Finite-dimensional approximation of a class of constrained nonlinear optimal control problems

SIAM Journal on Control and Optimization

Volumn 34, Issue 3, 1996, Pages 1001-1043

  Gunzburger, Max D ^a   Hou, L Steven ^b  

^a VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY   (United States)

^b YORK UNIVERSITY   (Canada)

Author keywords

Finite dimensional approximation; Finite element methods; Ginzburg Landau equations; Navier Stokes equations; Nonlinear partial differential equations; Optimal control; Von K rm n equations

Indexed keywords

APPROXIMATION THEORY; ERRORS; FINITE ELEMENT METHOD; LAGRANGE MULTIPLIERS; NAVIER STOKES EQUATIONS; NONLINEAR CONTROL SYSTEMS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; VECTORS;

FINITE DIMENSIONAL APPROXIMATION; GINZBURG-LANDAU EQUATIONS; NONLINEAR OPTIMAL CONTROL PROBLEMS; VON KARMAN EQUATIONS;

OPTIMAL CONTROL SYSTEMS;

EID: 0030150056     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012994262361     Document Type: Article

References (22)

1. R. ADAMS, Sobolev Spaces, Academic, New York, 1975.
2. I. BABUSKA AND A. AZIZ, Survey lectures on the mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A. Aziz, ed., Academic Press, New York, 1972, pp. 3-359.
3. H. BLUM AND R. RANNACHER, On the boundary value problem of the biharmonic operator on domains with angular corners, Math. Methods Appl. Sci., 2 (1980), pp. 556-581.
4. F. BREZZI, On the existence, uniqueness, and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Modél. Math. Anal. Numér., 8 (1974), pp. 129-151.
5. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer, New York, 1991.
6. F. BREZZI, J. RAPPAZ, AND P. RAVIART, Finite-dimensional approximation of nonlinear problems. Part I: Branches of nonsingular solutions, Numer. Math., 36 (1980), pp. 1-25.
7. J. CHAPMAN, Q. DU, M. GUNZBURGER, AND J. PETERSON, Simplified Ginzburg-Landau models for superconductivity valid for high kappa and high fields, Adv. Math. Sci. Appl., 5 (1995), pp. 193-218.
8. P. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.
9. P. CIARLET, Les Equations de von Karman, Springer, Berlin, 1980.
10. M. CROUZEIX AND J. RAPPAZ, On Numerical Approximation in Bifurcation Theory, Masson, Paris, 1990.
11. Q. DU, M. GUNZBURGER, AND J. PETERSON, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Rev., 34 (1992), pp. 54-81.
12. V. GEORGESCU, On the unique continuation property for Schödinger Hamiltonians, Helv. Phys. Acta, 52 (1979), pp. 655-670.
13. V. GIRAULT AND P. RAVIART, Finite Element Methods for Navier-Stokes Equations, Springer, Berlin, 1986.
14. M. GUNZBURGER, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms, Academic Press, Boston, MA, 1989.
15. M. GUNZBURGER AND L. HOU, Treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses, SIAM J. Numer. Anal., 29 (1992), pp. 390-424.
16. M. GUNZBURGER, L. HOU, AND T. SVOBODNY, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls, RAIRO Modél. Math. Anal. Numér., 25 (1991), pp. 711-748.
17. L. HOU AND T. SVOBODNY, Optimization problems for the Navier-Stokes equations with regular boundary controls, J. Math. Anal. Appl., 177 (1993), pp. 342-367.
18. J. LIONS, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Gauthier-Villars, Paris, 1969.
19. J. SAUT AND B. SCHEURER, Unique continuation and uniqueness of the Cauchy problem for elliptic operators with unbounded coefficients, in Applications of Nonlinear Partial Differential Equations, H. Brezis and J. Lions, eds., Longman, New York, 1982, pp. 260-275.
20. R. TEMAM, Navier-Stokes Equations, North-Holland, Amsterdam, 1979.
21. V. TIKHOMIROV, Fundamental Principles of the Theory of Extremal Problems, Wiley, Chichester, 1982.
22. M. TINKHAM, Introduction to Superconductivity, McGraw-Hill, New York, 1975.