Stackelberg-Nash Controllability for Abstract Stochastic Evolution Equations and Applications
Abdellatif Elgrou, Omar Oukdach
TL;DR
This work develops a rigorous framework for exact, null, and approximate Stackelberg-Nash controllability of abstract forward and backward stochastic evolution equations, incorporating dual leader-follower structures and Nash equilibrium analyses. Existence, uniqueness, and adjoint-based characterizations of Nash equilibria are established for both forward and backward systems, enabling reduction of Stackelberg-Nash controllability to coupled forward–backward observability problems. The authors prove observability inequalities via Carleman estimates, linking controllability to dual observability and providing constructive feedback controls of the form $v_i^* = -\frac{1}{\beta_i} R_i^{-1} B_i^* z_i$. Applications to forward and backward stochastic heat equations illustrate the theory, including a new Carleman estimate for the backward problem and explicit Nash feedbacks, highlighting the practical impact for hierarchical stochastic control in parabolic settings.
Abstract
This paper presents the concepts of exact, null, and approximate controllability in the Stackelberg-Nash sense for abstract forward and backward stochastic evolution equations, involving two types of controls: leaders and followers. We begin by proving the existence and uniqueness of the Nash equilibrium, as well as its characterization for fixed leader controls. We then establish a duality between these controllability concepts and the corresponding observability properties. Finally, we apply our theoretical results to the forward and backward stochastic heat equations. The results for the backward heat equation are obtained by deriving a new Carleman estimate.