Abstract

We analyze the connection between turnpike behaviors and strict dissipativity properties for discrete time finite dimensional linear quadratic optimal control problems. We first use strict dissipativity as a sufficient condition for the turnpike property. Next, we characterize strict dissipativity and the newly introduced property of strict pre-dissipativity in terms of the system matrices related to the linear quadratic problem. These characterizations then lead to new necessary conditions for the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities. One of the key novelties which distinguishes the results in the present paper from earlier ones on linear quadratic optimal control problems is the consideration of state and input constraints.

Keywords

  1. turnpike property
  2. linear quadratic optimal control
  3. dissipativity
  4. detectability
  5. Lyapunov matrix inequality
  6. long time behavior

MSC codes

  1. 49K15
  2. 49N10
  3. 49J15
  4. 93D20
  5. 93C15

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
B. D. O. Anderson and P. V. Kokotović, Optimal control problems over large time intervals, Automatica, 23 (1987), pp. 355--363.
2.
G. D. Birkhoff, Proof of the ergodic theorem, Proc. Natl. Acad. Sci. USA, 17 (1931), pp. 656--660.
3.
A. Boccia, L. Grüne, and K. Worthmann, Stability and feasibility of state constrained MPC without stabilizing terminal constraints, Systems Control Lett., 72 (2014), pp. 14--21.
4.
C. I. Byrnes and W. Lin, Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems, IEEE Trans. Automat. Control, 39 (1994), pp. 83--98.
5.
D. A. Carlson, A. B. Haurie, and A. Leizarowitz, Infinite Horizon Optimal Control---Deterministic and Stochastic Systems, 2nd ed., Springer-Verlag, Berlin, 1991.
6.
T. Damm, Rational Matrix Equations in Stochastic Control, Lecture Notes in Control and Inform. Sci. 297, Springer-Verlag, Berlin, 2004.
7.
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), pp. 1935--1957.
8.
R. Dorfman, P. A. Samuelson, and R. M. Solow, Linear Programming and Economic Analysis, Dover Publications, New York, 1987.
9.
L. Grüne, Economic receding horizon control without terminal constraints, Automatica, 49 (2013), pp. 725--734.
10.
L. Grüne, Approximation properties of receding horizon optimal control, Jahresber. Dtsch. Math.-Ver., 118 (2016), pp. 3--37.
11.
L. Grüne and M. A. Müller, On the relation between strict dissipativity and the turnpike property, Systems Control Lett., 90 (2016), pp. 45--53.
12.
L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Theory and Algorithms, 2nd ed., Springer-Verlag, Berlin, 2017.
13.
M. Gugat, E. Trélat, and E. Zuazua, Optimal Neumann control for the 1D wave equation: Finite horizon, infinite horizon, boundary tracking terms and the turnpike property, Systems Control Lett., 90 (2016), pp. 61--70.
14.
D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness. Texts in Appl. Math., Vol. 48, Springer-Verlag, Berlin, 2010. Corrected reprint [of MR2116013].
15.
L. W. McKenzie, Optimal economic growth, turnpike theorems and comparative dynamics, in Handbook of Mathematical Economics, Vol. III, North-Holland, Amsterdam, 1986, pp. 1281--1355.
16.
P. Moylan, Dissipative Systems and Stability, http://www.pmoylan.org (2014).
17.
A. Porretta and E. Zuazua, Long time versus steady state optimal control, SIAM J. Control Optim., 51 (2013), pp. 4242--4273.
18.
W. J. Rugh, Linear System Theory, 2nd ed., Prentice-Hall, Upper Saddle River, N.J., 1996.
19.
E. D. Sontag, Mathematical Control Theory, 2nd ed., Springer-Verlag, Berlin, 1998.
20.
E. Trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control, J. Differential Equations, 258 (2015), pp. 81--114.
21.
J. von Neumann, A model of general economic equilibrium, Rev. Econom. Stud., 13 (1945), pp. 1--9.
22.
J. C. Willems, Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Automat. Control, 16 (1971), pp. 621--634.
23.
J. C. Willems, Dissipative dynamical systems. I. General theory, Arch. Ration. Mech. Anal., 45 (1972), pp. 321--351.
24.
J. C. Willems, Dissipative dynamical systems. II. Linear systems with quadratic supply rates, Arch. Ration. Mech. Anal., 45 (1972), pp. 352--393.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 1282 - 1302
ISSN (online): 1095-7138

History

Submitted: 30 March 2017
Accepted: 24 January 2018
Published online: 3 April 2018

Keywords

  1. turnpike property
  2. linear quadratic optimal control
  3. dissipativity
  4. detectability
  5. Lyapunov matrix inequality
  6. long time behavior

MSC codes

  1. 49K15
  2. 49N10
  3. 49J15
  4. 93D20
  5. 93C15

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.