December  2021, 11(4): 771-796. doi: 10.3934/mcrf.2020046

Strict dissipativity for discrete time discounted optimal control problems

1. 

Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany

2. 

Institute of Automatic Control, Leibniz University Hannover, 30167 Hannover, Germany

3. 

Research School of Electrical, Energy and Materials Engineering, Australian National University, Canberra, ACT 2600, Australia

4. 

School of Electrical Engineering and Computing, University of Newcastle, Callaghan, NSW 2308, Australia

* Corresponding author: Lars Grüne

Received  September 2019 Revised  May 2020 Published  December 2021 Early access  November 2020

Fund Project: The research was supported by the Australian Research Council under grants DP160102138 and DP180103026 and by the Deutsche Forschungsgemeinschaft under grant Gr1569/13-2

The paradigm of discounting future costs is a common feature of economic applications of optimal control. In this paper, we provide several results for such discounted optimal control aimed at replicating the now well-known results in the standard, undiscounted, setting whereby (strict) dissipativity, turnpike properties, and near-optimality of closed-loop systems using model predictive control are essentially equivalent. To that end, we introduce a notion of discounted strict dissipativity and show that this implies various properties including the existence of available storage functions, required supply functions, and robustness of optimal equilibria. Additionally, for discount factors sufficiently close to one we demonstrate that strict dissipativity implies discounted strict dissipativity and that optimally controlled systems, derived from a discounted cost function, yield practically asymptotically stable equilibria. Several examples are provided throughout.

Citation: Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2021, 11 (4) : 771-796. doi: 10.3934/mcrf.2020046
References:
[1]

D. Acemoglu, Introduction to Modern Economic Growth, Princeton University Press, 2009. Google Scholar

[2]

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

[3]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[4]

S. BeckerL. Grüne and W. Semmler, Comparing accuracy of second order approximation and dynamic programming, Comput. Econ., 30 (2007), 65-91.  doi: 10.1007/s10614-007-9087-1.  Google Scholar

[5]

D. P. Bertsekas, Nonlinear Programming, 2nd edition, Athena Scientific, Belmont, Massachusetts, 1995.  Google Scholar

[6]

J. Blot and N. Hayek, Infinite-Horizon Optimal Control in the Discrete-Time Framework, SpringerBriefs in Optimization, Springer, New York, 2014. doi: 10.1007/978-1-4614-9038-8.  Google Scholar

[7]

W. A. Brock and L. Mirman, Optimal economic growth and uncertainty: The discounted case, J. Econom. Theory, 4 (1972), 479-513.  doi: 10.1016/0022-0531(72)90135-4.  Google Scholar

[8]

B. Brogliato, R. Lozano, B. Maschke and O. Egeland, Dissipative Systems Analysis and Control, Second edition, Springer-Verlag, London, Ltd., 2007. doi: 10.1007/978-1-84628-517-2.  Google Scholar

[9]

C. I. Byrnes and W. Lin, Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems, IEEE Trans. Automat. Control, 39 (1994), 83-98.  doi: 10.1109/9.273341.  Google Scholar

[10]

D. Cass, Optimum growth in an aggregative model of capital accumulation, Rev. Econ. Stud., 32 (1965), 233-240.  doi: 10.2307/2295827.  Google Scholar

[11]

T. DammL. GrüneM. 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.  Google Scholar

[12]

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

[13]

T. Faulwasser, C. M. Kellett and S. R. Weller, MPC-DICE: An open-source Matlab implementation of receding horizon solutions to DICE, in Proc. 1st IFAC Workshop on Integrated Assessment Modelling for Environmental Systems, Brescia, Italy, 2018,126–131. Google Scholar

[14]

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.  Google Scholar

[15]

V. GaitsgoryL. GrüneM. HögerC. M. Kellett and S. R. Weller, Stabilization of strictly dissipative discrete time systems with discounted optimal control, Automatica J. IFAC, 93 (2018), 311-320.  doi: 10.1016/j.automatica.2018.03.076.  Google Scholar

[16]

V. GaitsgoryL. Grüne and N. Thatcher, Stabilization with discounted optimal control, Systems Control Lett., 82 (2015), 91-98.  doi: 10.1016/j.sysconle.2015.05.010.  Google Scholar

[17]

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

[18]

