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March  2021, 11(1): 169-188. doi: 10.3934/mcrf.2020032

## On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems

 1 Mathematical Institute, University of Bayreuth, Germany 2 Institute of Applied Mathematics, Fundação Getúlio Vargas, Rio de Janeiro, Brasil

* Corresponding author: Roberto Guglielmi

Received  November 2019 Revised  March 2020 Published  March 2021 Early access  June 2020

Fund Project: The first author acknowledges support from the Deutsche Forschungsgemeinschaft via Grant GR 1569/16-1. The second author was partially supported by the project INdAM-GNAMPA 2019 on "Controllabilità di PDE in modelli fisici e in scienze della vita", and he wish to thanks also the Mathematical Institute of the University of Bayreuth for supporting his visit to the department

The paper is devoted to analyze the connection between turnpike phenomena and strict dissipativity properties for continuous-time finite dimensional linear quadratic optimal control problems. We characterize strict dissipativity properties of the dynamics 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 novelty of these results is the possibility to encompass the presence of state and input constraints.

Citation: Lars Grüne, Roberto Guglielmi. On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems. Mathematical Control and Related Fields, 2021, 11 (1) : 169-188. doi: 10.3934/mcrf.2020032
##### References:
 [1] B. D. O. Anderson and P. V. Kokotović, Optimal control problems over large time intervals, Automatica, 23 (1987), 355–363. doi: 10.1016/0005-1098(87)90008-2. [2] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441. [3] D.A. Carlson, A. B. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control - Deterministic and Stochastic Systems, 2$^nd$ edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76755-5. [4] T. Damm, Rational Matrix Equations in Stochastic Control, Lecture Notes in Control and Information Sciences, 297, Springer-Verlag, Berlin, 2004. [5] 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), 1935–1957. doi: 10.1137/120888934. [6] R. Dorfman, P. A. Samuelson and R. M. Solow, Linear Programming and Economic Analysis, Reprint of the 1958 original, Dover Publications, New York, 1987. [7] T. Faulwasser, M. Korda, C. N. Jones and D. Bonvin, On turnpike and dissipativity properties of continuous-time optimal control problems, Automatica, 81 (2017), 297–304. doi: 10.1016/j.automatica.2017.03.012. [8] L. Grüne, Economic receding horizon control without terminal constraints, Automatica, 49 (2013), 725–734. doi: 10.1016/j.automatica.2012.12.003. [9] L. Grüne, Approximation properties of receding horizon optimal control, Jahresber. DMV, 118 (2016), 3–37. doi: 10.1365/s13291-016-0134-5. [10] L. Grüne and R. Guglielmi, Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems, SIAM J. Control and Optim., 56 (2018), 1282–1302. doi: 10.1137/17M112350X. [11] L. Grüne and M. A. Müller, On the relation between strict dissipativity and the turnpike property, Syst. Contr. Lett., 90, (2016), 45–53. doi: 10.1016/j.sysconle.2016.01.003. [12] L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Theory and Algorithms, 2nd edition, Springer-Verlag, London, 2017. doi: 10.1007/978-3-319-46024-6. [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, Syst. Control Lett., 90 (2016), 61–70. doi: 10.1016/j.sysconle.2016.02.001. [14] D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I, Texts in Applied Mathematics, 48, Springer, Heidelberg, 2010. [15] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994. [16] 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. [17] L. W. McKenzie, Optimal economic growth, turnpike theorems and comparative dynamics, in Handbook of Mathematical Economics, Vol. Ⅲ, Amsterdam, North-Holland, 1 (1986), 1281–1355. [18] P. Moylan, Dissipative Systems and Stability, 2014. [19] A. Porretta and E. Zuazua, Long time versus steady state optimal control, SIAM J. Control Optim., 51 (2013), 4242–4273. doi: 10.1137/130907239. [20] J. B. Rawlings and R. Amrit, Optimizing process economic performance using model predictive control, in Nonlinear Model Predictive Control, (eds L. Magni and D. M. Raimondo and F. Allgöwer), Lecture Notes in Control and Information Science, 384, Springer-Verlag, (2009), 119–138. doi: 10.1007/978-3-642-01094-1_10. [21] N. Sakamoto, D. Pighin and E. Zuazua, The turnpike property in nonlinear optimal control - A geometric approach, 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, (2019), 2422–2427. doi: 10.1109/CDC40024.2019.9028863. [22] E. D. Sontag, Mathematical Control Theory, 2nd edition, Springer Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7. [23] E. Trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control, J. Differ. Equ., 258 (2015), 81–114. doi: 10.1016/j.jde.2014.09.005. [24] E. Trélat and C. Zhang, Integral and measure-turnpike property for infinite dimensional optimal control problems, Math. Control Signals Systems, 30 (2018), Art. 3, 34 pp. doi: 10.1007/s00498-018-0209-1. [25] J. von Neumann, A model of general economic equilibrium, The Review of Economic Studies, 13 (1945), 1–9. doi: 10.2307/2296111. [26] J. C. Willems, Dissipative dynamical systems. Ⅰ. General theory, Arch. Rational Mech. Anal., 45 (1972), 321–351. doi: 10.1007/BF00276493. [27] J. C. Willems, Dissipative dynamical systems. Ⅱ. Linear systems with quadratic supply rates, Arch. Rational Mech. Anal., 45 (1972), 352–393. doi: 10.1007/BF00276494. [28] J. C. Willems, Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Autom. Control, 16 (1971), 621–634. doi: 10.1109/tac.1971.1099831. [29] A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006. [30] A. J. Zaslavski, Turnpike Phenomenon and Infinite Horizon Optimal Control, Springer International, 2014. doi: 10.1007/978-3-319-08828-0.

