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Finite element approximation of time optimal control problems for parabolic equations with Dirichelt boundary conditions

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System Modeling and Optimization

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 38))

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Abstract

Finite element approximation of the time optimal control problem with Dirichlet boundary conditions is considered. Convergence of optimal controls as well as rate of convergence is discussed. An approximation using subspaces which are not required to satisfy zero boundary conditions is also considered.

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References

  1. I. Babuska and A. Aziz, „The Mathematics Foundations of the Finite Element Method with Applications to Partical Differential Equations“, Academic Press, New York, 1972.

    Google Scholar 

  2. A.V. Babakrishnan, „Applied Functional Analysis“, Springer-Verlag, Berlin, 1976.

    Google Scholar 

  3. J. Bramble, A. Schatz Reyleigh-Ritz Galerkin Methods for Dirichlet Problem using subspaces without Bound Cond. Comm on Pure and Applied Math. Vol XXIII 653–675 (1970).

    Google Scholar 

  4. H. Fattorini, The Time Optimal Problem for Boundary Control of the Heat Equation Calculus of Variations and Control Theory. D.L. Russell, ed. Academic Press, New York. pp. 305–320, 1976.

    Google Scholar 

  5. D. Fujiwara, Concrete Characterization of the domains of fractional powers of some elliptic differential operators of the second order. Proc. Japan Acad. 43 (1967), 82–86.

    Google Scholar 

  6. I. Lasiecka. Boundary Control of Parabolic Systems: Finite element approximation. Appl. Math. Optim. 6, 31–62, (1980).

    Google Scholar 

  7. I. Lasiecka. Unified theory for abstract parabolic boundary problems. Applied Math. Optim. 6, 287–333, (1980).

    Google Scholar 

  8. J. Nitsche, Uberin Variationsprinzip zur Losung von Dirichlet-Problems: App. Math. Sem. Univ. Hamburg 36 (1971).

    Google Scholar 

  9. G. Schmidt, N. Weck. On the Behavior of solutions to elliptic and parabolic equations-with applications to boundary control for parabolic equations. SIAM J. Control and Optim. Vol. 16, 4, 493–538, 1978.

    Google Scholar 

  10. T. Seidman. Approximation methods for distributed systems. Mathematics Program at UMBC, Research Report, 79-18.

    Google Scholar 

  11. R.S. Varga. Functional Analysis and Approximation Theory in Numerical Analysis. Rep. Conf. Ser. Appl. Math. publ. by SIAM, Philadelphia, 1971.

    Google Scholar 

  12. D. Washburn. A bound on the boundary input map for parabolic equations with applications to time optimal control. SIAM J. Control and Optim. Vol. 17, No. 5, 1979.

    Google Scholar 

  13. M. Zlamal. "Curved elements in the finite element method I", SIAM J. Number. Anal. 10, 229–240, 1973.

    Google Scholar 

  14. I. Lasiecka, Ritz Salerkin Approximation of time-optimal control problem for parabolic systems with Dirichlet boundary conditions (submitted to SIAM J. Control).

    Google Scholar 

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R. F. Drenick F. Kozin

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© 1982 Springer-Verlag

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Lasiecka, I. (1982). Finite element approximation of time optimal control problems for parabolic equations with Dirichelt boundary conditions. In: Drenick, R.F., Kozin, F. (eds) System Modeling and Optimization. Lecture Notes in Control and Information Sciences, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0006155

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  • DOI: https://doi.org/10.1007/BFb0006155

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11691-2

  • Online ISBN: 978-3-540-39459-4

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