L. Grüne, C. M. Kellett and S. R. Weller, On a discounted notion of strict dissipativity, in Proc. 10th IFAC Symposium on Nonlinear Control Systems, 2016,247–252. Google Scholar

[19]

L. GrüneC. M. Kellett and S. R. Weller, On the relation between turnpike properties for finite and infinite horizon optimal control problems, J. Optim. Theory Appl., 173 (2017), 727-745.  doi: 10.1007/s10957-017-1103-6.  Google Scholar

[20]

L. Grüne and M. A. Müller, On the relation between strict dissipativity and turnpike properties, Systems Control Lett., 90 (2016), 45-53.  doi: 10.1016/j.sysconle.2016.01.003.  Google Scholar

[21]

L. Grüne and A. Panin, On non-averaged performance of economic MPC with terminal conditions, in Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, 2015, 4332–4337. Google Scholar

[22]

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

[23]

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

[24]

L. GrüneW. Semmler and M. Stieler, Using nonlinear model predictive control for dynamic decision problems in economics, J. Econom. Dynam. Control, 60 (2015), 112-133.  doi: 10.1016/j.jedc.2015.08.010.  Google Scholar

[25]

Interagency Working Group on Social Cost of Carbon, United States Government, Technical Support Document: Technical Update of the Social Cost of Carbon for Regulatory Impact Analysis - Under Executive Order 12866, Technical report, 2013. Google Scholar

[26]

C. M. KellettS. R. WellerT. FaulwasserL. Grüne and W. Semmler, Feedback, dynamics, and optimal control in climate economics, Annu. Rev. Control, 47 (2019), 7-20.  doi: 10.1016/j.arcontrol.2019.04.003.  Google Scholar

[27]

T. C. Koopmans, On the concept of optimal economic growth, in The Economic Approach to Development Planning, Rand McNally, Chicago, IL, 1965,225–287. Google Scholar

[28]

R. Lopezlena and J. M. A. Scherpen, Energy functions for dissipativity-based balancing of discrete-time nonlinear systems, Math. Control Signals Systems, 18 (2006), 345-368.  doi: 10.1007/s00498-006-0007-z.  Google Scholar

[29]

P. Moylan, Dissipative Systems and Stability [Online], 2014. Google Scholar

[30]

M. A. MüllerD. Angeli and F. Allgöwer, On necessity and robustness of dissipativity in economic model predictive control, IEEE Trans. Automat. Control, 60 (2015), 1671-1676.  doi: 10.1109/TAC.2014.2361193.  Google Scholar

[31]

M. A. Müller and L. Grüne, On the relation between dissipativity and discounted dissipativity, in Proc. 56th IEEE Conf. Decis. Control, Melbourne, Australia, 2017, 5570–5575. Google Scholar

[32]

W. D. Nordhaus, An optimal transition path for controlling greenhouse gases, Science, 258 (1992), 1315-1319.  doi: 10.1126/science.258.5086.1315.  Google Scholar

[33]

W. D. Nordhaus, Revisiting the social cost of carbon, Proc. Natl. Acad. Scie. USA (PNAS), 114 (2017), 1518-1523.  doi: 10.1073/pnas.1609244114.  Google Scholar

[34]

R. Postoyan, L. Buşoniu, D. Nešić and J. Daafouz, Stability of infinite-horizon optimal control with discounted cost, in Proc. 53rd IEEE Conf. Decis. Control doi: 10.1109/CDC.2014.7039995.  Google Scholar

[35]

R. PostoyanL. BusoniuD. Nešić and J. Daafouz, Stability analysis of discrete-time infinite-horizon optimal control with discounted cost, IEEE Trans. Automat. Control, 62 (2017), 2736-2749.  doi: 10.1109/TAC.2016.2616644.  Google Scholar

[36]

F. P. Ramsey, A Mathematical Theory of Saving, Econ. J., 38 (1928), 543-559.  doi: 10.2307/2224098.  Google Scholar

[37]

M. S. Santos and J. Vigo-Aguiar, Analysis of a numerical dynamic programming algorithm applied to economic models, Econometrica, 66 (1998), 409-426.  doi: 10.2307/2998564.  Google Scholar

[38]

A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications, North-Holland, Amsterdam, 1987. Google Scholar

[39]

N. Stern, The Economics of Climate Change: The Stern Review, Cambridge University Press, 2007. doi: 10.1017/CBO9780511817434.  Google Scholar

[40]

