Zubov’s method for controlled diffusions with state constraints

Lars Grüne (1) , Athena Picarelli (2)

1 Mathematisches Institut [Bayreuth]
2 MI - Mathematical Institute [Oxford]

We consider a controlled stochastic system in presence of state-constraints. Under the assumption of exponential stabilizability of the system near a target set, we aim to characterize the set of points which can be asymptotically driven by an admissible control to the target with positive probability. We show that this set can be characterized as a level set of the optimal value function of a suitable unconstrained optimal control problem which in turn is the unique viscosity solution of a second order PDE which can thus be interpreted as a generalized Zubov equation.

Mots clés: Controllability for diffusion systems; Hamilton-Jacobi-Bellman equations; viscosity solutions; stochastic optimal control

Domaines: Mathématiques [math]

Lars Grüne, Athena Picarelli. Zubov’s method for controlled diffusions with state constraints. Nonlinear Differential Equations and Applications, 2015, 22 (6), pp.1765-1799. ⟨10.1007/s00030-015-0343-0⟩. ⟨hal-01620010⟩