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

Fleig Arthur; Grüne Lars

doi:10.3934/mcrf.2020053

Article Contents

2021, Volume 11, Issue 4: 935-964. Doi: 10.3934/mcrf.2020053 open access

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

Fleig Arthur^1, , and
Grüne Lars^2,

1.
Chair of Serious Games, University of Bayreuth, 95440 Bayreuth, Germany
2.
Chair of Applied Mathematics, University of Bayreuth, 95440 Bayreuth, Germany

^* Corresponding author: Arthur Fleig
^* Corresponding author: Arthur Fleig

Received: June 30, 2019

Revised: February 29, 2020

Early access: December 2020

Published: December 2021

The first author was supported by DFG grant GR 1569/15-1

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Abstract

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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Mathematics Subject Classification: Primary: 35Q84, 35Q93, 60G15; Secondary: 49N35, 93C15.

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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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Figure 2. (Non-)Convexity of the reduced cost $ \hat{\ell}_{2F}(\Sigma,K) $ depending on $ \varsigma^2 $ (left) and on $ \gamma $ (right)

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Figure 3. Modified cost $ \tilde{\ell}_{L^2}(\Sigma,K) $ for Example 2. The optimal equilibrium $ (\Sigma^e,K^e) $ is illustrated by the orange circle. The white area represents negative values; the black diamond marks the minimum of the depicted area

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Figure 4. Modified cost $ \tilde{\ell}_{L^2}(\Sigma,K) $ for Example 3. The optimal equilibrium $ (\Sigma^e,K^e) $ is illustrated by the orange circle. The white area represents negative values; the black diamond marks the minimum of the depicted area

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Figure 5. Modified cost $ \tilde{\ell}_{W^2}(\Sigma,K) $ for Example 4 zoomed in (left) and zoomed out (right). The optimal equilibrium $ (\Sigma^e,K^e) $ is illustrated by the orange circle. The white area on the right plot is due to control constraints (26)

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Figure 6. 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 2; state $ \Sigma $ (left) and control $ K $ (right)

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Figure 7. 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).

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Figure 8. New modified cost $ \tilde{\ell}_{W^2}^s(\Sigma,K) $ for Examples 2 (left) and 3 (right). The optimal equilibrium $ (\Sigma^e,K^e) $ is illustrated by the orange circle. The white area on the right plot is due to the control constraints (26)

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