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Continuous dependence of the value function on a class of one-dimensional autonomous optimal control problems and existence of solutions. (English) Zbl 1429.49020

Summary: We give in this paper the proof of existence of solutions for a class of one-dimensional autonomous optimal control problems. The second result in the paper is about convergent subsequences of solutions to problems, we show that the value function depends continuously on the problems.

MSC:

49K05 Optimality conditions for free problems in one independent variable
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[1] J. P. Aubin and H. Frankowska, Set valued analysis, Birkhaeuser, Boston, 1990. · Zbl 0713.49021
[2] H. Bresis, Analyse fonctionnelle, théorie et applications, Masson, Paris, 1983. · Zbl 0511.46001
[3] H. Bresis, Functional analysis, Sobolev spaces and partial differential equations, DOI 10.1007/978-0-387-70914-7 Springer, 2010. · Zbl 1220.46002 · doi:10.1007/978-0-387-70914-7
[4] X. Cabre, Elliptic PDEs in probability and geometry, Discrete Contin. Dyn. Syst. Ser. A, 20(3) (2008), 425-457. · Zbl 1158.35033 · doi:10.3934/dcds.2008.20.425
[5] J. M. Coron, Phantom tracking method, homogeneity and rapid stabilization, Math. Control Relat. Fields, 3(3) (2013), 303-322. · Zbl 1272.93094 · doi:10.3934/mcrf.2013.3.303
[6] J. M. Coron, Controllability and nonlinearity, ESAIM, Proc., 22 (2008), 21-39. · Zbl 1146.93007 · doi:10.1051/proc:072203
[7] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), 1-42. · Zbl 0599.35024 · doi:10.1090/S0002-9947-1983-0690039-8
[8] M. G. Crandall, L. C. Evans, and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Math. Soc., 282(2) (1984), 487-502. · Zbl 0543.35011 · doi:10.1090/S0002-9947-1984-0732102-X
[9] M. I. Kamien and N. L. Schwartz, Dynamic optimisation: The calculus of variations and Optimal control in economic and management, North-Holland, Amesterdam, 1981, pp. 164. · Zbl 0455.49002
[10] N. V. Krylov, Fully nonlinear second order elliptic equations: Recent development, Ann. Sc. Norm. Pisa, 25(3-4) (1997), 569-595. · Zbl 1033.35036
[11] P. L. Lions, Generalized solutions of Hamilton-Jacobi equations, Pitman Research Notes in Mathematics, Longman Scientific and Technical, Harbour, 69 (1982). · Zbl 0497.35001
[12] V. P. Maslov, On a new principle of superposition for optimisation problems, Russian Maths. Survey, 42(3) (1987), 43-84. · Zbl 0707.35138 · doi:10.1070/RM1987v042n03ABEH001439
[13] J. Cristiana Silva, F. M. Delfim Torres, and E. Trélat, On optimal control and its applications, Bol. Soc. Port. Mat., 61 (2009), 11-37. · Zbl 1207.49002
[14] E. Trélat, Optimal control and applications to aerospace: Some results and challenges, J. Optim. Theory Appl., 154(3) (2012), 713-758. · Zbl 1257.49019 · doi:10.1007/s10957-012-0050-5
[15] E. Trélat, Global subanalytic solutions of Hamilton-Jacobi type equations, Ann. I. H. Poincaré, AN23 (2006), 363-387. · Zbl 1094.35020 · doi:10.1016/j.anihpc.2005.05.002
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