# American Institute of Mathematical Sciences

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The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives
April  2011, 29(2): 453-466. doi: 10.3934/dcds.2011.29.453

## Semiconcavity of the value function for a class of differential inclusions

 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.
Citation: Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 453-466. doi: 10.3934/dcds.2011.29.453
##### References:
 [1] J.-P. Aubin and H. Frankowska, "Set-Valued Analysis,", Birkhäuser, (1990). [2] P. Cannarsa and H. Frankowska, Some characterizations of optimal trajectories in control theory,, SIAM J. Control Optim., 29 (1991), 1322. doi: doi:10.1137/0329068. [3] P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,", Birkhäuser, (2004). [4] F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983). [5] F. H. Clarke, "Necessary Conditions in Dynamic Optimization,", Memoir of the American Mathematical Society, 816 (2005). [6] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Springer, (1998). [7] A. Ornelas, Parametrization of Carathéodory multifunctions,, Rend. Sem. Mat. Univ. Padova, 83 (1990), 33. [8] C. Knuckles and P. R. Wolenski, $C^1$ selections of multifunctions in one dimension,, Real Analysis Exchange, 22 (1997), 655. [9] A. Pliś, Accessible sets in control theory,, in, (1975), 646. [10] R. T. Rockafellar and R. Wets, "Variational Analysis,", Springer-Verlag, (1998). doi: doi:10.1007/978-3-642-02431-3.

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##### References:
 [1] J.-P. Aubin and H. Frankowska, "Set-Valued Analysis,", Birkhäuser, (1990). [2] P. Cannarsa and H. Frankowska, Some characterizations of optimal trajectories in control theory,, SIAM J. Control Optim., 29 (1991), 1322. doi: doi:10.1137/0329068. [3] P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,", Birkhäuser, (2004). [4] F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983). [5] F. H. Clarke, "Necessary Conditions in Dynamic Optimization,", Memoir of the American Mathematical Society, 816 (2005). [6] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Springer, (1998). [7] A. Ornelas, Parametrization of Carathéodory multifunctions,, Rend. Sem. Mat. Univ. Padova, 83 (1990), 33. [8] C. Knuckles and P. R. Wolenski, $C^1$ selections of multifunctions in one dimension,, Real Analysis Exchange, 22 (1997), 655. [9] A. Pliś, Accessible sets in control theory,, in, (1975), 646. [10] R. T. Rockafellar and R. Wets, "Variational Analysis,", Springer-Verlag, (1998). doi: doi:10.1007/978-3-642-02431-3.
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