Optimal control and semicontinuous viscosity solutions

Barron, E. N.; Jensen, R.

doi:10.1090/S0002-9939-1991-1076572-8

Optimal control and semicontinuous viscosity solutions
HTML articles powered by AMS MathViewer

by E. N. Barron and R. Jensen

Proc. Amer. Math. Soc. 113 (1991), 397-402

DOI: https://doi.org/10.1090/S0002-9939-1991-1076572-8

PDF | Request permission

Abstract:

We study the value function for an optimal control problem with upper semicontinuous terminal data. We prove that the upper semicontinuous envelope of the value function is the unique semicontinuous viscosity solution of the Bellman equation and that it coincides with the value function obtained when using relaxed controls.

References

G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 4, 557–579 (English, with French summary). MR 921827, DOI 10.1051/m2an/1987210405571
E. N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians, Comm. Partial Differential Equations 15 (1990), no. 12, 1713–1742. MR 1080619, DOI 10.1080/03605309908820745
Lamberto Cesari, Optimization—theory and applications, Applications of Mathematics (New York), vol. 17, Springer-Verlag, New York, 1983. Problems with ordinary differential equations. MR 688142, DOI 10.1007/978-1-4613-8165-5
Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8
Hitoshi Ishii, Perron’s method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987), no. 2, 369–384. MR 894587, DOI 10.1215/S0012-7094-87-05521-9
N. N. Krasovskiĭ and A. I. Subbotin, Game-theoretical control problems, Springer Series in Soviet Mathematics, Springer-Verlag, New York, 1988. Translated from the Russian by Samuel Kotz. MR 918771, DOI 10.1007/978-1-4612-3716-7
J. Warga, Optimal control of differential and functional equations, Academic Press, New York-London, 1972. MR 0372708