Abstract
We consider infinite-dimensional nonlinear programming problems which consist of minimizing a functionf 0(u) under a target set constraint. We obtain necessary conditions for minima that reduce to the Kuhn-Tucker conditions in the finite-dimensional case. Among other applications of these necessary conditions and related results, we derive Pontryagin’s maximum principle for a class of control systems described by semilinear equations in Hilbert space and study convergence properties of sequences of near-optimal controls for these systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J.-P. Aubin and I. Ekeland,Applied Nonlinear Analysis, Wiley-Interscience, New York, 1984.
F. Clarke, The maximum principle under minimum hypotheses,SIAM J. Control Optim.,14 (1976), 1078–1091.
F. Clarke,Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.
F. Clarke and P. Löwen, State constraints in optimal control: a proximal normal analysis approach,SIAM J. Control Optim. (to appear).
I. Ekeland, Sur les problémes variationnels,C. R. Acad. Sci. Paris,275 (1972), 1057–1059.
I. Ekeland, Nonconvex minimization problems,Bull. Amer. Math. Soc. (N.S.),1 (1979), 443–474.
H. O. Fattorini, A unified theory of necessary conditions for nonlinear nonconvex control systems,Appl. Math. Optim.,15 (1987), 141–185.
H. O. Fattorini,Optimal Control of Nonlinear Systems: Convergence of Suboptimal Controls, I, pp. 159–199, Lecture Notes in Pure and Applied Mathematics, Vol. 108, Marcel Dekker, New York, 1987.
H. O. Fattorini,Optimal Control of Nonlinear Systems: Convergence of Suboptimal Controls, II, pp. 230–266, Lecture Notes on Control and Information Sciences, Vol. 97, Springer-Verlag, Berlin, 1987.
H. O. Fattorini,Convergence of Suboptimal Controls for Point Targets, pp. 91–107. International Series of Numerical Mathematics, Vol. 78, Birkhäuser, Basel, 1987.
H. O. Fattorini, Some remarks on convergence of suboptimal controls, inSoftware Optimization, pp. 359–363, A. V. Balakrishnan Anniversary Volume, New York, 1988.
H. O. Fattorini, Convergence of suboptimal controls: the point target case,SIAM J. Control Optim. (to appear).
H. O. Fattorini, Constancy of the Hamiltonian in infinite-dimensional systems, inProceedings of the 4th International Conference on Control of Distributed Parameter Systems, Vorau, 1988, pp. 123–133, International Series of Numerical Mathematics, Vol. 91, Birkhäuser, Basel, 1989.
H. O. Fattorini and H. Frankowska, Necessary conditions for infinite-dimensional control problems,Proceedings of the 8th International Conference on Analysis and Optimization of Systems, Antibes-Juan Les Pins, June 1988, Lecture Notes on Control and Information Sciences, Springer-Verlag, Berlin (to appear).
H. O. Fattorini and H. Frankowska, Explict convergence estimates for suboptimal controls, I, II,Problems Control Inform. Theory (to appear).
H. Frankowska, The maximum principle for differential inclusions with end point constraints,SIAM J. Control Optim.,25 (1987), 145–157.
H. Frankowska,On the Linearization of Nonlinear Control Systems and Exact Reachability, pp. 132–143, Lecture Notes on Control and Information Sciences, Vol. 114, Springer-Verlag, Berlin, 1988.
D. G. Luenberger,Linear and Nonlinear Programming (2nd edn.), Addison-Wesley, Reading, MA, 1984.
Author information
Authors and Affiliations
Additional information
The work of this author was supported in part by the National Science Foundation under Grant DMS-8701877.
Rights and permissions
About this article
Cite this article
Fattorini, H.O., Frankowska, H. Necessary conditions for infinite-dimensional control problems. Math. Control Signal Systems 4, 41–67 (1991). https://doi.org/10.1007/BF02551380
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02551380