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April  2011, 29(2): 453-466. doi: 10.3934/dcds.2011.29.453

Semiconcavity of the value function for a class of differential inclusions

Piermarco Cannarsa 1, and  Peter R. Wolenski 2,

1. 

Dipartimento di Matematica, Via della Ricerca Scientifica 1, Università di Roma 'Tor Vergata’, 00133 Roma, Italy
2. 

Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918, United States

Received  September 2009 Revised  March 2010 Published  October 2010

We provide intrinsic sufficient conditions on a multifunction $F$ and endpoint data φ so that the value function associated to the Mayer problem is semiconcave.
Keywords: differential inclusions, Semiconcave functions, optimal control, Mayer problems..
Mathematics Subject Classification: Primary: 34A60, 49J53; Secondary: 49J1.
Citation: Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 453-466. doi: 10.3934/dcds.2011.29.453
References:
[1]

J.-P. Aubin and H. Frankowska, "Set-Valued Analysis," Birkhäuser, Boston, 1990.  Google Scholar

[2]

P. Cannarsa and H. Frankowska, Some characterizations of optimal trajectories in control theory, SIAM J. Control Optim., 29 (1991), 1322-1347. doi: doi:10.1137/0329068.  Google Scholar

[3]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Birkhäuser, Boston, 2004.  Google Scholar

[4]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983.  Google Scholar

[5]

F. H. Clarke, "Necessary Conditions in Dynamic Optimization," Memoir of the American Mathematical Society, 816, 2005.  Google Scholar

[6]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Springer, New York, 1998.  Google Scholar

[7]

A. Ornelas, Parametrization of Carathéodory multifunctions, Rend. Sem. Mat. Univ. Padova, 83 (1990), 33-44.  Google Scholar

[8]

C. Knuckles and P. R. Wolenski, $C^1$ selections of multifunctions in one dimension, Real Analysis Exchange, 22 (1997), 655-676.  Google Scholar

[9]

A. Pliś, Accessible sets in control theory, in "International Conference on Differential Equations" (Los Angeles, 1974), Academic Press, (1975), 646-650.  Google Scholar

[10]

R. T. Rockafellar and R. Wets, "Variational Analysis," Springer-Verlag, Berlin Heidelberg, 1998. doi: doi:10.1007/978-3-642-02431-3.  Google Scholar

show all references

References:
[1]

J.-P. Aubin and H. Frankowska, "Set-Valued Analysis," Birkhäuser, Boston, 1990.  Google Scholar

[2]

P. Cannarsa and H. Frankowska, Some characterizations of optimal trajectories in control theory, SIAM J. Control Optim., 29 (1991), 1322-1347. doi: doi:10.1137/0329068.  Google Scholar

[3]

P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Birkhäuser, Boston, 2004.  Google Scholar

[4]

F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983.  Google Scholar

[5]

F. H. Clarke, "Necessary Conditions in Dynamic Optimization," Memoir of the American Mathematical Society, 816, 2005.  Google Scholar

[6]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, "Nonsmooth Analysis and Control Theory," Springer, New York, 1998.  Google Scholar

[7]

A. Ornelas, Parametrization of Carathéodory multifunctions, Rend. Sem. Mat. Univ. Padova, 83 (1990), 33-44.  Google Scholar

[8]

C. Knuckles and P. R. Wolenski, $C^1$ selections of multifunctions in one dimension, Real Analysis Exchange, 22 (1997), 655-676.  Google Scholar

[9]

A. Pliś, Accessible sets in control theory, in "International Conference on Differential Equations" (Los Angeles, 1974), Academic Press, (1975), 646-650.  Google Scholar

[10]

R. T. Rockafellar and R. Wets, "Variational Analysis," Springer-Verlag, Berlin Heidelberg, 1998. doi: doi:10.1007/978-3-642-02431-3.  Google Scholar

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