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The time-optimal control problem in Banach spaces

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Abstract

We show here that a variant of Pontryagin's maximum principle holds for time-optimal trajectories of the systemu′ = Au + f that begin and end at the domain ofA. HereA is the infinitesimal generator of a strongly continuous semigroupS(·) in the reflexive Banach spaceE. This partly generalizes a result of Balakrishnan ([1]) where no conditions are assumed on the trajectories but whereS(t) is supposed to beonto for allt ≥ 0. A weakened version of the maximum principle is then proved forall time-optimal trajectories whenA generates an analytic semigroup andE is a Hilbert space. Finally, the results are modified to handle two examples involving the heat equation where the spaceE is not reflexive.

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References

  1. A. V. Balakrishnan, Optimal control problem in Banach spaces,Journal of the Society for Industrial and Applied Mathematics Series A: On Control 3 (1965) 152–180.

    Google Scholar 

  2. A. V. Balakrishnan, Introduction to Optimization Theory in a Hilbert space,Lecture Notes in Operations Research and Mathematical Systems, no. 42, Springer, Berlin, 1971.

    Google Scholar 

  3. R. Bellman, I. Glicksberg andO. Gross, On the “bang-bang” control problems,Quarterly of Applied Mathematics 14 (1956), 11–18.

    Google Scholar 

  4. S. Bochner andA. E. Taylor, Linear functionals on certain spaces of abstractly-valued functions,Annals of Mathematics 39 (1938), 913–944.

    Google Scholar 

  5. A. G. Butkovskii, Theory of optimal control of system with distributed parameters (Russian,Nauka, Moscow, 1965. English translation: Distributed control systems Elsevier, New York, 1969.

    Google Scholar 

  6. P. L. Butzer andH. Berens, Semi-groups of operators and approximation, Springer, New York, 1967.

    Google Scholar 

  7. L. De Simon, Un' applicazione della teoria degli integrali singulari allo studio delle equazioni differenziali lineari astratte del primo ordine,Rendiconti del Seminario Matematico della Universitá de Padova XXXIV (1964), 205–223.

    Google Scholar 

  8. J. Dieudonné, Sur le théorème de Lebesgue-Nikodym (V),Canadian Journal of Mathematics 3 (1951), 129–139.

    Google Scholar 

  9. N. Dunford andJ. T. Schwartz, Linear operators, part I,Interscience, New York, 1957.

  10. N. Dunford andJ. T. Schwartz, Linear operators, part II, Wiley, New York, 1963.

    Google Scholar 

  11. Yu. B. Egorov, Optimal control in Banach spaces (Russian),Doklady Akademii NAUK SSSR 50 (1963), 241–244. English translation.

    Google Scholar 

  12. Yu. B. Egorov, Some necessary conditions for optimality in Banach spaces,Mathematicheskii Sbornik 64 (1964), 79–101.

    Google Scholar 

  13. H. O. Fattorini, Time-optimal control of solutions of operational differential equations,Journal of the Society for Industrial and Applied Mathematics Series A: On Control 2 (1964), 54–59.

    Google Scholar 

  14. A. Friedman, Optimal control in Banach spaces,Journal of Mathematical Analysis and Applications 19 (1967), 35–55.

    Google Scholar 

  15. A. Friedman, Optimal control in Banach spaces with fixed end-points,Journal of Mathematical Analysis and Applications 24 (1968), 161–181.

    Google Scholar 

  16. A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N.J., 1964.

    Google Scholar 

  17. E. Hille andR. S. Phillips, Functional analysis and semi-groups,American Mathematical Society Colloquium Publications, vol. XXXI, Providence, 1957.

  18. J. L. Lions, Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles, Dunod, Paris, 1968.

    Google Scholar 

  19. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze andE. F. Mischenko, The mathematical theory of optimal processes (Russian), Goztekhizdat, Moscow, 1961, English translation: Wiley, New York, 1962.

    Google Scholar 

  20. A. I. Tikhonov andA. A. Samarskii,Equations of mathematical physics (Russian), Goztekhizdat, Moscow, 1953. English translation:

    Google Scholar 

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Authors and Affiliations

  1. Departments of Mathematics and System Science, University of California, 90024, Los Angeles, California, USA

    H. O. Fattorini

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  1. H. O. Fattorini
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This work was partly supported by the National Science Foundation under contract GP-27973.

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Fattorini, H.O. The time-optimal control problem in Banach spaces. Appl Math Optim 1, 163–188 (1974). https://doi.org/10.1007/BF01449028

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