A control theoretic approach to the approximant solution of some PDEs
Italo Capuzzo Dolcetta, Maurizio Falcone · OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) · 1994
It is well known that the notion of value function provides, via Dynamic Programming, a bridge between deterministic and stochastic optimal control (and differential games) theory and a wide class of first and second order nonlinear PDE`s. Several efforts has been directed to establishing this rather formal connection on rigorous bases by different tools of non smooth analysis and PDE theory. Our aim here is to point out how this circle of ideas can be usefully exploited to the purpose of building approximate viscosity solutions of such PDE`s. The schemes that we propose apply, generally speaking, to operators which are generators of nonlinear semigroups. They are obtained by suitable discrete versions of the dynamic programming principle and can be interpreted as {open_quotes}methods of characteristics{close_quotes} (in the Ito`s sense, in case of second order equations). These schemes, for which convergence results and estimates are available, seem to present some advantages with respect to finite difference or finite element methods in handling degenerate operators or irregular domains. Some explicit implementation of the method will be presented for equations of Bellman type and for equations connected with shape from shading and edge detection problems in computer vision.