Strict dissipativity for discrete time discounted optimal control problems

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 an appropriate available storage function 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.

[1]  Dragan Nesic,et al.  Stability of infinite-horizon optimal control with discounted cost , 2014, 53rd IEEE Conference on Decision and Control.

[2]  Steven R. Weller,et al.  MPC-DICE: An open-source Matlab implementation of receding horizon solutions to DICE , 2018 .

[3]  F. Ramsey,et al.  THE MATHEMATICAL THEORY OF SAVING , 1928 .

[4]  Jan C. Willems,et al.  Dissipative Dynamical Systems , 2007, Eur. J. Control.

[5]  Vladimir Gaitsgory,et al.  Stabilization with discounted optimal control , 2015, Syst. Control. Lett..

[6]  William D. Nordhaus,et al.  Revisiting the social cost of carbon , 2017, Proceedings of the National Academy of Sciences.

[7]  Joël Blot,et al.  Infinite-Horizon Optimal Control in the Discrete-Time Framework , 2013 .

[8]  David Cass,et al.  Optimum Growth in an Aggregative Model of Capital Accumulation , 1965 .

[9]  Tjalling C. Koopmans,et al.  On the concept of optimal economic growth , 1963 .

[10]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[11]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[12]  Lars Grüne,et al.  On a discounted notion of strict dissipativity , 2016 .

[13]  Manuel S. Santos,et al.  Analysis of a Numerical Dynamic Programming Algorithm Applied to Economic Models , 1998 .

[14]  Lars Grüne,et al.  On the relation between dissipativity and discounted dissipativity , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[15]  Lars Grüne,et al.  Economic receding horizon control without terminal constraints , 2013, Autom..

[16]  Lars Grüne,et al.  Asymptotic stability and transient optimality of economic MPC without terminal conditions , 2014 .

[17]  Lars Grüne,et al.  Comparing accuracy of second-order approximation and dynamic programming , 2006 .

[18]  Vladimir Gaitsgory,et al.  Stabilization of strictly dissipative discrete time systems with discounted optimal control , 2018, Autom..

[19]  Leonard J. Mirman,et al.  Optimal economic growth and uncertainty: The discounted case , 1972 .

[20]  Karl Worthmann,et al.  An Exponential Turnpike Theorem for Dissipative Discrete Time Optimal Control Problems , 2014, SIAM J. Control. Optim..

[21]  Romain Postoyan,et al.  Stability Analysis of Discrete-Time Infinite-Horizon Optimal Control With Discounted Cost , 2017, IEEE Transactions on Automatic Control.

[22]  Daron Acemoglu,et al.  Introduction to Modern Economic Growth , 2009 .

[23]  David Angeli,et al.  On Average Performance and Stability of Economic Model Predictive Control , 2012, IEEE Transactions on Automatic Control.

[24]  David Angeli,et al.  On Necessity and Robustness of Dissipativity in Economic Model Predictive Control , 2015, IEEE Transactions on Automatic Control.

[25]  N. Stern.  The Economics of Climate Change: Implications of Climate Change for Development , 2007 .

[26]  Knut Sydsæter,et al.  Optimal control theory with economic applications , 1987 .

[27]  Steven R. Weller,et al.  A receding horizon control approach to estimating the social cost of carbon in the presence of emissions and temperature uncertainty , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[28]  Diana Bohm,et al.  L2 Gain And Passivity Techniques In Nonlinear Control , 2016 .

[29]  Lars Grüne,et al.  On the Relation Between Turnpike Properties for Finite and Infinite Horizon Optimal Control Problems , 2017, J. Optim. Theory Appl..

[30]  W. Nordhaus.  An Optimal Transition Path for Controlling Greenhouse Gases , 1992, Science.

[31]  Lars Grüne,et al.  Using Nonlinear Model Predictive Control for Dynamic Decision Problems in Economics , 2015 .

[32]  Wei Lin,et al.  Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems , 1994, IEEE Trans. Autom. Control..

[33]  Lars Grüne,et al.  On non-averaged performance of economic MPC with terminal conditions , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[34]  Can Zhang,et al.  Integral and measure-turnpike properties for infinite-dimensional optimal control systems , 2017, Math. Control. Signals Syst..

[35]  Lars Grüne,et al.  Feedback and Optimal Control in Climate Economics , 2018, ArXiv.

[36]  L. Grüne,et al.  Nonlinear Model Predictive Control : Theory and Algorithms. 2nd Edition , 2011 .

[37]  Bernhard Maschke,et al.  Dissipative Systems Analysis and Control , 2000 .

[38]  Moritz Diehl,et al.  A Lyapunov Function for Economic Optimizing Model Predictive Control , 2011, IEEE Transactions on Automatic Control.