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Necessary conditions for optimal control problems with state constraints


Authors: R. B. Vinter and H. Zheng
Journal: Trans. Amer. Math. Soc. 350 (1998), 1181-1204
MSC (1991): Primary 49K24
DOI: https://doi.org/10.1090/S0002-9947-98-02129-1
MathSciNet review: 1458337
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Abstract: Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded, nonconvex sets and satisfies a mild `one-sided' Lipschitz continuity hypothesis. We recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by Ioffe. For problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one-sided Lipschitz hypothesis is replaced by a milder, local hypothesis. A recent `dualization' theorem permits us to infer a strengthened form of the Hamiltonian inclusion from the Euler Lagrange condition. The necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by Loewen and Rockafellar to achieve related necessary conditions for state constrained problems, and improve on available results in certain respects even when specialized to the state constraint free case.

Proofs make use of recent `decoupling' ideas of the authors, which reduce the optimization problem to one to which Pontryagin's maximum principle is applicable, and a refined penalization technique to deal with the dynamic constraint.


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  • 1. F. H. Clarke, Optimal Solutions to Differential Inclusions, J. Optim. Theory Appl. 19 (1976), 469-478. MR 54:13367
  • 2. F. H. Clarke, ``Optimization and Nonsmooth Analysis, John Wiley, New York, 1983. MR 85m:49002
  • 3. F. H. Clarke, ``Methods of Dynamic and Nonsmooth Optimization'', (CBMS-NSF Regional conference series in applied mathematics, vol 57) SIAM Publication, Philadelphia, 1989. MR 91j:49001
  • 4. A. D. Ioffe, Euler-Lagrange and Hamiltonian Formalisms in Dynamic Optimization, Trans. Amer. Math. Soc. 349 (1997), 2871-2900. MR 97i:49028
  • 5. A. D. Ioffe and R. T. Rockafellar, The Euler and Weierstrass Conditions for Nonsmooth Variational Problems, Calc. Var. Partial Differential Equations 4 (1996), 59-87. MR 96k:49031
  • 6. P. D. Loewen, ``Optimal Control Via Nonsmooth Analysis'', CRM Procedings and Lecture Notes, American Mathematical Society, Providence, 1993. MR 94h:49003
  • 7. P. D. Loewen and R. T. Rockafellar, Optimal Control of Unbounded Differential Inclusions, SIAM J. Control Optim. 32 (1994), 442-470. MR 95h:49043
  • 8. P. D. Loewen and R. T. Rockafellar, New Necessary Conditions for the Generalized Problem of Bolza, SIAM J. Control. Optim. 34 (1996), 1496-1511. MR 97d:49021
  • 9. B. S. Mordukhovich, Maximum Principle in Problems of Time Optimal Control with Nonsmooth Constraints, J. Appl. Math. Mech. 40 (1976), 960-969. MR 58:7284
  • 10. B. S. Mordukhovich,, Generalized Differential Calculus for Nonsmooth and Set-Valued Mappings, J. Math. Anal. Appl. 183 (1994), 250-288. MR 95i:49029
  • 11. B. S. Mordukhovich, Discrete Approximations and Refined Euler-Lagrange Conditions for Nonconvex Differential Inclusions, SIAM J. Control Optim. 33 (1995), 882-915. MR 96d:49028
  • 12. R. T. Rockafellar, Equivalent Subgradient Versions of Hamiltonian and Euler Lagrange Equations in Variational Analysis, SIAM J. Control. Optim. 34 (1996), 1300-1314. CMP 96:14
  • 13. R. B. Vinter and G. Pappas, A Maximum Principle for Nonsmooth Optimal Control Problems with State Constraints, J. Math. Anal. Appl. 89 (1982), 212-232. MR 84b:49026
  • 14. R. B. Vinter and H. Zheng, The Extended Euler-Lagrange Condition for Nonconvex Variational Problems, SIAM J. Control Optim. 35 (1997), 56-77. CMP 97:07

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Additional Information

R. B. Vinter
Affiliation: Department of Electrical and Electronic Engineering and Centre for Process Systems Engineering, Imperial College, Exhibition Road, London SW7 2BT, UK
Email: r.vinter@ic.ac.uk

H. Zheng
Affiliation: Department of Business Studies, University of Edinburgh, 50 George Square, Edinburgh EH9 9JY, Scotland, UK
Email: zheng@maths.ed.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-98-02129-1
Keywords: Euler Lagrange condition, Hamiltonian inclusion, state constraint, nonconvex differential inclusion, nonsmooth analysis
Received by editor(s): May 6, 1996
Additional Notes: This research was carried out with financial support provided by EPSRC
Article copyright: © Copyright 1998 American Mathematical Society

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