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doi: 10.3934/mcrf.2022016
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Averaged turnpike property for differential equations with random constant coefficients

1. 

Friedrich-Alexander-Universität Erlangen-Nürnberg, Faculty of Sciences, Department of Data Science, Chair for Dynamics, Control and Numerics, Cauerstr. 11, 91058 Erlangen, Germany

2. 

Universidad Técnica Federico Santa María, Departamento de Matemática, Casilla 110-V, Valparaíso, Chile

3. 

Universidad de Santiago de Chile (USACH), Facultad de Ciencia, Departamento de Matemática y Ciencia de la Computación, Casilla 307-Correo 2, Santiago, Chile

* Corresponding author: Sebastián Zamorano

Received  March 2021 Revised  February 2022 Early access April 2022

This paper studies the integral turnpike and turnpike in average for a class of random ordinary differential equations. We prove that, under suitable assumptions on the matrices that define the system, the optimal solutions for an optimal distributed control tracking problem remain, in an averaged sense, sufficiently close to the associated random stationary optimal solution for the majority of the time horizon.

Citation: Martín Hernández, Rodrigo Lecaros, Sebastián Zamorano. Averaged turnpike property for differential equations with random constant coefficients. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022016
References:
[1]

L. D. R. BealD. C. HillR. A. Martin and J. D. Hedengren, Gekko optimization suite, Processes, 6 (2018). 

[2]

P. G. Bergmann, Propagation of radiation in a medium with random inhomogeneities, Physical Review, 70 (1946), 486-492. 

[3]

L. T. Biegler, An overview of simultaneous strategies for dynamic optimization, Chemical Engineering and Processing: Process Intensification, 46 (2007), 1043-1053. 

[4]

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

[5]

G. Carey and B. A. Finlayson, Orthogonal collocation on finite elements, Chemical Engineering Science, 30 (1975), 587-596. 

[6]

R. Dorfman, P. A. Samuelson and R. M. Solow, Linear Programming and Economic Analysis, A Rand Corporation Research Study. McGraw-Hill Book Co., Inc., New York-Toronto-London, 1958.

[7]

C. Esteve, C. Kouhkouh, D. Pighin and E. Zuazua, The turnpike property and the long-time behavior of the Hamilton-Jacobi equation, preprint, 2020, arXiv: 2006.10430.

[8]

C. Esteve-YagüeB. GeshkovskiD. Pighin and E. Zuazua, Turnpike in Lipschitz-nonlinear optimal control, Nonlinearity, 35 (2022), 1652-1701.  doi: 10.1088/1361-6544/ac4e61.

[9]

T. FaulwasserM. KordaC. N. Jones and D. Bonvin, On turnpike and dissipativity properties of continuous-time optimal control problems, Automatica J. IFAC, 81 (2017), 297-304.  doi: 10.1016/j.automatica.2017.03.012.

[10]

L. GrüneM. 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.

[11]

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

[12]

A. Ibañez, Optimal control of the Lotka-Volterra system: Turnpike property and numerical simulations, J. Biol. Dyn., 11 (2017), 25-41.  doi: 10.1080/17513758.2016.1226435.

[13]

R. Kress, Linear Integral Equations, third edition, Springer, New York, 2014. doi: 10.1007/978-1-4614-9593-2.

[14]

J. Lohéac and E. Zuazua, From averaged to simultaneous controllability, Ann. Fac. Sci. Toulouse Math., 25 (2016), 785-828.  doi: 10.5802/afst.1511.

[15]

A. PesareM. Palladino and M. Falcone, Convergence of the value function in optimal control problems with unknown dynamics, 2021 European Control Conference (ECC), (2021), 2426-2431. 

[16]

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

[17]

A. Porretta and E. Zuazua, Remarks on long time versus steady state optimal control, in Mathematical Paradigms of Climate Science, volume 15 of Springer INdAM Ser., Springer, 2016, 67–89.

[18] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York-London, 1973. 
[19]

T. T. Soong, Probabilistic Modeling and Analysis in Science and Engineering, John Wiley & Sons, Inc., New York, 1981.

[20]

E. Trélat and C. Zhang, Integral and measure-turnpike properties for infinite-dimensional optimal control systems, Math. Control Signals Systems, 30 (2018), Art. 3, 34 pp. doi: 10.1007/s00498-018-0209-1.

[21]

E. TrélatC. 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.

[22]

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.

[23]

M. Warma and S. Zamorano, Exponential turnpike property for fractional parabolic equations with non-zero exterior data, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 1, 35 pp. doi: 10.1051/cocv/2020076.

[24]

S. Zamorano, Turnpike property for two-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 20 (2018), 869-888.  doi: 10.1007/s00021-018-0382-5.