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.  Google Scholar

[41]

A. van der Schaft, $L_2$-Gain and Passivity Techniques in Nonlinear Control, Lecture Notes in Control and Information Sciences, vol. 218, Springer-Verlag London, Ltd., London, 1996. doi: 10.1007/3-540-76074-1.  Google Scholar

[42]

S. R. Weller, S. Hafeez and C. M. Kellett, A receding horizon control approach to estimating the social cost of carbon in the presence of emissions and temperature uncertainty, in Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, 2015, 5384–5390. doi: 10.1109/CDC.2015.7403062.  Google Scholar

[43]

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

show all references

References:
[1]

D. Acemoglu, Introduction to Modern Economic Growth, Princeton University Press, 2009. Google Scholar

[2]

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

[3]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[4]

S. BeckerL. Grüne and W. Semmler, Comparing accuracy of second order approximation and dynamic programming, Comput. Econ., 30 (2007), 65-91.  doi: 10.1007/s10614-007-9087-1.  Google Scholar

[5]

D. P. Bertsekas, Nonlinear Programming, 2nd edition, Athena Scientific, Belmont, Massachusetts, 1995.  Google Scholar

[6]

J. Blot and N. Hayek, Infinite-Horizon Optimal Control in the Discrete-Time Framework, SpringerBriefs in Optimization, Springer, New York, 2014. doi: 10.1007/978-1-4614-9038-8.  Google Scholar

[7]

W. A. Brock and L. Mirman, Optimal economic growth and uncertainty: The discounted case, J. Econom. Theory, 4 (1972), 479-513.  doi: 10.1016/0022-0531(72)90135-4.  Google Scholar

[8]

B. Brogliato, R. Lozano, B. Maschke and O. Egeland, Dissipative Systems Analysis and Control, Second edition, Springer-Verlag, London, Ltd., 2007. doi: 10.1007/978-1-84628-517-2.  Google Scholar

[9]

C. I. Byrnes and W. Lin, Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems, IEEE Trans. Automat. Control, 39 (1994), 83-98.  doi: 10.1109/9.273341.  Google Scholar

[10]

D. Cass, Optimum growth in an aggregative model of capital accumulation, Rev. Econ. Stud., 32 (1965), 233-240.  doi: 10.2307/2295827.  Google Scholar

[11]

T. DammL. GrüneM. 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.  Google Scholar

[12]

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

[13]

T. Faulwasser, C. M. Kellett and S. R. Weller, MPC-DICE: An open-source Matlab implementation of receding horizon solutions to DICE, in Proc. 1st IFAC Workshop on Integrated Assessment Modelling for Environmental Systems, Brescia, Italy, 2018,126–131. Google Scholar

[14]

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.  Google Scholar

[15]

V. GaitsgoryL. GrüneM. HögerC. M. Kellett and S. R. Weller, Stabilization of strictly dissipative discrete time systems with discounted optimal control, Automatica J. IFAC, 93 (2018), 311-320.  doi: 10.1016/j.automatica.2018.03.076.  Google Scholar

[16]

V. GaitsgoryL. Grüne and N. Thatcher, Stabilization with discounted optimal control, Systems Control Lett., 82 (2015), 91-98.  doi: 10.1016/j.sysconle.2015.05.010.  Google Scholar

[17]

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

[18]

L. Grüne, C. M. Kellett and S. R. Weller, On a discounted notion of strict dissipativity, in Proc. 10th IFAC Symposium on Nonlinear Control Systems, 2016,247–252. Google Scholar

[19]

L. GrüneC. M. Kellett and S. R. Weller, On the relation between turnpike properties for finite and infinite horizon optimal control problems, J. Optim. Theory Appl., 173 (2017), 727-745.  doi: 10.1007/s10957-017-1103-6.  Google Scholar

[20]

L. Grüne and M. A. Müller, On the relation between strict dissipativity and turnpike properties, Systems Control Lett., 90 (2016), 45-53.  doi: 10.1016/j.sysconle.2016.01.003.  Google Scholar

[21]

L. Grüne and A. Panin, On non-averaged performance of economic MPC with terminal conditions, in Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, 2015, 4332–4337. Google Scholar

[22]

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

[23]

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

[24]

L. GrüneW. Semmler and M. Stieler, Using nonlinear model predictive control for dynamic decision problems in economics, J. Econom. Dynam. Control, 60 (2015), 112-133.  doi: 10.1016/j.jedc.2015.08.010.  Google Scholar

