On Dissipativity of the Fokker–Planck Equation for the Ornstein–Uhlenbeck Process

Strict dissipativity analysis for classes of optimal control problems involving probability density functions

“…In order to allow for a fast computation of the optimal open-loop trajectories, in MPC implementations simple but less accurate schemes are often preferred to more expensive high-order methods. For these reasons, as in [14], we perform our analysis for the forward Euler approximation of the ODE system (9). This discretization both maintains the bilinear structure and defines a scheme that is frequently used in practice.…”

“…Motivated by these facts, we decided to analyze strict dissipativity for certain classes of this optimal control problem. This analysis was started in [14], where objectives involving the L 2 -norm for penalizing the distance to the desired equilibrium PDF was investigated -the results from this paper will be briefly summarized below. Here we extend and complement this analysis to alternative cost functions including the Wasserstein distance W 2 .…”

“…While both the L 2 and the W 2 cost are perfectly suited for being used in a nonlinear setting for general PDFs -the resulting control scheme performs excellently in numerical tests -it turned out that for a rigorous mathematical analysis the problem must be simplified. Hence, as in [14], we perform our analysis for linear SDE dynamics governed by the Ornstein-Uhlenbeck process, linear feedback controllers, and Gaussian PDFs. While this is clearly a restricted setting, we believe that the insights from this analysis are nevertheless very valuable for the general nonlinear setting, in particular those results that clarify that certain storage functions are not appropriate for the strict dissipativity analysis.…”

“…In Section 3 we introduce strict dissipativity and briefly summarize the main results for MPC schemes that can be derived from this property, in order to motivate our subsequent analysis. Section 4 collects a few auxiliary results and summarizes the main results from [14] for the L 2 cost before we present our new results for the W 2 cost and the 2F cost in Section 5. We end the paper with concluding remarks in Section 6.…”

“…which is the standard norm used in costs for optimal control problems governed by parabolic PDEs [33]. This choice of the cost was analyzed in [14]. We can express the L 2 norm in terms of Σ and µ, which proves useful when focusing on the ODE system (6) instead:…”