M. (2009). Finite element approximation of elliptic control problems with constraints on the gradient. Numer. Math. 111(3):335–350. DOI: 10.1007/s00211-008-0185-3.
  • Ortner, C., Wollner, W. (2011). A priori error estimates for optimal control problems with pointwise constraints on the gradient of the state. Numer. Math. 118(3):587–600. DOI: 10.1007/s00211-011-0360-9.
  • Wollner, W. (2012). Optimal control of elliptic equations with pointwise constraints on the gradient of the state in nonsmooth polygonal domains. SIAM J. Control Optim. 50(4):2117–2129. DOI: 10.1137/110836419.
  • Brenner, S. C., Sung, L.-Y., Wollner, W. (2020). Finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state. Optim Eng. published online 18 February 2020. doi.10.1007/s11081-020-09491-1).
  • Ekeland, I., Témam, R. (1999). Convex Analysis and Variational Problems. Classics in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
  • Kinderlehrer, D., Stampacchia, G. (2000). An Introduction to Variational Inequalities and Their Applications. Philadelphia, PA: Society for Industrial and Applied Mathematics.
  • Pierre, M., Sokołowski, J. (1996). Differentiability of projection and applications. In Control of Partial Differential Equations and Applications (Laredo, 1994), Volume 174 of Lecture Notes in Pure and Applied Mathematics. New York: Dekker, pp. 231–240.
  • Liu, W., Gong, W., Yan, N. (2009). A new finite element approximation of a state-constrained optimal control problem. J. Comput. Math. 27:97–114.
  • Brenner, S. C., Davis, C. B., Sung, L.-Y. (2014). A partition of unity method for a class of fourth order elliptic variational inequalities. Comp. Methods Appl. Mech. Eng. 276:612–626. DOI: 10.1016/j.cma.2014.04.004.
  • Brenner, S. C., Gedicke, J., Sung, L.-Y. (2018). C0 interior penalty methods for an elliptic distributed optimal control problem on nonconvex polygonal domains with pointwise state constraints. SIAM J. Numer. Anal. 56(3):1758–1785. DOI: 10.1137/17M1140649.
  • Brenner, S. C., Gudi, T., Porwal, K., Sung, L.-Y. (2018). A Morley finite element method for an elliptic distributed optimal control problem with pointwise state and control constraints. ESAIM: COCV. 24(3):1181–1206. DOI: 10.1051/cocv/2017031.
  • Brenner, S. C., Oh, M., Pollock, S., Porwal, K., Schedensack, M., Sharma, N. (2016). A C0 interior penalty method for elliptic distributed optimal control problems in three dimensions with pointwise state constraints. In Brenner, S. C., eds., Topics in Numerical Partial Differential Equations and Scientific Computing, Volume 160 of the IMA Volumes in Mathematics and Its Applications. Cham: Springer, pp. 1–22.
  • Brenner, S. C., Sung, L.-Y., Zhang, Y. (2013). A quadratic C0 interior penalty method for an elliptic optimal control problem with state constraints. In: Karakashian, O., Feng, X., and Xing, Y., eds., Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, Volume 157 of the IMA Volumes in Mathematics and Its Applications. Cham: Springer, pp. 97–132. (2012 John H. Barrett Memorial Lectures).
  • Brenner, S. C., Sung, L.-Y., Zhang, Y. (2019). C0 interior penalty methods for an elliptic state-constrained optimal control problem with Neumann boundary condition. J. Comput. Appl. Math. 350:212–232. DOI: 10.1016/j.cam.2018.10.015.
  • Gong, W., Yan, N. (2011). A mixed finite element scheme for optimal control problems with pointwise state constraints. J. Sci. Comput. 46(2):182–203. DOI: 10.1007/s10915-010-9392-z.
  • Ito, K., Kunisch, K. (2008). Lagrange Multiplier Approach to Variational Problems and Applications. Philadelphia, PA: Society for Industrial and Applied Mathematics.
  • Rodrigues, J.-F. (1987). Obstacle Problems in Mathematical Physics. Amsterdam: North-Holland Publishing Co.
  • Brenner, S. C., Scott, L. R. (2008). The Mathematical Theory of Finite Element Methods, 3rd ed. New York: Springer-Verlag.
  • Ciarlet, P. G. (1978). The Finite Element Method for Elliptic Problems. Amsterdam: North-Holland.
  • Brenner, S. C., Sung, L.-Y. (2017). A new convergence analysis of finite element methods for elliptic distributed optimal control problems with pointwise state constraints. SIAM J. Control Optim. 55(4):2289–2304. DOI: 10.1137/16M1088090.
  • Nečas, J. (2012). Direct Methods in the Theory of Elliptic Equations. Heidelberg: Springer.
  • Girault, V., Raviart, P.-A. (1986). Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Berlin: Springer-Verlag.

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