Skip to main content
SpringerLink
Log in

Optimal control problems with state constraints for semilinear distributed-parameter systems

  • Contributed Papers
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We consider optimal control problems for distributed-parameter systems described by semilinear equations, with constraints on the control and on the state, and an exact pointwise target condition. As an application of a general theory of nonlinear programming problems in Banach spaces, a version of the Pontryagin maximum principle is obtained.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bonnans, J. F., andCasas, E.,A Boundary Pontryagin's Principle for the Optimal Control of State-Constrained Elliptic Systems, International Series of Numerical Mathematics, Birkhäuser, Basel, Switzerland, Vol. 107, pp. 241–249, 1992.

    Google Scholar 

  2. Casas, E.,Optimal Control in Coefficients of Elliptic Equations with State Constraints, Applied Mathematics and Optimization, Vol. 26, pp. 21–37, 1992.

    Google Scholar 

  3. Casas, E., andFernandez, L. A.,Optimal Control of Semilinear Elliptic Equations with Pointwise Constraints on the Gradient of the State, Applied Mathematics and Optimization, Vol. 27, pp. 35–56, 1993.

    Google Scholar 

  4. Alt, W., andMackenroth U.,Convergence of Finite Element Approximations to State Constrained Convex Parabolic Boundary Control Problems, SIAM Journal of Control and Optimization, Vol. 27, pp. 718–736, 1989.

    Google Scholar 

  5. Goldberg, H., andTröltzsch, F.,Second-Order Sufficient Optimality Conditions for a Class of Nonlinear Parabolic Boundary Control Problems, SIAM Journal on Control and Optimization, Vol. 31, pp. 1007–1025, 1993.

    Google Scholar 

  6. Mackenroth, U.,Optimalitätsbedingungen und Dualität bei zustandsrestringierten Parabolischen Kontrollproblemen, Mathematische Operationsforschung und Statistik, Series Optimization, Vol. 12, pp. 65–89, 1981.

    Google Scholar 

  7. Mackenroth, U.,Time-Optimal Parabolic Boundary Control Problems with State Constraints, Numerical Functional Analysis and Optimization, Vol. 3, pp. 285–300, 1981.

    Google Scholar 

  8. Mackenroth, U.,On Some Elliptic Control Problems with State Constraints, Optimization, Vol. 17, pp. 595–607, 1986.

    Google Scholar 

  9. Mackenroth, U.,Convex Parabolic Boundary Control Problems with Pointwise State Constraints, Journal of Mathematical Analysis and Applications, Vol. 48, pp. 102–116, 1986.

    Google Scholar 

  10. Roubicek, T.,Optimal Control of a Stefan Problem with State-Space Constraints, Numerische Mathematik, Vol. 50, pp. 723–744, 1987.

    Google Scholar 

  11. Tiba, D.,Optimal Control of Nonsmooth Distributed-Parameter Systems, Lecture Notes in Mathematics, Springer, Heidelberg, Germany, Vol. 1459, 1990.

    Google Scholar 

  12. Tiba, D.,Nonqualified Convex Control Problems, Preprint, Iasi University, Iasi, Romania, 1993.

    Google Scholar 

  13. Tröltzsch, F.,On Some Parabolic Boundary Control Problems with Constraints on the Control and Functional-Controls on the State, Zeitschrift für Analysis und Ihre Anwendungen, Vol. 1, pp. 1–13, 1982.

    Google Scholar 

  14. Tröltzsch, F.,Optimality Conditions for Parabolic Control Problems and Applications, Teubner, Leipzig, Germany, 1984.

    Google Scholar 

  15. White, L. W.,Control of a Hyperbolic Problem with Pointwise Stress Constraints, Journal of Optimization Theory and Applications, Vol. 41, pp. 359–369, 1983.

    Google Scholar 

  16. Fattorini, H. O., andFrankowska, H.,Infinite-Dimensional Control Problems with State Constraints, Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, Vol. 154, pp. 52–62, 1991.

    Google Scholar 

  17. Fattorini, H. O.,The Maximum Principle for Nonlinear Nonconvex Systems in Infinite-Dimensional Spaces, Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, Vol. 75, pp. 162–178, 1985.

