Optimal control and stability of elliptic systems with integral cost functional

• Autorzy: Bors, D. , Skowron, A. , Walczak, S.
• Języki publikacji: EN

Abstrakty

EN

We consider a system of resonant elliptic equations and using variational methods derive suffcient conditions for solutions of the system to depend continuously on parameters. This result is employed to obtain existence of solution for some optimal control problem.

Słowa kluczowe

EN

Epi/hypo-convergence; saddle points; elliptic boundary problem; stability; optimal control

• Czasopismo: Systems Science
• Rocznik: 2007
• Tom: Vol. 33, no 4
• Strony: 13-26
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

Bors, D.

autor

Skowron, A.

autor

Walczak, S.

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

Bibliografia

• [1] Attouch H., Wets R., Approximation and convergence in nonlinear optimization, Nonlinear Programming, Vol. 4, 1981, pp. 367-394.
• [2] Attouch H., Wets R., A convergence theory for saddle functions, Trans. Amer. Math. Soc., Vol. 280, 1983, pp. 1-44.
• [3] Aze D., Attouch H., Wets R., Convergence of convex-concave saddle functions: applications to convex programming and mechanics, Annales de I’I. H.P., Vol. 5, No. 6, 1988, pp. 537-572.
• [4] Bors D., Skowron A., Walczak S., On the stability of critical points of funclionals and solutions to boundary value problem with applications to optimal control, accepted for publication in post-conference volume of Computer Methods and Systems 2005.
• [5] Cavazzuti E., Maine caratterizzazioni della Γ-convergenza multipla, Ann. Mat. Pura Appl., Vol. 4, 1982, pp. 69-112.
• [6] Cavazzuti E., Γ-convergenze multiple, convergenze di punti di sella e di max-min, Boll. Un. Mat. Ital., Vol. 1-B, No. 6, 1982, pp. 251-274.
• [7] Dal Maso G., An Introduction to Γ-convergence, Birkhäuser, Boston, 1993.
• [8] Greco G., Saddle topology and Min-Max Theorems, Tech. Report. Univ. Trento, 1984.
• [9] Idczak D., Rogowski A., On generalization of Krasnoselskii’s theorem, J. Austral. Math. Soc., Vol. 72, 2002, pp. 389-394.
• [10] Ioffe A.D., On lower semicontinuity of integral functionals, SIAM J. Control Optimization, Vol. 15, No. 6, 1977, pp. 991-1000.
• [11] Jakszto M., Skowron A., Existence of optimal control via continuous dependence on parameters, Comput. Math. Appl.. Vol. 46, 2003, pp. 1657-1669.
• [12] Kisielewicz M., Differential Inclusions and Optimal Controls, Kluwer Academic Publishers, Dordrecht, Boston, 1991.
• [13] Ledzewicz U., Walczak S., Optimal control of systems governed by some elliptic equations, Discrete Contin. Dynam. Systems, Vol. 5, No. 2, 1999, pp. 279-290.
• [14] Polyanin A.D., Linear Partial Differential Equations for Engineers and Scientists, CRC Press, Boca Raton, 2002.
• [15] Rabinowitz P.M., Minimax methods in critical point theory with applications to differential equations, American Mathematical Society, 1992.
• [16] Skowron A., Minimax results for semicoercive functionals with application to differential equations, accepted for publication in Acta Math. Hung.
• [17] Walczak S., On the continuous dependence on parameters of solutions of the Dirichlet problem, Acad. Roy. Belg. Bull. Cl. Sci., Vol. 6, 1995, pp. 247-261 and 263-273.
• [18] Walczak S., Well-posed and ill-posed optimal control problems, Jota, Vol. 109, No. 1, 2001, pp. 169-185.