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December  2021, 11(4): 935-964. doi: 10.3934/mcrf.2020053

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

Arthur Fleig 1,, and  Lars Grüne 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

Received  July 2019 Revised  March 2020 Published  December 2021 Early access  December 2020

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

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.

Keywords: Model predictive control, optimal control, dissipativity analysis, Fokker-Planck equation, stochastic processes, probability density function.
Mathematics Subject Classification: Primary: 35Q84, 35Q93, 60G15; Secondary: 49N35, 93C15.
Citation: Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control and Related Fields, 2021, 11 (4) : 935-964. doi: 10.3934/mcrf.2020053
References:
[1]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109.  doi: 10.1137/100790069.

[2]

D. Angeli, R. Amrit and J. B. Rawlings, On average performance and stability of economic model predictive control, IEEE Trans. Autom. Control, 57 (2012), 1615-1626.  doi: 10.1109/TAC.2011.2179349.

[3]

M. Annunziato and A. Borzì, Optimal control of probability density functions of stochastic processes, Math. Model. Anal., 15 (2010), 393-407.  doi: 10.3846/1392-6292.2010.15.393-407.

[4]

M. Annunziato and A. Borzì, A Fokker-Planck control framework for multidimensional stochastic processes, J. Comput. Appl. Math., 237 (2013), 487-507.  doi: 10.1016/j.cam.2012.06.019.

[5]

J.-D. Benamou and G. Carlier, Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theory Appl., 167 (2015), 1-26.  doi: 10.1007/s10957-015-0725-9.

[6]

M. Bongini, M. Fornasier, F. Rossi and F. Solombrino, Mean-field Pontryagin maximum principle, J. Optim. Theory Appl., 175 (2017), 1-38.  doi: 10.1007/s10957-017-1149-5.

[7]

T. Breiten, K. Kunisch and L. Pfeiffer, Control strategies for the Fokker-Planck equation, ESAIM: COCV, 24 (2018), 741-763.  doi: 10.1051/cocv/2017046.

[8]

T. Breiten and L. Pfeiffer, On the turnpike property and the receding-horizon method for linear-quadratic optimal control problems, SIAM Journal on Control and Optimization, 58 (2020), 1077-1102.  doi: 10.1137/18M1225811.

[9]

T. Damm, L. Grüne, M. Stieler and K. Worthmann, An exponential turnpike theorem for dissipative discrete time optimal control problems, SIAM J. Control Optim., 52 (2014), 1935-1957.  doi: 10.1137/120888934.

[10]

M. Diehl, R. Amrit and J. B. Rawlings, A Lyapunov function for economic optimizing model predictive control, IEEE Trans. Autom. Control, 56 (2011), 703-707.  doi: 10.1109/TAC.2010.2101291.

[11]

T. Faulwasser, L. Grüne and M. A. Müller, Economic nonlinear model predictive control, Foundations and Trends® in Systems and Control, 5 (2018), 1–98.

[12]

A. Fleig and L. Grüne, Estimates on the minimal stabilizing horizon length in model predictive control for the Fokker-Planck equation, IFAC-PapersOnLine, 49 (2016), 260-265.  doi: 10.1016/j.ifacol.2016.07.451.

[13]

A. Fleig and L. Grüne, $L^2$-tracking of Gaussian distributions via model predictive control for the Fokker-Planck equation, Vietnam J. Math., 46 (2018), 915-948.  doi: 10.1007/s10013-018-0309-8.

[14]

A. Fleig and L. Grüne, On dissipativity of the Fokker-Planck equation for the OrnsteinUhlenbeck process, in IFAC-PapersOnLine, 3rd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2019, 52 (2019), 13-18,

[15]

C. R. Givens and R. M. Shortt, A class of wasserstein metrics for probability distributions, Michigan Math. J., 31 (1984), 231-240.  doi: 10.1307/mmj/1029003026.

[16]

L. Grüne, Economic receding horizon control without terminal constraints, Automatica, 49 (2013), 725-734.  doi: 10.1016/j.automatica.2012.12.003.

[17]

L. Grüne and R. Guglielmi, Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems, SIAM J. Cont. Optim., 56 (2018), 1282-1302.  doi: 10.1137/17M112350X.

[18]

L. Grüne and M. A. Müller, On the relation between strict dissipativity and the turnpike property, Syst. Contr. Lett., 90 (2016), 45-53.  doi: 10.1016/j.sysconle.2016.01.003.

[19]

L. Grüne and J. Pannek, Nonlinear Model Predictive Control, Theory and Algorithms, 2nd edition, Springer, London, 2017. doi: 10.1007/978-3-319-46024-6.

[20]

L. Grüne, M. Schaller and A. Schiela, Sensitivity analysis of optimal control for a class of parabolic PDEs motivated by model predictive control, SIAM J. Control Optim., 57 (2019), 2753-2774.  doi: 10.1137/18M1223083.

