Semiconcavity of the value function for exit time problems with nonsmooth target. (English) Zbl 1064.49024

Sinestrari, Carlo

Commun. Pure Appl. Anal. 3, No. 4, 757-774 (2004).

The author proves a new semiconcavity theorem for the value function \[ V(x):=\inf_{u(.)}\int_0^{\tau(x;u(.))}L(y(t))dt, \;x\in {\mathcal R}:=\text{dom}(V(.)) \] of the exit time problem defined by the control system \[ y'(t)=f(y(t),u(t)), \;u(t)\in U, \;x(0)=x\in R^n, \;\tau(x;u(.)):=\inf\{t\geq 0; \;y(t)\in {\mathcal K}\} \] where, in contrast with previous work on this topic, the target \({\mathcal K}\subset R^n\) is an arbitrary closed set with compact boundary while the “vectograms” \(f(x,U)\) are assumed to be smooth and convex.
The main result of the paper may also be interpreted as a regularity result for the viscosity solutions of the associated HJB equation \[ H(x,DV(x))=0, \;x\in {\mathcal R}\setminus {\mathcal K}, \;H(x,p):= \inf_{u\in U}[-<f(x,u),p>-L(x)], \] \[ V(x)=0, x\in\partial{\mathcal K}, \;\lim_{x\to \partial{\mathcal R}}V(x)= +\infty \] and may be related to some of the previous results on this topic.

Reviewer: Ştefan Mirică (Bucureşti)

MSC:

49L20	Dynamic programming in optimal control and differential games
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35D10	Regularity of generalized solutions of PDE (MSC2000)
26B25	Convexity of real functions of several variables, generalizations

Keywords:

optimal control; exit time problem; dynamic programming; value function; semiconcavity; Hamilton-Jacobi equation; viscosity solutions