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Existence and Lipschitz regularity of solutions to Bolza problems in optimal control
Author(s):
P.
Cannarsa;
H.
Frankowska;
E.
M.
Marchini
Journal:
Trans. Amer. Math. Soc.
361
(2009),
4491-4517.
MSC (2000):
Primary 49J15, 49J30, 49K15, 49K30
Posted:
April 16, 2009
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Abstract:
In this paper we investigate the existence and Lipschitz continuity of optimal trajectories for the autonomous Bolza problem in control theory. The main feature of our results is that they relax the usual fast growth condition for the Lagrangian. Furthermore, we show that optimal solutions do satisfy the maximum principle.
References:
-
- 1.
- J.-P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. MR 1048347 (91d:49001)
- 2.
- A. Cellina, The classical problem of the calculus of variations in the autonomous case: Relaxation and Lipschitzianity of solutions, Trans. Amer. Math. Soc., 356 (2004), 1, 415-426. MR 2020039 (2004k:49087)
- 3.
- A. Cellina, A. Ferriero, Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 6, 911-919. MR 2008683 (2004f:49069)
- 4.
- L. Cesari, Optimization, Theory and Applications, Springer, Berlin, 1983. MR 688142 (85c:49001)
- 5.
- F. H. Clarke, An indirect method in the Calculus of Variations, Trans. Amer. Math. Soc., 336 (1993), 2, 655-673. MR 1118823 (93f:49002)
- 6.
- F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983. MR 709590 (85m:49002)
- 7.
- F. H. Clarke, R. B. Vinter, Regularity properties of optimal controls, SIAM J. Control Optim., 28 (1990), 4, 980-997. MR 1051634 (91b:49033)
- 8.
- N. Dunford and J. Schwartz, Linear operators, Part I, General Theory, Interscience, New York, 1967. MR 1009162 (90g:47001a)
- 9.
- H. Frankowska, E. M. Marchini, Lipschitzianity of optimal trajectories for the Bolza optimal control problem, Calc. Var. Partial Differential Equations, 27 (2006), 4, 467-492. MR 2263674 (2007f:49021)
- 10.
- A. V. Sarychev, D. F. M. Torres, Lipschitzian regularity of the minimizers for optimal control problems with control-affine dynamics, Appl. Math. Optim., 41 (2000), 2, 237-254. MR 1731420 (2000m:49048)
- 11.
- D. F. M. Torres, Lipschitzian regularity of the minimizing trajectories for nonlinear optimal control problems, Math. Control Signals Systems, 16 (2003), 2-3, 158-174. MR 2006825 (2004i:49047)
- 12.
- R. B. Vinter, Optimal control, Birkhäuser, Boston, Basel, Berlin, 2000. MR 1756410 (2001c:49001)
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Additional Information:
P.
Cannarsa
Affiliation:
Dipartimento di Matematica, Università di Roma ``Tor Vergata'', Via della Ricerca Scientifica 1, 00133 Roma, Italy
Email:
cannarsa@axp.mat.uniroma2.it
H.
Frankowska
Affiliation:
Combinatoire et Optimisation, Université Pierre et Marie Curie (Paris 6) case 189, 4 place Jussieu, 75252 Paris cedex 05, France
Email:
frankowska@math.jussieu.fr
E.
M.
Marchini
Affiliation:
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
Email:
elsa.marchini@polimi.it
DOI:
10.1090/S0002-9947-09-04765-5
PII:
S 0002-9947(09)04765-5
Keywords:
Optimal control,
Bolza problem,
existence of minimizers,
Lipschitz optimal trajectory,
Lipschitz costate
Received by editor(s):
January 27, 2006
Posted:
April 16, 2009
Additional Notes:
This work was supported in part by European Community's Human Potential Programme under contract HPRN-CT-2002-00281, Evolution Equations. This research was completed in part while the first and third authors visited the CREA, École Polytechnique, Paris and also while the second author visited the Dipartimento di Matematica, Università di Roma ``Tor Vergata''
The third author acknowledges the financial support provided through the European Community's Human Potential Programme under contract HPRN-CT-2002-00281, Evolution Equations.
Copyright of article:
Copyright
2009,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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