# American Institute of Mathematical Sciences

• Previous Article
The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives
• DCDS Home
• This Issue
• Next Article
Existence of minimizers for nonautonomous highly discontinuous scalar multiple integrals with pointwise constrained gradients
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, 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
 [1] Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455 [2] Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051 [3] Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028 [4] Maciej Smołka. Asymptotic behaviour of optimal solutions of control problems governed by inclusions. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 641-652. doi: 10.3934/dcds.1998.4.641 [5] Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116 [6] Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations & Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024 [7] Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066 [8] Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107 [9] Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989 [10] Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036 [11] Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020053 [12] Tomasz Kapela, Piotr Zgliczyński. A Lohner-type algorithm for control systems and ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 365-385. doi: 10.3934/dcdsb.2009.11.365 [13] Jacson Simsen, Mariza Stefanello Simsen, José Valero. Convergence of nonautonomous multivalued problems with large diffusion to ordinary differential inclusions. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2347-2368. doi: 10.3934/cpaa.2020102 [14] Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013 [15] Clara Carlota, António Ornelas. The DuBois-Reymond differential inclusion for autonomous optimal control problems with pointwise-constrained derivatives. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 467-484. doi: 10.3934/dcds.2011.29.467 [16] Ishak Alia. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach. Mathematical Control & Related Fields, 2020, 10 (4) : 785-826. doi: 10.3934/mcrf.2020020 [17] Bavo Langerock. Optimal control problems with variable endpoints. Conference Publications, 2003, 2003 (Special) : 507-516. doi: 10.3934/proc.2003.2003.507 [18] Alberto Bressan, Yunho Hong. Optimal control problems on stratified domains. Networks & Heterogeneous Media, 2007, 2 (2) : 313-331. doi: 10.3934/nhm.2007.2.313 [19] M'hamed Kesri. Structural stability of optimal control problems. Communications on Pure & Applied Analysis, 2005, 4 (4) : 743-756. doi: 10.3934/cpaa.2005.4.743 [20] Piermarco Cannarsa, Hélène Frankowska, Elsa M. Marchini. On Bolza optimal control problems with constraints. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 629-653. doi: 10.3934/dcdsb.2009.11.629

2019 Impact Factor: 1.338