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Regularity along optimaltrajectories of the value function of a Mayer problem

Published online by Cambridge University Press:  15 October 2004

Carlo Sinestrari
Carlo Sinestrari*
Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma, Italy; sinestra@mat.uniroma2.it.
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Abstract

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

Keywords

Optimal controlvalue functionsemiconcavity.

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 4 , October 2004 , pp. 666 - 676
Copyright
© EDP Sciences, SMAI, 2004

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References

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