Viscosity solutions for the dynamic programming equations. (English) Zbl 0760.49017

Tataru, Daniel

Appl. Math. Optimization 25, No. 2, 109-126 (1992).

This paper extends previous work on Hamilton-Jacobi equations of the form \(u+H(x,u,Du)=0\), \(u_ t+H(t,x,u,Du)=0\), by adding an unbounded nonlinear term \(\langle Ax,Du\rangle\), so that the above equations become \(u+\langle Ax,Du\rangle+H(x,u,Du)=0\) and \(u_ t+\langle Ax,Du\rangle+H(t,x,u,Du)=0\).
Then it is shown that the minimum time problem for an evolution equation of the form \(\dot x+Ax=Bu\), \(x(0)=y\), \(x(\tau)=0\), is a viscosity solution (with respect to \(y\)) if it is continuous in \(y\). Moreover, with a special unbounded nonlinear term, it is shown that the Neumann problem in convex domains may be interpreted as a Hamilton-Jacobi equation with this nonlinear term.

Reviewer: S.P.Banks (Sheffield)

MSC:

49L20	Dynamic programming in optimal control and differential games
49L99	Hamilton-Jacobi theories
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49J15	Existence theories for optimal control problems involving ordinary differential equations
35F20	Nonlinear first-order PDEs

Keywords:

Hamilton-Jacobi equations; minimum time problem; evolution equation; Neumann problem in convex domains

References:

[1]	V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff, Leyden, 1976. · Zbl 0328.47035
[2]	V. Barbu, The dynamic programming equation for the time optimal problem in infinite dimension (to appear). · Zbl 0737.49022
[3]	P. Cannarsa, G. Da Prato, Some Results on Non-Linear Optimal Control Problems and Hamilton-Jacobi Equations in Infinite Dimensions, Preprints di Matematica, No. 16, Scuola Normale Superiore, Pisa, 1988. · Zbl 0717.49022
[4]	M. G. Crandall, 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
[5]	M. G. Crandall, P. L. Lions, Hamilton-Jacobi equations in infinite dimension, I?III, J. Funct. Anal.: (I) 62 (1985), 379-396, (II) 65 (1986), 368-405, (III) 68 (1986), 214-247. · Zbl 0627.49013 · doi:10.1016/0022-1236(85)90011-4
[6]	M. G. Crandall, P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions, IV, Hamiltonians with unbounded linear term, J. Funct. Anal. 90 (1990), 237-283. · Zbl 0739.49016 · doi:10.1016/0022-1236(90)90084-X
[7]	P. L. Lions, Neumann type boundary condition for Hamilton-Jacobi equations, Duke J. Math. 52 (1985), 793-820. · Zbl 0599.35025 · doi:10.1215/S0012-7094-85-05242-1
[8]	D. Tataru, Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear term, J. Math. Anal. (1991). · Zbl 0815.47075