Regularity along optimal trajectories of the value function of a Mayer problem

Carlo Sinestrari

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 4, page 666-676
  • ISSN: 1292-8119

Abstract

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We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system.

How to cite

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Sinestrari, Carlo. "Regularity along optimal trajectories of the value function of a Mayer problem." ESAIM: Control, Optimisation and Calculus of Variations 10.4 (2010): 666-676. <http://eudml.org/doc/90750>.

@article{Sinestrari2010,
abstract = { We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system. },
author = {Sinestrari, Carlo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal control; value function; semiconcavity.; optimal control; semiconcavity},
language = {eng},
month = {3},
number = {4},
pages = {666-676},
publisher = {EDP Sciences},
title = {Regularity along optimal trajectories of the value function of a Mayer problem},
url = {http://eudml.org/doc/90750},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Sinestrari, Carlo
TI - Regularity along optimal trajectories of the value function of a Mayer problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 4
SP - 666
EP - 676
AB - We consider an optimal control problem of Mayer type and prove that, under suitable conditions on the system, the value function is differentiable along optimal trajectories, except possibly at the endpoints. We provide counterexamples to show that this property may fail to hold if some of our conditions are violated. We then apply our regularity result to derive optimality conditions for the trajectories of the system.
LA - eng
KW - Optimal control; value function; semiconcavity.; optimal control; semiconcavity
UR - http://eudml.org/doc/90750
ER -

References

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  1. P. Albano and P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations. Arch. Ration. Mech. Anal.162 (2002) 1-23.  
  2. M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi equations. Birkhäuser, Boston (1997).  
  3. P. Cannarsa and H. Frankowska, Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim.29 (1991) 1322-1347.  
  4. P. Cannarsa, C. Pignotti and C. Sinestrari, Semiconcavity for optimal control problems with exit time. Discrete Contin. Dyn. Syst.6 (2000) 975-997.  
  5. P. Cannarsa and C. Sinestrari, Convexity properties of the minimum time function. Calc. Var.3 (1995) 273-298.  
  6. P. Cannarsa and C. Sinestrari, On a class of nonlinear time optimal control problems. Discrete Contin. Dyn. Syst.1 (1995) 285-300.  
  7. P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations and optimal control. Birkhäuser, Boston (2004).  
  8. P. Cannarsa and H.M. Soner, Generalized one-sided estimates for solutions of Hamilton-Jacobi equations and applications. Nonlinear Anal.13 (1989) 305-323.  
  9. P. Cannarsa and M. E. Tessitore, On the behaviour of the value function of a Mayer optimal control problem along optimal trajectories, in Control and estimation of distributed parameter systems (Vorau, 1996). Internat. Ser. Numer. Math.126 81-88 (1998).  
  10. F.H. Clarke and R.B. Vinter, The relationship between the maximum principle and dynamic programming. SIAM J. Control Optim.25 (1987) 1291-1311.  
  11. W.H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation. J. Diff. Eq.5 (1969) 515-530.  
  12. N.N. Kuznetzov and A.A. Siskin, On a many dimensional problem in the theory of quasilinear equations. Z. Vycisl. Mat. i Mat. Fiz.4 (1964) 192-205.  
  13. P.L. Lions, Generalized solutions of Hamilton-Jacobi equations. Pitman, Boston (1982).  
  14. R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970).  
  15. X.Y. Zhou, Maximum principle, dynamic programming and their connection in deterministic control. J. Optim. Theory Appl.65 (1990) 363-373.  

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