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Controllability and extremality in nonconvex differential inclusions

  • H. D. Tuan
Contributed Papers

Abstract

LetF:[0, T]×R n →2 R n be a set-valued map with compact values; let η:R n →R m be a locally Lipschitzian map,z(t) a given trajectory, andR the reachable set atT of the differential inclusion\(\dot x(t) \in F(t,x(t))\). We prove sufficient conditions for η(z(T))∈intR and establish necessary conditions in maximum principle form for η(z(T))∈(R). As a consequence of these results, we show that every boundary trajectory is simultaneously a Pontryagin extremal, Lagrangian extremal, and relaxed Lagrangian extremal.

Key Words

Differential inclusions η-local controllability η-extremality Kaskosz maximum principle 

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References

  1. 1.
    Warga, J.,Optimization and Controllability without Differentiability Assumptions, SIAM Journal on Control and Optimization, Vol. 22, pp. 837–855, 1983.Google Scholar
  2. 2.
    Warga, J. Controllability, Extremality, and Abnormality in Nonsmooth Optimal Control, Journal of Optimization Theory and Applications, Vol. 41, pp. 239–260, 1983.Google Scholar
  3. 3.
    Clarke, F. H.,Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, New York, 1983.Google Scholar
  4. 4.
    Kaskosz, B., andLojasiewisz, S., Jr.,A Maximum Principle for Generalized Control Systems, Nonlinear Analysis, Vol. 9, pp. 109–130, 1985.Google Scholar
  5. 5.
    Kaskosz, B., andLojasiewsz, S., Jr.,On a Nonsmooth Nonconvex Control Systems, Journal of Mathematical Analysis and Applications, Vol. 136, pp. 39–53, 1988.Google Scholar
  6. 6.
    Kaskosz, B.,A Maximum Principle in Relaxed Controls, Nonlinear Analysis, Vol. 14, pp. 357–367, 1990.Google Scholar
  7. 7.
    Kaskosz, B., andLojasiewisz, S., Jr.,Lagrange-Type Extremal Trajectories in Differential Inclusions, Systems and Control Letters, Vol. 19, pp. 241–247, 1992.Google Scholar
  8. 8.
    Kaskosz, B., andLojasiewisz, S., Jr.,Boundary Trajectories of Systems with Unbounded Controls, Journal of Optimization Theory and Applications, Vol. 70, pp. 539–559, 1991.Google Scholar
  9. 9.
    Lojasiewisz, S., Jr.,Invariance of Extremals, Nonlinear Controllability and Optimal Control, Edited by H. J. Sussman, Marcel Dekker, New York, pp. 98–116, 1990.Google Scholar
  10. 10.
    Loewen, P. D., andRockafellar, R. T.,The Adjoint Arc in Nonsmooth Optimization, Transactions of the American Mathematical Society, Vol. 325, pp. 39–72, 1991.Google Scholar
  11. 11.
    Warga, J.,An Extension of the Kaskosz Maximum Principle, Applied Mathematics and Optimization, Vol. 22, pp. 61–74, 1990.Google Scholar
  12. 12.
    Frankowska, H., andKaskosz, B.,A Maximum Principle for Differential Inclusion Problems with State Constraints, Systems and Control Letters, Vol. 11, pp. 189–194, 1988.Google Scholar
  13. 13.
    Colombo, R. M., Fryszkowski, A., Rzezuchowski, T. andStaicu, V.,Continuous Selections of Solution Sets of Lipschitzian Differential Inclusions, Funkcialaj Ekvacioj, Vol. 34, pp. 321–330, 1991.Google Scholar
  14. 14.
    Fryszkowski, A., andRzezuchowski, T.,Continuous Versions of the Filippov-Wazewski Relaxation Theorem, Journal of Differential Equations, Vol. 94, pp. 254–265, 1991.Google Scholar
  15. 15.
    Aubin, J. P., andFrankowska, H.,Set-Valued Analysis, Birkhauser, Boston, Massachusetts, 1990.Google Scholar
  16. 16.
    Polovikin, E. S., andSmirnov, V. G.,An Approach to Differentiation of Set-Valued Maps and Necessary Optimality Conditions for Differential Inclusions, Differentialnye Uravnenia, Vol. 22, pp. 944–954, 1986 (in Russian).Google Scholar
  17. 17.
    Tuan, H. D.,f-Local Controllability and Extremal Trajectories in Systems under Uncertainty Governed by Inclusions, Preprint, Institute of Mathematics, Honoi, Vietnam, 1992.Google Scholar
  18. 18.
    Frankowska, H., andKaskosz, B.,Linearization and Boundary Trajectories of Nonsmooth Control Systems, Canadian Journal of Mathematics, Vol. 40, pp. 589–609, 1988.Google Scholar
  19. 19.
    Tuan, H. D.,On Stability in Local Controllability Problems for Discrete Non-linear Systems, Optimization, Vol. 29, pp. 157–172, 1994.Google Scholar
  20. 20.
    Frankowska, H.,Contingent Cones to Reachable Sets of Control Systems, SIAM Journal on Control and Optimization, Vol. 27, pp. 170–198, 1989.Google Scholar
  21. 21.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  22. 22.
    Loewen, P. D., andVinter, R. B.,Pontryagin-Type Necessary Conditions for Differential Inclusions Problems, Systems and Control Letters, Vol. 9, pp. 263–265, 1987.Google Scholar
  23. 23.
    Loewen, P. D., Clarke, F. H., andVinter, R. B.,Differential Inclusions with Free Time, Annales de l'Institut Henri Poincaré, Analyse Nonlineaire, Vol. 5, pp. 573–593, 1988.Google Scholar
  24. 24.
    Rowland, J. D. L., andVinter, R. B.,Pontryagin-Type Condition for Differential Inclusions with Free time, Journal of Mathematical Analysis and Applications, Vol. 165, pp. 587–597, 1992.Google Scholar
  25. 25.
    Vinter, R. B., andPapas, G.,A Maximum Principle for Nonsmooth Optimal Control Problems with State Constraints, Journal of Mathematical Analysis and Applications, Vol. 44, pp. 212–232, 1982.Google Scholar

Copyright information

© Plenum Publishing Corporation 1995

Authors and Affiliations

  • H. D. Tuan
    • 1
  1. 1.Department of Electronic and Mechanical EngineeringNagoya UniversityNagoyaJapan

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