On the Stability of Receding Horizon Control for Continuous-Time Stochastic Systems

We study the stability of receding horizon control for continuous-time non-linear stochastic differential equations. We illustrate the results with a simulation example in which we employ receding horizon control to design an investment strategy to repay a debt.

💡 Research Summary

The paper addresses a fundamental gap in the theory of Model Predictive Control (MPC), also known as Receding Horizon Control (RHC), for continuous‑time stochastic systems. While discrete‑time MPC has been extensively studied, the stability of its continuous‑time counterpart under stochastic dynamics has received far less attention. The authors consider a nonlinear stochastic differential equation (SDE) of the form

 dx(t)=f(x(t),u(t))dt+σ(x(t),u(t))dW(t),

where x(t)∈ℝⁿ is the state, u(t)∈U (U compact) is the control input, and W(t) is a standard Wiener process. The drift f and diffusion σ are assumed globally Lipschitz, guaranteeing existence and uniqueness of strong solutions.

The control objective is expressed through a finite‑horizon cost functional

 J_T(x₀,u)=E