show all references

##### References:
 [1] B. D. O. Anderson and P. V. Kokotović, Optimal control problems over large time intervals, Automatica, 23 (1987), 355–363. doi: 10.1016/0005-1098(87)90008-2. [2] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004. doi: 10.1017/CBO9780511804441. [3] D.A. Carlson, A. B. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control - Deterministic and Stochastic Systems, 2$^nd$ edition, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76755-5. [4] T. Damm, Rational Matrix Equations in Stochastic Control, Lecture Notes in Control and Information Sciences, 297, Springer-Verlag, Berlin, 2004. [5] 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), 1935–1957. doi: 10.1137/120888934. [6] R. Dorfman, P. A. Samuelson and R. M. Solow, Linear Programming and Economic Analysis, Reprint of the 1958 original, Dover Publications, New York, 1987. [7] T. Faulwasser, M. Korda, C. N. Jones and D. Bonvin, On turnpike and dissipativity properties of continuous-time optimal control problems, Automatica, 81 (2017), 297–304. doi: 10.1016/j.automatica.2017.03.012. [8] L. Grüne, Economic receding horizon control without terminal constraints, Automatica, 49 (2013), 725–734. doi: 10.1016/j.automatica.2012.12.003. [9] L. Grüne, Approximation properties of receding horizon optimal control, Jahresber. DMV, 118 (2016), 3–37. doi: 10.1365/s13291-016-0134-5. [10] L. Grüne and R. Guglielmi, Turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems, SIAM J. Control and Optim., 56 (2018), 1282–1302. doi: 10.1137/17M112350X. [11] L. Grüne and M. A. Müller, On the relation between strict dissipativity and the turnpike property, Syst. Contr. Lett., 90, (2016), 45–53. doi: 10.1016/j.sysconle.2016.01.003. [12] L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Theory and Algorithms, 2nd edition, Springer-Verlag, London, 2017. doi: 10.1007/978-3-319-46024-6. [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, Syst. Control Lett., 90 (2016), 61–70. doi: 10.1016/j.sysconle.2016.02.001. [14] D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I, Texts in Applied Mathematics, 48, Springer, Heidelberg, 2010. [15] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994. [16] 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. [17] L. W. McKenzie, Optimal economic growth, turnpike theorems and comparative dynamics, in Handbook of Mathematical Economics, Vol. Ⅲ, Amsterdam, North-Holland, 1 (1986), 1281–1355. [18] P. Moylan, Dissipative Systems and Stability, 2014. [19] A. Porretta and E. Zuazua, Long time versus steady state optimal control, SIAM J. Control Optim., 51 (2013), 4242–4273. doi: 10.1137/130907239. [20] J. B. Rawlings and R. Amrit, Optimizing process economic performance using model predictive control, in Nonlinear Model Predictive Control, (eds L. Magni and D. M. Raimondo and F. Allgöwer), Lecture Notes in Control and Information Science, 384, Springer-Verlag, (2009), 119–138. doi: 10.1007/978-3-642-01094-1_10. [21] N. Sakamoto, D. Pighin and E. Zuazua, The turnpike property in nonlinear optimal control - A geometric approach, 2019 IEEE 58th Conference on Decision and Control (CDC), Nice, France, (2019), 2422–2427. doi: 10.1109/CDC40024.2019.9028863. [22] E. D. Sontag, Mathematical Control Theory, 2nd edition, Springer Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7. [23] E. Trélat and E. Zuazua, The turnpike property in finite-dimensional nonlinear optimal control, J. Differ. Equ., 258 (2015), 81–114. doi: 10.1016/j.jde.2014.09.005. [24] E. Trélat and C. Zhang, Integral and measure-turnpike property for infinite dimensional optimal control problems, Math. Control Signals Systems, 30 (2018), Art. 3, 34 pp. doi: 10.1007/s00498-018-0209-1. [25] J. von Neumann, A model of general economic equilibrium, The Review of Economic Studies, 13 (1945), 1–9. doi: 10.2307/2296111. [26] J. C. Willems, Dissipative dynamical systems. Ⅰ. General theory, Arch. Rational Mech. Anal., 45 (1972), 321–351. doi: 10.1007/BF00276493. [27] J. C. Willems, Dissipative dynamical systems. Ⅱ. Linear systems with quadratic supply rates, Arch. Rational Mech. Anal., 45 (1972), 352–393. doi: 10.1007/BF00276494. [28] J. C. Willems, Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Autom. Control, 16 (1971), 621–634. doi: 10.1109/tac.1971.1099831. [29] A. J. Zaslavski, Turnpike Properties in the Calculus of Variations and Optimal Control, Springer, New York, 2006. [30] A. J. Zaslavski, Turnpike Phenomenon and Infinite Horizon Optimal Control, Springer International, 2014. doi: 10.1007/978-3-319-08828-0.
Schematic sketch of Theorem 8.1
Schematic sketch of Theorem 8.4
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