[25]

Interagency Working Group on Social Cost of Carbon, United States Government, Technical Support Document: Technical Update of the Social Cost of Carbon for Regulatory Impact Analysis - Under Executive Order 12866, Technical report, 2013. Google Scholar

[26]

C. M. KellettS. R. WellerT. FaulwasserL. Grüne and W. Semmler, Feedback, dynamics, and optimal control in climate economics, Annu. Rev. Control, 47 (2019), 7-20.  doi: 10.1016/j.arcontrol.2019.04.003.  Google Scholar

[27]

T. C. Koopmans, On the concept of optimal economic growth, in The Economic Approach to Development Planning, Rand McNally, Chicago, IL, 1965,225–287. Google Scholar

[28]

R. Lopezlena and J. M. A. Scherpen, Energy functions for dissipativity-based balancing of discrete-time nonlinear systems, Math. Control Signals Systems, 18 (2006), 345-368.  doi: 10.1007/s00498-006-0007-z.  Google Scholar

[29]

P. Moylan, Dissipative Systems and Stability [Online], 2014. Google Scholar

[30]

M. A. MüllerD. Angeli and F. Allgöwer, On necessity and robustness of dissipativity in economic model predictive control, IEEE Trans. Automat. Control, 60 (2015), 1671-1676.  doi: 10.1109/TAC.2014.2361193.  Google Scholar

[31]

M. A. Müller and L. Grüne, On the relation between dissipativity and discounted dissipativity, in Proc. 56th IEEE Conf. Decis. Control, Melbourne, Australia, 2017, 5570–5575. Google Scholar

[32]

W. D. Nordhaus, An optimal transition path for controlling greenhouse gases, Science, 258 (1992), 1315-1319.  doi: 10.1126/science.258.5086.1315.  Google Scholar

[33]

W. D. Nordhaus, Revisiting the social cost of carbon, Proc. Natl. Acad. Scie. USA (PNAS), 114 (2017), 1518-1523.  doi: 10.1073/pnas.1609244114.  Google Scholar

[34]

R. Postoyan, L. Buşoniu, D. Nešić and J. Daafouz, Stability of infinite-horizon optimal control with discounted cost, in Proc. 53rd IEEE Conf. Decis. Control doi: 10.1109/CDC.2014.7039995.  Google Scholar

[35]

R. PostoyanL. BusoniuD. Nešić and J. Daafouz, Stability analysis of discrete-time infinite-horizon optimal control with discounted cost, IEEE Trans. Automat. Control, 62 (2017), 2736-2749.  doi: 10.1109/TAC.2016.2616644.  Google Scholar

[36]

F. P. Ramsey, A Mathematical Theory of Saving, Econ. J., 38 (1928), 543-559.  doi: 10.2307/2224098.  Google Scholar

[37]

M. S. Santos and J. Vigo-Aguiar, Analysis of a numerical dynamic programming algorithm applied to economic models, Econometrica, 66 (1998), 409-426.  doi: 10.2307/2998564.  Google Scholar

[38]

A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications, North-Holland, Amsterdam, 1987. Google Scholar

[39]

N. Stern, The Economics of Climate Change: The Stern Review, Cambridge University Press, 2007. doi: 10.1017/CBO9780511817434.  Google Scholar

[40]

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.  Google Scholar

[41]

A. van der Schaft, $L_2$-Gain and Passivity Techniques in Nonlinear Control, Lecture Notes in Control and Information Sciences, vol. 218, Springer-Verlag London, Ltd., London, 1996. doi: 10.1007/3-540-76074-1.  Google Scholar

[42]

S. R. Weller, S. Hafeez and C. M. Kellett, A receding horizon control approach to estimating the social cost of carbon in the presence of emissions and temperature uncertainty, in Proc. 54th IEEE Conf. Decis. Control, Osaka, Japan, 2015, 5384–5390. doi: 10.1109/CDC.2015.7403062.  Google Scholar

[43]

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

Figure 1.  Illustration of the steady-states (blue solid line), level sets of $ \ell $ (black ellipses), and the additional constraint $ g_{ad} $ (red dashed) of the example in Section 8.4. The optimal steady-state $ (x^e,u^e)=(0,0) $ for the undiscounted case is marked with a circle
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