viability theory in stocastic control

Classical viability theory studies whether a controlled dynamical system can be kept inside a prescribed constraint set over time. Given a state space XXX, a control system xt+1=f(xt,ut)xₜ₊₁=f(x_t,u_t)xt+1=f(xt,ut), and a safe set K⊂XK⊂ XK⊂X, the viability kernel is the set of initial states from which at least one admissible control strategy can keep the system inside KKK for all future times. However, in real systems, two fundamental complications arise: Uncertainty: system evolution is stochastic rather than deterministic. Partial observability: the controller does not directly observe the state. Stochastic viability kernels generalize classical viability theory to this setting.