Motivated by the stability and performance analysis of model predictive control schemes, we investigate strict dissipativity for a class of optimal control problems involving probability density functions. The dynamics are governed by a Fokker-Planck partial differential equation. However, for the particular classes under investigation involving linear dynamics, linear feedback laws, and Gaussian probability density functions, we are able to significantly simplify these dynamics. This enables us to perform an in-depth analysis of strict dissipativity for different cost functions.
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Figure 1. The state cost parts of the $ L^2 $, $ W^2 $, and 2F stage costs, i.e., (13) (in terms of $ \mu $ and $ \Sigma $ for $ d = 1 $), (15), and (16) for $ \gamma = 0 $, respectively. The desired state was set to $ (\bar{\mu},\bar{\Sigma}) = (0,1) $. The orange dot in the respective plots marks the minimum
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The state cost parts of the
(Non-)Convexity of the reduced cost
Modified cost
Modified cost
Modified cost
Open loop optimal trajectories for various horizons
Open loop optimal trajectories for various horizons N between 1 and 60 and MPC closed loop trajectories for two different initial conditions, indicating turnpike behavior in Example 3; state Σ (left) and control K (right).
New modified cost