Continuous dependence of solutions of elliptic BVPs on parameters

• Autorzy: Orpel A.
• Języki publikacji: EN

Abstrakty

EN

The continuous dependence of solutions for a certain class of elliptic PDE on functional parameters is studied in this paper. The main result is as follow: the sequence {k}k∈N of solutions of the Dirichlet problem discussed here (corresponding to parameters {uk}k∈N) converges weakly to x0 (corresponding to u0) in W1,q 0 (Ω, R), provided that {uk}k∈N tends to u0 a.e. in Ω. Our investigation covers both sub and superlinear cases. We apply this result to some optimal control problems.

Słowa kluczowe

EN

continuous dependence on parameters; elliptic Dirichlet problems; optimal control problem

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2006
• Tom: Vol. 26, no. 2
• Strony: 351--359
• Opis fizyczny: Bibliogr. 13 poz.

Twórcy

autor

Orpel A.

• University of Łódź, Faculty of Mathematics, ul. Banacha 22, 90-238 Łódź, Poland,

Bibliografia

• [1] Idczak D., Optimal Control of a Coercive Dirichlet Problem, SIAM J. Control Optim. 36 (1998) 4, 1250-1267.
• [2] Idczak D., Stability in Semilinear Problems, J. Differential Equations 162 (2000) 1, 64-90.
• [3] Idczak D., Majewski M., Walczak S., Stability analysis of one and two-dimensional continuous systems with parameters, Proceedings of the Fourteenth International Symposium of Mathematical Theory of Networks and Systems MTNS 2000, June 19-23, 2000 Perpignan, France.
• [4] Idczak D., Majewski M., Walczak S., Stability of solutions to an optimal control problem for a continuous Fornasini-Marchesini system, The Second International Workshop on Multidimensional (nD) Systems (Czocha Castle, 2000), 201-208, Tech. Univ. Press, Zielona Gora, 2000.
• [5] Idczak D., Majewski M., Walczak S., N-dimensional continuous systems with the Darboux-Goursat and Dirichlet boundary data, Proceedings of the 9th IEEE International Conference on Electronics, Circuits and Systems. September 15-18, 2002, Dubrovnik, Croatia.
• [6] Ledzewicz U., Schattler H., Walczak S., Stability of elliptic optimal control problems, Comput. Math. Appl. 41 (2001) 10-11, 1245-1256.
• [7] Mawhin J., Problemes de Dirichlet Variationnels Non Lineares, Les Presses de l'Universite dr Montreal, 1987.
• [8] Mawhin J., Willem M., Critical point theory and Hamiltonian systems, New York 1989.
• [9] Nowakowski A., Rogowski A., Dependence on Parameters for the Dirichlet problem with Superlinear Nonlinearities, Topological Methods in Nonlinear Analysis, 16 (2000) 1, 145-160.
• [10] Orpel A., On the existence of positive solutions and their continuous dependence on functional parameters for some class of elliptic problems, J. Differential Equations, 204 (2004), pp. 247-264
• [11] Walczak S., On the continuous dependence on parameters of solutions of the Dirichlet problem. Part I Coercive case, Part II The case of saddle points, Bull. Classe Sci. l'Acad. Royale Belgique, 7-12 (1995), 263-273.
• [12] Walczak S., Continuous dependence on parameters and boundary data for nonlinear P.D.E., coercive case, Differential and Integral Equations 11 (1998), 35-46.
• [13] Walczak S., Superlinear variational and boundary value problems with parameters, Nonlinear Anal. 43 (2001) 2. Ser. A: Theory Methods 183-198.