Abstract
For a general controlled diffusion process and an arbitrary closed set K we study the viability, or weak invariance, or controlled invariance, of K, that is, the existence of a control for each initial point in K keeping the trajectory forever in K. By viscosity solutions methods we prove a simple necessary and sufficient condition involving only a deterministic second-order normal cone to K and the data of the diffusion process. We also give an extension to stochastic differential games.
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Bardi, M., Jensen, R. A Geometric Characterization of Viable Sets for Controlled Degenerate Diffusions. Set-Valued Analysis 10, 129–141 (2002). https://doi.org/10.1023/A:1016596318432
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DOI: https://doi.org/10.1023/A:1016596318432