[21]

L. Grüne and M. Stieler, Asymptotic stability and transient optimality of economic MPC without terminal conditions, J. Proc. Control, 24 (2014), 1187-1196. 

[22]

L. Grüne, M. Schaller and A. Schiela, Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations, J. Differ. Equ., 268 (2020), 7311-7341.  doi: 10.1016/j.jde.2019.11.064.

[23]

W. Hahn, Stability of Motion, Springer, 1967.

[24]

A. Porretta and E. Zuazua, Long time versus steady state optimal control, SIAM J. Control Optim., 51 (2013), 4242-4273.  doi: 10.1137/130907239.

[25]

S. Primak, V. Kontorovich and V. Lyandres, Stochastic Methods and Their Applications to Communications, John Wiley & Sons, Inc., Hoboken, NJ, 2004.

[26]

P. E. Protter, Stochastic Integration and Differential Equations, vol. 21 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2005. doi: 10.1007/978-3-662-10061-5.

[27]

J. B. Rawlings, D. Bonné, J. B. Jørgensen, A. N. Venkat and S. B. Jørgensen, Unreachable setpoints in model predictive control, IEEE Transactions on Automatic Control, 53 (2008), 2209-2215.  doi: 10.1109/TAC.2008.928125.

[28]

J. B. Rawlings, D. Q. Mayne and M. M. Diehl, Model Predictive Control: Theory and Design, 2nd edition, Nob Hill Publishing, 2017.

[29]

H. Risken, The Fokker-Planck Equation, vol. 18 of Springer Series in Synergetics, 2nd edition, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3.

[30]

S. Roy, M. Annunziato and A. Borzì, A Fokker-Planck feedback control-constrained approach for modelling crowd motion, J. Comput. Theor. Transp., 45 (2016), 442-458.  doi: 10.1080/23324309.2016.1189435.

[31]

E. Trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control, J. Differential Equations, 258 (2015), 81-114.  doi: 10.1016/j.jde.2014.09.005.

[32]

E. Trélat, C. Zhang and E. Zuazua, Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces, SIAM J. Control Optim., 56 (2018), 1222-1252.  doi: 10.1137/16M1097638.

[33]

F. Tröltzsch, Optimal Control of Partial Differential Equations, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[34]

J. C. Willems, Dissipative dynamical systems. I. General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351.  doi: 10.1007/BF00276493.

show all references

References:
[1]

Y. Achdou, F. Camilli and I. Capuzzo-Dolcetta, Mean field games: Numerical methods for the planning problem, SIAM J. Control Optim., 50 (2012), 77-109.  doi: 10.1137/100790069.

[2]

D. Angeli, R. Amrit and J. B. Rawlings, On average performance and stability of economic model predictive control, IEEE Trans. Autom. Control, 57 (2012), 1615-1626.  doi: 10.1109/TAC.2011.2179349.

[3]

M. Annunziato and A. Borzì, Optimal control of probability density functions of stochastic processes, Math. Model. Anal., 15 (2010), 393-407.  doi: 10.3846/1392-6292.2010.15.393-407.

[4]

M. Annunziato and A. Borzì, A Fokker-Planck control framework for multidimensional stochastic processes, J. Comput. Appl. Math., 237 (2013), 487-507.  doi: 10.1016/j.cam.2012.06.019.

[5]

J.-D. Benamou and G. Carlier, Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theory Appl., 167 (2015), 1-26.  doi: 10.1007/s10957-015-0725-9.

[6]

M. Bongini, M. Fornasier, F. Rossi and F. Solombrino, Mean-field Pontryagin maximum principle, J. Optim. Theory Appl., 175 (2017), 1-38.  doi: 10.1007/s10957-017-1149-5.

[7]

T. Breiten, K. Kunisch and L. Pfeiffer, Control strategies for the Fokker-Planck equation, ESAIM: COCV, 24 (2018), 741-763.  doi: 10.1051/cocv/2017046.

[8]

T. Breiten and L. Pfeiffer, On the turnpike property and the receding-horizon method for linear-quadratic optimal control problems, SIAM Journal on Control and Optimization, 58 (2020), 1077-1102.  doi: 10.1137/18M1225811.

[9]

T. Damm, L. Grüne, M. Stieler and K. Worthmann, An exponential turnpike theorem for dissipative discrete time optimal control problems, SIAM J. Control Optim., 52 (2014), 1935-1957.  doi: 10.1137/120888934.

[10]

M. Diehl, R. Amrit and J. B. Rawlings, A Lyapunov function for economic optimizing model predictive control, IEEE Trans. Autom. Control, 56 (2011), 703-707.  doi: 10.1109/TAC.2010.2101291.

