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The necessary optimality conditions for a nonlinear stationary system whose state functional is not differentiable with respect to the control

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Abstract

We consider a control system described by a nonlinear elliptic equation. Its control-state mapping is extendedly differentiable but not Gateaux differentiable for large values of the domain dimension and the nonlinearity index. We obtain the necessary optimality condition for various state functionals.

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References

  1. S. Ya. Serovaiskii, “Calculation of Functional Gradients and Extended Differentiation of Operators,” J. of Inverse and Ill-Posed Problems 13(4), 383–396 (2005).

    Article  MathSciNet  Google Scholar 

  2. S. Ya. Serovaiskii, Optimization and Differentiation, Vol. 1: Minimization of Functionals. Stationary Systems (Print-S, Almaty, 2006) [in Russian].

    Google Scholar 

  3. I. Ekeland and R. Temam, Convex Analysis and Variational Problems (North-Holland, Amsterdam, 1976; Mir, Moscow, 1979).

    MATH  Google Scholar 

  4. F. Clarke, Optimization and Nonsmooth Analysis (Wiley, New York, 1983; Nauka, Moscow, 1988).

    MATH  Google Scholar 

  5. V. F. Dem’yanov and A. M. Rubinov, Foundations of Nonsmooth Analysis and Quasidifferential Calculus (Nauka, Moscow, 1990) [in Russian].

    MATH  Google Scholar 

  6. L. W. Neustadt, “An Abstract Variational Theory with Application to a Broad Class of Optimization Problems,” SIAM J. Control 4(3), 505–527 (1966).

    Article  MATH  MathSciNet  Google Scholar 

  7. L. W. Neustadt, “An Abstract Variational Theory with Application to a Broad Class of Optimization Problems. II,” SIAM J. Control 5(1), 90–137 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  8. A. D. Ioffe and V. M. Tikhomirov, Theory of Extremal Problems (Nauka, Moscow, 1974) [in Russian].

    Google Scholar 

  9. A. S. Matveev and V. A. Yakubovich, The Abstract Theory of Optimal Control (St.-Petersburg Univ., St.-Petersburg, 1994) [in Russian].

    Google Scholar 

  10. A. V. Dmitruk, A. A. Milyutin, and N. P. Osmolovskii, “Lyusternik’s Theorem and the Theory of Extrema,” Usp. Mat. Nauk 35(6), 11–46 (1980).

    MathSciNet  Google Scholar 

  11. J.-L. Lions, Contrôle des Systèmes Distribués Singuliers (Bordas, Paris, 1983; Nauka, Moscow, 1987).

    MATH  Google Scholar 

  12. U. E. Raitum, Optimal Control Problems for Elliptic Equations (Zinatne, Riga, 1989) [in Russian].

    Google Scholar 

  13. J.-J. Alibert and J.-P. Raymond, “Boundary Control of Semilinear Elliptic Equations with Discontinuous Leading Coefficients and Unbounded Controls,” Numer. Funct. Anal. Optim. 18(3–4), 235–250 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  14. N. Arada and J.-P. Raymond, “State-Constrained Relaxed Problems for Semilinear Elliptic Equations,” J. Math. Anal. Appl. 223(1), 248–271 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  15. A. V. Fursikov, Optimal Control of Distributed Systems: Theory and Applications (Nauchnaya Kniga, Novosobirsk, 1999) [in Russian].

    MATH  Google Scholar 

  16. M. D. Voisei, “Mathematical Programming Problems Governed by Nonlinear Elliptic PDEs,” SIAM J. Optim. 18(4), 1231–1249 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  17. J.-L. Lions, Quelques M éthodes de Résolution des Problèmes aux Limites non Linéaires (Dunod, Paris, 1969; Mir, Moscow, 1972).

    Google Scholar 

  18. L. V. Kantorovich and G. P. Akilov, Functional Analysis (Nauka, Moscow, 1977) [in Russian].

    Google Scholar 

  19. J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis (Wiley, New York, 1984; Mir, Moscow, 1988).

    MATH  Google Scholar 

  20. J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer-Verlag, 1971; Mir, Moscow, 1972).

  21. A. V. Arutyunov, “Second Order Necessary Conditions in Optimal Control Problems,” Dokl. Phys. 371(1), 10–13 (2000).

    MathSciNet  Google Scholar 

  22. V. I. Averbukh and O. G. Smolyanov, “The Theory of Differentiation in Linear Topological Spaces,” Usp. Mat. Nauk 22(6), 201–260 (1967).

    Google Scholar 

  23. H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie-Verlag, Berlin, 1974; Mir, Moscow, 1978).

    MATH  Google Scholar 

  24. J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications (Dunod, Paris, 1968; Mir, Moscow, 1971).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Kazakh National University, ul. Masanchi 39/47, Almaty, 050012, Republic of Kazakhstan

    S. Ya. Serovaiskii

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  1. S. Ya. Serovaiskii
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Correspondence to S. Ya. Serovaiskii.

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Original Russian Text © S.Ya. Serovaiskii, 2010, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2010, No. 6, pp. 32–46.

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Serovaiskii, S.Y. The necessary optimality conditions for a nonlinear stationary system whose state functional is not differentiable with respect to the control. Russ Math. 54, 26–38 (2010). https://doi.org/10.3103/S1066369X10060046

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