Iterative Solution Methods for a Class of State and Control Constrained Optimal Control Problems

ABSTRACT

Iterative methods for solving discrete optimal control problems are constructed and investigated. These discrete problems arise when approximating by finite difference method or by finite element method the optimal control problems which contain a linear elliptic boundary value problem as a state equation, control in the righthand side of the equation or in the boundary conditions, and point-wise constraints for both state and control functions. The convergence of the constructed iterative methods is proved, the implementation problems are discussed, and the numerical comparison of the methods is executed.

Iterative methods for solving discrete optimal control problems are constructed and investigated. These discrete problems arise when approximating by finite difference method or by finite element method the optimal control problems which contain a linear elliptic boundary value problem as a state equation, control in the righthand side of the equation or in the boundary conditions, and point-wise constraints for both state and control functions. The convergence of the constructed iterative methods is proved, the implementation problems are discussed, and the numerical comparison of the methods is executed.

Cite this paper

E. Laitinen and A. Lapin, "Iterative Solution Methods for a Class of State and Control Constrained Optimal Control Problems,"*Applied Mathematics*, Vol. 3 No. 12, 2012, pp. 1862-1867. doi: 10.4236/am.2012.312253.

E. Laitinen and A. Lapin, "Iterative Solution Methods for a Class of State and Control Constrained Optimal Control Problems,"

References

[1] J.-L. Lions, “Optimal Control of Systems Gouverhed by Partial Differential Equations,” Springer-Verlag, New York, 1971. doi:10.1007/978-3-642-65024-6

[2] P. Neittaanmaki, J. Sprekels and D. Tiba, “Optimization of Elliptic Systems. Theory and Applications,” SpringerVerlag, New York, 2006.

[3] P. H. Ciarlet and J.-L. Lions, “Handbook of Mumerical Analysis: Finite Element Methods,” North-Holland, Amsterdam, 1991.

[4] A. Lapin, “Preconditioned Uzawa-Type Methods for Finite-Dimensional Constrained Saddle Point Problems,” Lobachevskii Journal of Mathematics, Vol. 31, No. 4, 2010, pp. 309-322.

[5] R. Glowinski, J.-L. Lions and R. Tremolieres, “Numeral Analysis of Variational Inequalities,” Translated and Re-vised Edition, North-Holland Publishing Company, Amsterdam, New York and Oxford, 1981.

[6] A. Lapin, “Iterative Solution Methods for Mesh Variational Inequalities,” Kazan State University, Kazan, 2008.

[7] I. Hlavac?k, J. Haslinger, J. Nec?s and J. Lovis?k, “Solution of Variational Inequalities in Mechanics,” Springer Verlag, New York, 1988.

[8] A. V. Lapin, “Methods of Upper Relaxation Type for the Sum of a Quadratic and a Convex Functional,” Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, No. 8, 1993, pp. 30-39.

[9] A. Lapin and M. Khasanov, “State-Constrained Optimal Control of an Elliptic Equation with Its Right-Hand Side Used as Control Function,” Lobachevskii Journal of Mathematics, Vol. 32, No. 4, 2011, pp. 453-462. doi:10.1134/S1995080211040287

[10] E. Laitinen and A. Lapin, “Iterative Solution Methods for the Large Scale Constrained Saddle Point Problems,” In: S. Repin, T. Tiihonen and T. Tuovinen, Eds., Numerical Methods for Differential Equations, Optimization, and Technological Problems, Springer, New York, 2012, pp. 19-39.

[1] J.-L. Lions, “Optimal Control of Systems Gouverhed by Partial Differential Equations,” Springer-Verlag, New York, 1971. doi:10.1007/978-3-642-65024-6

[2] P. Neittaanmaki, J. Sprekels and D. Tiba, “Optimization of Elliptic Systems. Theory and Applications,” SpringerVerlag, New York, 2006.

[3] P. H. Ciarlet and J.-L. Lions, “Handbook of Mumerical Analysis: Finite Element Methods,” North-Holland, Amsterdam, 1991.

[4] A. Lapin, “Preconditioned Uzawa-Type Methods for Finite-Dimensional Constrained Saddle Point Problems,” Lobachevskii Journal of Mathematics, Vol. 31, No. 4, 2010, pp. 309-322.

[5] R. Glowinski, J.-L. Lions and R. Tremolieres, “Numeral Analysis of Variational Inequalities,” Translated and Re-vised Edition, North-Holland Publishing Company, Amsterdam, New York and Oxford, 1981.

[6] A. Lapin, “Iterative Solution Methods for Mesh Variational Inequalities,” Kazan State University, Kazan, 2008.

[7] I. Hlavac?k, J. Haslinger, J. Nec?s and J. Lovis?k, “Solution of Variational Inequalities in Mechanics,” Springer Verlag, New York, 1988.

[8] A. V. Lapin, “Methods of Upper Relaxation Type for the Sum of a Quadratic and a Convex Functional,” Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, No. 8, 1993, pp. 30-39.

[9] A. Lapin and M. Khasanov, “State-Constrained Optimal Control of an Elliptic Equation with Its Right-Hand Side Used as Control Function,” Lobachevskii Journal of Mathematics, Vol. 32, No. 4, 2011, pp. 453-462. doi:10.1134/S1995080211040287

[10] E. Laitinen and A. Lapin, “Iterative Solution Methods for the Large Scale Constrained Saddle Point Problems,” In: S. Repin, T. Tiihonen and T. Tuovinen, Eds., Numerical Methods for Differential Equations, Optimization, and Technological Problems, Springer, New York, 2012, pp. 19-39.