On a discounted notion of strict dissipativity

Abstract: Recent results in the literature have provided connections between the so-called turnpike property, near optimality of closed-loop solutions using model predictive control schemes, and strict dissipativity. An important feature of these results is that strict dissipativity provides a checkable condition for the other two properties. These results relate to optimal control problems with undiscounted stage cost. Motivated by applications in economics, we consider optimal control problems with discounted stage cost and define a notion of discounted strict dissipativity. As in the undiscounted case, we show that discounted strict dissipativity provides a checkable condition for various properties of the solutions of the optimal control problem associated with the appropriately defined discounted available storage function.

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

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

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

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

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

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

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

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

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

[10]  W. Nordhaus.  Estimates of the Social Cost of Carbon: Concepts and Results from the DICE-2013R Model and Alternative Approaches , 2014, Journal of the Association of Environmental and Resource Economists.

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

[12]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

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

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

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

[16]  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).

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

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

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

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

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