Viscosity solutions of HJB equations with unbounded data and characteristic points
The author refines and extends to the ``unbounded case'' certain previous results concerning viscosity-type properties of the value functions of optimal control problems which consist in the minimization of functionals of the form \[ J(t,x,\alpha(.)):=\int_0^t e^{-\int_0^tk(y(\sigma),\alpha(\sigma)) \,d\sigma} l((y(s),\alpha(s))ds+ e^{-\int_0^tk(y(\sigma),\alpha(\sigma)) \,d\sigma} g(y(t)) \] subject to: \[ y'(s)=f(y(s),\alpha(s)), \;\alpha(s)\in A\subseteq R^m \;a.e.(0,t_x), \;y(0)=x, \;x\in \Theta\subseteq R^n. \] As stated in the Introduction, ``the main goal is to characterize the minimal and maximal (viscosity) solutions of the'' associated HJB equation \[ H(x,u(x),Du(x))=0, \;H(x,u,p):= \sup_{a\in A} [-\langle p,f(x,u)\rangle- l(x,a)+u.k(x,a)] \] ``as value functions of the optimal control problems'' which are either of ``exit-time'' type or of ``infinite horizon'' type. As in some previous papers, to treat the case of unbounded data the author assumes the existence of a continuous ``recession function'' \(\Phi^\infty(x,w):= \lim_{\rho\to 0_+}\rho^p\Phi(x,\rho^{-1} w)\) associated to each of the unbounded data \(\Phi\in\{f,l,k\}\) to introduce a certain ``space-time'' (continuous) Hamiltonian.
scientific article; zbMATH DE number 2102075