    Google Scholar 

  18. Fattorini, H. O.,A Unified Theory of Necessary Conditions for Nonlinear Nonconvex Control Systems, Applied Mathematics and Optimization, Vol. 15, pp. 141–185, 1987.

    Google Scholar 

  19. Fattorini, H. O., andFrankowska, H.,Necessary Conditions for Infinite Dimensional Control Problems, Lecture Notes in Control and Information Sciences, Springer, Heidelberg, Germany, Vol. 111, pp. 381–392, 1990.

    Google Scholar 

  20. Fattorini, H. O., andFrankowska, H.,Necessary Conditions for Infinite Dimensional Control Problems, Mathematics of Control, Signals, and Systems, Vol. 4, pp. 41–67, 1991.

    Google Scholar 

  21. Frankowska, H.,Some Inverse Mapping Theorems, Annales de l'Institut Henri Poincaré, Vol. 7, pp. 183–234, 1990.

    Google Scholar 

  22. Fattorini, H. O., Optimal Control Problems for Distributed Parameter Systems Governed by Semilinear Parabolic Equations in LI and L∞ Spaces, Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, Vol. 149, pp. 68–80, 1991.

    Google Scholar 

  23. Fattorini, H. O.,Optimal Control Problems for Distributed Parameter Systems in Banach Spaces, Applied Mathematics and Optimization, Vol. 28, pp. 225–257, 1993.

    Google Scholar 

  24. Hille, E., andPhillips, R. S.,Functional Analysis and Semigroups, Colloquium Publications, American Mathematical Society, Providence, Rhode Island, Vol. 31, 1957.

    Google Scholar 

  25. Li, X., andYao, Y.,Maximum Principle of Distributed Parameter Systesm with Time Lags, Lecture Notes in Control and Information Sciences, Springer, Heidelberg, Germany, Vol. 75, pp. 410–427, 1985.

    Google Scholar 

  26. Diestel, J., andUhl, J. J.,Vector Measures, American Mathematical Society, Providence, Rhode Island, 1977.

    Google Scholar 

  27. Bartle, R. G.,A General Bilinear Vector Integral, Studia Mathematica, Vol. 15, pp. 337–352, 1956.

    Google Scholar 

  28. Dunford, N., andSchwartz, J. T.,Linear Operators, Interscience-Wiley, New York, New York, 1957.

    Google Scholar 

  29. Adams, R. A. Sobolev Spaces, Academic Press, New York, New York, 1975.

    Google Scholar 

  30. Fattorini, H. O.,The Time Optimal Problem for Distributed Control of Systems Described by the Wave Equation, Control Theory of Systems Governed by Partial Differential Equations, Edited by M. J. Balas and J. Wingate, Academic Press, New York, New York, pp. 151–175, 1977.

    Google Scholar 

  31. Fattorini, H. O., andFrankowska, H.,Explicit Convergence Estimates for Suboptimal Controls, I, Problems of Control and Information Theory, Vol. 19, pp. 3–29, 1990.

    Google Scholar 

  32. Fattorini, H. O., andFrankowska, H.,Explicit Convergence Estimates for Suboptimal Controls, II, Problems of Control and Information Theory, Vol. 19, pp. 69–93, 1990.

    Google Scholar 

  33. Fattorini, H. O.,Existence Theory and the Maximum Principle for Relaxed Infinite-Dimensional Optimal Control Problems, SIAM Journal on Control and Optimization, Vol. 32, pp. 311–331, 1994.

    Google Scholar 

  34. Fattorini, H. O.,Optimal Control Problems with State Constraints for Parabolic Distributed-Parameter Systems, Discrete and Continuous Dynamic Systems, Vol. 1, pp. 77–102, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, California

    H. O. Fattorini (Professor)

Authors
  1. H. O. Fattorini
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by F. A. Potra

This research was partly supported by the National Science Foundation under Grant DMS-92-21819.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fattorini, H.O. Optimal control problems with state constraints for semilinear distributed-parameter systems. J Optim Theory Appl 88, 25–59 (1996). https://doi.org/10.1007/BF02192021

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02192021

Key Words

Search

Navigation