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1995 | 62 | 2 | 111-121
Tytuł artykułu

Convergence of optimal solutions in control problems for hyperbolic equations

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EN
Abstrakty
EN
A sequence of optimal control problems for systems governed by linear hyperbolic equations with the nonhomogeneous Neumann boundary conditions is considered. The integral cost functionals and the differential operators in the equations depend on the parameter k. We deal with the limit behaviour, as k → ∞, of the sequence of optimal solutions using the notions of G- and Γ-convergences. The conditions under which this sequence converges to an optimal solution for the limit problem are given.
Twórcy
autor
  • Institute of Computer Science Jagiellonian University Nawojki 11 30-072 Kraków, Poland
Bibliografia
  • [1] A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978.
  • [2] G. Buttazzo and G. Dal Maso, Γ-convergence and optimal control problems, J. Optim. Theory. Appl. 38 (1982), 385-407.
  • [3] F. Colombini et S. Spagnolo, Sur la convergence de solutions d'équations paraboliques, J. Math. Pures Appl. 56 (1977), 263-306.
  • [4] F. Colombini et S. Spagnolo, On convergence of solutions of hyperbolic equations, Comm. Partial Differential Equations 3 (1978), 77-91.
  • [5] E. De Giorgi, Convergence problems for functionals and operators, in: Proc. Internat. Meeting on Recent Methods in Nonlinear Analysis, E. De Giorgi, E. Magenes and U. Mosco (eds.), Pitagora, Bologna, 1979, 131-188.
  • [6] E. De Giorgi e T. Franzoni, Su un tipo di convergenza variazionale, Rend. Sem. Mat. Brescia 3 (1979), 63-101.
  • [7] E. De Giorgi e S. Spagnolo, Sulla convergenza degli integrali della energia per operatori ellitici del secondo ordine, Boll. Un. Mat. Ital. 8 (1973), 391-411.
  • [8] Z. Denkowski and S. Migórski, Control problems for parabolic and hyperbolic equations via the theory of G and Γ convergence, Ann. Mat. Pura Appl. (4) 149 (1987), 23-39.
  • [9] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.
  • [10] J. L. Lions, Some Methods in the Mathematical Analysis of Systems and their Control, Science Press, Beijing and Gordon and Breach, New York, 1981.
  • [11] J. L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems, Vol. I, Springer, Berlin, 1972.
  • [12] P. Marcellini e C. Sbordone, Dualità e perturbazione di funzionali integrali, Ricerche Mat. 26 (1977), 383-421.
  • [13] S. Migórski, Convergence of optimal solutions in control problems for hyperbolic equations, preprint CSJU 1/1991, Jagiellonian University, Kraków.
  • [14] S. Migórski, Asymptotic behaviour of optimal solutions in control problems for elliptic equations, Riv. Mat. Pura Appl. 11 (1992), 7-28.
  • [15] S. Migórski, On asymptotic limits of control problems with parabolic and hyperbolic equations, Riv. Mat. Pura Appl. 12 (1992), 33-50.
  • [16] S. Migórski, Sensitivity analysis of distributed parameter optimal control problems for nonlinear parabolic equations, J. Optim. Theory Appl. 87 (1995), to appear.
  • [17] E. Sanchez-Palencia, Nonhomogeneous Media and Vibration Theory, Lecture Notes in Phys. 127, Springer, Berlin, 1980.
  • [18] S. Spagnolo, Sulla convergenza di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 577-597.
  • [19] S. Spagnolo, Convergence in energy for elliptic operators, in: Proc. Third Symp. Numer. Solutions PDE (College Park, 1975), Academic Press, San Diego, 1976, 469-498.
  • [20] V. V. Zhikov, S. M. Kozlov, O. A. Oleĭnik and Kha T'en Ngoan, Averaging and G-convergence of differential operators, Russian Math. Surveys 34 (1979), 69-147.
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