[25]

A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Nonconvex Optimization and Its Applications, Springer US, 2006.

[26]

E. Zuazua, Averaged control, Automatica J. IFAC, 50 (2014), 3077-3087.  doi: 10.1016/j.automatica.2014.10.054.

[27]

E. Zuazua, Large time control and turnpike properties for wave equations, Annual Reviews in Control, 44 (2017), 199-210. 

show all references

References:
[1]

L. D. R. BealD. C. HillR. A. Martin and J. D. Hedengren, Gekko optimization suite, Processes, 6 (2018). 

[2]

P. G. Bergmann, Propagation of radiation in a medium with random inhomogeneities, Physical Review, 70 (1946), 486-492. 

[3]

L. T. Biegler, An overview of simultaneous strategies for dynamic optimization, Chemical Engineering and Processing: Process Intensification, 46 (2007), 1043-1053. 

[4]

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

[5]

G. Carey and B. A. Finlayson, Orthogonal collocation on finite elements, Chemical Engineering Science, 30 (1975), 587-596. 

[6]

R. Dorfman, P. A. Samuelson and R. M. Solow, Linear Programming and Economic Analysis, A Rand Corporation Research Study. McGraw-Hill Book Co., Inc., New York-Toronto-London, 1958.

[7]

C. Esteve, C. Kouhkouh, D. Pighin and E. Zuazua, The turnpike property and the long-time behavior of the Hamilton-Jacobi equation, preprint, 2020, arXiv: 2006.10430.

[8]

C. Esteve-YagüeB. GeshkovskiD. Pighin and E. Zuazua, Turnpike in Lipschitz-nonlinear optimal control, Nonlinearity, 35 (2022), 1652-1701.  doi: 10.1088/1361-6544/ac4e61.

[9]

T. FaulwasserM. KordaC. N. Jones and D. Bonvin, On turnpike and dissipativity properties of continuous-time optimal control problems, Automatica J. IFAC, 81 (2017), 297-304.  doi: 10.1016/j.automatica.2017.03.012.

[10]

L. GrüneM. 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.

[11]

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

[12]

A. Ibañez, Optimal control of the Lotka-Volterra system: Turnpike property and numerical simulations, J. Biol. Dyn., 11 (2017), 25-41.  doi: 10.1080/17513758.2016.1226435.

[13]

R. Kress, Linear Integral Equations, third edition, Springer, New York, 2014. doi: 10.1007/978-1-4614-9593-2.

[14]

J. Lohéac and E. Zuazua, From averaged to simultaneous controllability, Ann. Fac. Sci. Toulouse Math., 25 (2016), 785-828.  doi: 10.5802/afst.1511.

[15]

A. PesareM. Palladino and M. Falcone, Convergence of the value function in optimal control problems with unknown dynamics, 2021 European Control Conference (ECC), (2021), 2426-2431. 

[16]

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

[17]

A. Porretta and E. Zuazua, Remarks on long time versus steady state optimal control, in Mathematical Paradigms of Climate Science, volume 15 of Springer INdAM Ser., Springer, 2016, 67–89.

[18] T. T. Soong, Random Differential Equations in Science and Engineering, Academic Press, New York-London, 1973. 
[19]

T. T. Soong, Probabilistic Modeling and Analysis in Science and Engineering, John Wiley & Sons, Inc., New York, 1981.

[20]

E. Trélat and C. Zhang, Integral and measure-turnpike properties for infinite-dimensional optimal control systems, Math. Control Signals Systems, 30 (2018), Art. 3, 34 pp. doi: 10.1007/s00498-018-0209-1.

[21]

E. TrélatC. 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.

[22]

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.

[23]

M. Warma and S. Zamorano, Exponential turnpike property for fractional parabolic equations with non-zero exterior data, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 1, 35 pp. doi: 10.1051/cocv/2020076.

[24]

S. Zamorano, Turnpike property for two-dimensional Navier-Stokes equations, J. Math. Fluid Mech., 20 (2018), 869-888.  doi: 10.1007/s00021-018-0382-5.

[25]

A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Nonconvex Optimization and Its Applications, Springer US, 2006.

[26]

E. Zuazua, Averaged control, Automatica J. IFAC, 50 (2014), 3077-3087.  doi: 10.1016/j.automatica.2014.10.054.

[27]

E. Zuazua, Large time control and turnpike properties for wave equations, Annual Reviews in Control, 44 (2017), 199-210. 

Figure 1.  Evolutionary v/s stationary systems
Figure 2.  Evolutionary v/s stationary systems
Figure 3.  Evolutionary v/s stationary systems
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