Discontinuous solutions of deterministic optimal stopping time problems. (English) Zbl 0629.49017

The deterministic optimal stopping time problem with continuous control as well consists of minimizing the cost \[ J(x,v,\vartheta,\psi)\equiv \int^{\vartheta}_{0}f(y_ x(s),v(s))e^{-\lambda s}ds+\psi (y_ x(\vartheta))e^{-\lambda \vartheta} \] over stopping times \(\vartheta\geq 0\) and controls \(V(\cdot)\in L^{\infty}({\mathbb{R}}^+;V)\) subject to the state y evolving according to \(dy_ x(s)+b(y_ x(s),v(s))ds=0\), \(y_ x(0)=x\in {\mathbb{R}}^ N\). The obstacle is \(\psi\) (\(\cdot)\) and the value function for a given obstacle is denoted \(u[\psi](x)=\inf_{\vartheta,v}J(x,v,\vartheta,\psi)\). When \(\psi\) is bounded and continuous, it is well known that u is characterized as the unique viscosity solution of the Bellman variational inequality \(\max \{H(x,u,Du),u-\psi \}=0\) in \({\mathbb{R}}^ N\) where \(H(x,t,p)=\sup \{b(x,v)\cdot p+\lambda t-f(x,v);v\in V\}\) The objective of this paper is to obtain a similar characterization for u if the obstacle is any arbitrary, possibly discontinuous, but bounded function. An application in which the obstacle is discontinuous is the problem of minimum exit time from a domain.
The study proceeds in this paper by using the upper and lower semicontinuous envelopes of the value function.
Reviewer: E.Barron


49L20 Dynamic programming in optimal control and differential games
35F20 Nonlinear first-order PDEs
49J40 Variational inequalities
60G40 Stopping times; optimal stopping problems; gambling theory
35D05 Existence of generalized solutions of PDE (MSC2000)
49J45 Methods involving semicontinuity and convergence; relaxation
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