[11]

T. Faulwasser, L. Grüne and M. A. Müller, Economic nonlinear model predictive control, Foundations and Trends® in Systems and Control, 5 (2018), 1–98.

[12]

A. Fleig and L. Grüne, Estimates on the minimal stabilizing horizon length in model predictive control for the Fokker-Planck equation, IFAC-PapersOnLine, 49 (2016), 260-265.  doi: 10.1016/j.ifacol.2016.07.451.

[13]

A. Fleig and L. Grüne, $L^2$-tracking of Gaussian distributions via model predictive control for the Fokker-Planck equation, Vietnam J. Math., 46 (2018), 915-948.  doi: 10.1007/s10013-018-0309-8.

[14]

A. Fleig and L. Grüne, On dissipativity of the Fokker-Planck equation for the OrnsteinUhlenbeck process, in IFAC-PapersOnLine, 3rd IFAC Workshop on Control of Systems Governed by Partial Differential Equations CPDE 2019, 52 (2019), 13-18,

[15]

C. R. Givens and R. M. Shortt, A class of wasserstein metrics for probability distributions, Michigan Math. J., 31 (1984), 231-240.  doi: 10.1307/mmj/1029003026.

[16]

L. Grüne, Economic receding horizon control without terminal constraints, Automatica, 49 (2013), 725-734.  doi: 10.1016/j.automatica.2012.12.003.

[17]

L. Grüne and R. Guglielmi, Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems, SIAM J. Cont. Optim., 56 (2018), 1282-1302.  doi: 10.1137/17M112350X.

[18]

L. Grüne and M. A. Müller, On the relation between strict dissipativity and the turnpike property, Syst. Contr. Lett., 90 (2016), 45-53.  doi: 10.1016/j.sysconle.2016.01.003.

[19]

L. Grüne and J. Pannek, Nonlinear Model Predictive Control, Theory and Algorithms, 2nd edition, Springer, London, 2017. doi: 10.1007/978-3-319-46024-6.

[20]

L. Grüne, M. Schaller and A. Schiela, Sensitivity analysis of optimal control for a class of parabolic PDEs motivated by model predictive control, SIAM J. Control Optim., 57 (2019), 2753-2774.  doi: 10.1137/18M1223083.

[21]

L. Grüne and M. Stieler, Asymptotic stability and transient optimality of economic MPC without terminal conditions, J. Proc. Control, 24 (2014), 1187-1196. 

[22]

L. Grüne, M. Schaller and A. Schiela, Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations, J. Differ. Equ., 268 (2020), 7311-7341.  doi: 10.1016/j.jde.2019.11.064.

[23]

W. Hahn, Stability of Motion, Springer, 1967.

[24]

A. Porretta and E. Zuazua, Long time versus steady state optimal control, SIAM J. Control Optim., 51 (2013), 4242-4273.  doi: 10.1137/130907239.

[25]

S. Primak, V. Kontorovich and V. Lyandres, Stochastic Methods and Their Applications to Communications, John Wiley & Sons, Inc., Hoboken, NJ, 2004.

[26]

P. E. Protter, Stochastic Integration and Differential Equations, vol. 21 of Stochastic Modelling and Applied Probability, Springer-Verlag, Berlin, 2005. doi: 10.1007/978-3-662-10061-5.

[27]

J. B. Rawlings, D. Bonné, J. B. Jørgensen, A. N. Venkat and S. B. Jørgensen, Unreachable setpoints in model predictive control, IEEE Transactions on Automatic Control, 53 (2008), 2209-2215.  doi: 10.1109/TAC.2008.928125.

[28]

J. B. Rawlings, D. Q. Mayne and M. M. Diehl, Model Predictive Control: Theory and Design, 2nd edition, Nob Hill Publishing, 2017.

[29]

H. Risken, The Fokker-Planck Equation, vol. 18 of Springer Series in Synergetics, 2nd edition, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61544-3.

[30]

S. Roy, M. Annunziato and A. Borzì, A Fokker-Planck feedback control-constrained approach for modelling crowd motion, J. Comput. Theor. Transp., 45 (2016), 442-458.  doi: 10.1080/23324309.2016.1189435.

[31]

E. Trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control, J. Differential Equations, 258 (2015), 81-114.  doi: 10.1016/j.jde.2014.09.005.

[32]

E. Trélat, C. Zhang and E. Zuazua, Steady-state and periodic exponential turnpike property for optimal control problems in Hilbert spaces, SIAM J. Control Optim., 56 (2018), 1222-1252.  doi: 10.1137/16M1097638.

[33]

F. Tröltzsch, Optimal Control of Partial Differential Equations, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[34]

J. C. Willems, Dissipative dynamical systems. I. General theory, Arch. Rational Mech. Anal., 45 (1972), 321-351.  doi: 10.1007/BF00276493.

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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