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2011, 1(2): 245-260. doi: 10.3934/naco.2011.1.245

Stability of a turnpike phenomenon for a class of optimal control systems in metric spaces

Alexander J. Zaslavski 1,

1. 

Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel

Received  December 2010 Revised  March 2011 Published  June 2011

We study a turnpike property of approximate solutions of a discrete-time control system with a compact metric space of states. In our recent work we prove this turnpike property and show that it is stable under perturbations of an objective function. In the present paper we improve this turnpike result by showing that it also holds for those solutions defined on a finite interval (domain) which are approximately optimal on all subintervals of the domain that have a fixed length which does not depend on the length of the whole domain.
Keywords: infinite horizon problem, optimal solution, program., Good function.
Mathematics Subject Classification: 49J9.
Citation: Alexander J. Zaslavski. Stability of a turnpike phenomenon for a class of optimal control systems in metric spaces. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 245-260. doi: 10.3934/naco.2011.1.245
References:
[1]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I , Physica D, 8 (1983), 381-422. doi: doi:10.1016/0167-2789(83)90233-6.  Google Scholar

[2]

J. Blot, Infinite-horizon Pontryagin principles without invertibility, J. Nonlinear Convex Anal., 10 (2009), 177-189.  Google Scholar

[3]

J. Blot and P. Cartigny, Optimality in infinite-horizon variational problems under sign conditions, J. Optim. Theory Appl., 106 (2000), 411-419. doi: doi:10.1023/A:1004611816252.  Google Scholar

[4]

P. Cartigny and P. Michel, On a sufficient transversality condition for infinite horizon optimal control problems, Automatica J. IFAC, 39 (2003), 1007-1010. doi: doi:10.1016/S0005-1098(03)00060-8.  Google Scholar

[5]

D. Gale, On optimal development in a multi-sector economy, Review of Economic Studies, 34 (1967), 1-18. doi: doi:10.2307/2296567.  Google Scholar

[6]

A. Leizarowitz, Infinite horizon autonomous systems with unbounded cost, Appl. Math. and Opt., 13 (1985), 19-43. doi: doi:10.1007/BF01442197.  Google Scholar

[7]

A. Leizarowitz, Tracking nonperiodic trajectories with the overtaking criterion, Appl. Math. and Opt., 14 (1986), 155-171. doi: doi:10.1007/BF01442233.  Google Scholar

[8]

A. Leizarowitz and V. J. Mizel, One dimensional infinite horizon variational problems arising in continuum mechanics, Arch. Rational Mech. Anal., 106 (1989), 161-194. doi: doi:10.1007/BF00251430.  Google Scholar

[9]

V. Lykina, S. Pickenhain and M. Wagner, Different interpretations of the improper integral objective in an infinite horizon control problem, J. Math. Anal. Appl, 340 (2008), 498-510. doi: doi:10.1016/j.jmaa.2007.08.008.  Google Scholar

[10]

V. L. Makarov and A. M. Rubinov, "Mathematical Theory of Economic Dynamics and Equilibria,'' Springer-Verlag, New York, 1977.  Google Scholar

[11]

M. A. Mamedov and S. Pehlivan, Statistical convergence of optimal paths, Math. Japon., 52 (2000), 51-55.  Google Scholar

[12]

M. A. Mamedov and S. Pehlivan, Statistical cluster points and turnpike theorem in nonconvex problems, J. Math. Anal. Appl., 256 (2001), 686-693. doi: doi:10.1006/jmaa.2000.7061.  Google Scholar

[13]

L. W. McKenzie, Turnpike theory, Econometrica 44 (1976), 841-866. doi: doi:10.2307/1911532.  Google Scholar

[14]

B. Mordukhovich, Minimax design for a class of distributed parameter systems, Automat. Remote Control, 50 (1990), 1333-1340. Google Scholar

[15]

B. Mordukhovich and I. Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations, in "Optimal Control, Stabilization and Nonsmooth Analysis,'' Lecture Notes Control Inform. Sci. Springer, 2004, 121-132.  Google Scholar

[16]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincare, Analyse Nonlineare, 3 (1986), 229-272.  Google Scholar

[17]

S. Pickenhain, V. Lykina and M. Wagner, On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems, Control Cybernet. 37 (2008), 451-468.  Google Scholar

[18]

A. M. Rubinov, Economic dynamics, J. Soviet Math., 26 (1984), 1975-2012. doi: doi:10.1007/BF01084444.  Google Scholar

[19]

P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule, American Economic Review, 55 (1965), 486-496. Google Scholar

[20]

A. J. Zaslavski, Optimal programs on infinite horizon 1, SIAM Journal on Control and Optimization, 33 (1995), 1643-1660. doi: doi:10.1137/S036301299325726X.  Google Scholar

[21]

A. J. Zaslavski, Optimal programs on infinite horizon 2, SIAM Journal on Control and Optimization, 33 (1995), 1661-1686. doi: doi:10.1137/S0363012993257271.  Google Scholar

[22]

A. J. Zaslavski, "Turnpike Properties in the Calculus of Variations and Optimal Control,'' Springer, New York, 2006.  Google Scholar

[23]

A. J. Zaslavski, Turnpike results for a discrete-time optimal control system arising in economic dynamics, Nonlinear Analysis, 67 (2007), 2024-2049. doi: doi:10.1016/j.na.2006.08.029.  Google Scholar

[24]

A. J. Zaslavski, Two turnpike results for a discrete-time optimal control system, Nonlinear Analysis, 71 (2009), 902-909. doi: doi:10.1016/j.na.2008.12.053.  Google Scholar

[25]

A. J .Zaslavski, Stability of a turnpike phenomenon for a discrete-time optimal control system, J. Optim. Theory Appl., 145 (2010), 597-612. doi: doi:10.1007/s10957-010-9677-2.  Google Scholar

show all references

References:
[1]

S. Aubry and P. Y. Le Daeron, The discrete Frenkel-Kontorova model and its extensions I , Physica D, 8 (1983), 381-422. doi: doi:10.1016/0167-2789(83)90233-6.  Google Scholar

[2]

J. Blot, Infinite-horizon Pontryagin principles without invertibility, J. Nonlinear Convex Anal., 10 (2009), 177-189.  Google Scholar

[3]

J. Blot and P. Cartigny, Optimality in infinite-horizon variational problems under sign conditions, J. Optim. Theory Appl., 106 (2000), 411-419. doi: doi:10.1023/A:1004611816252.  Google Scholar

[4]

P. Cartigny and P. Michel, On a sufficient transversality condition for infinite horizon optimal control problems, Automatica J. IFAC, 39 (2003), 1007-1010. doi: doi:10.1016/S0005-1098(03)00060-8.  Google Scholar

[5]

D. Gale, On optimal development in a multi-sector economy, Review of Economic Studies, 34 (1967), 1-18. doi: doi:10.2307/2296567.  Google Scholar

[6]

A. Leizarowitz, Infinite horizon autonomous systems with unbounded cost, Appl. Math. and Opt., 13 (1985), 19-43. doi: doi:10.1007/BF01442197.  Google Scholar

[7]

A. Leizarowitz, Tracking nonperiodic trajectories with the overtaking criterion, Appl. Math. and Opt., 14 (1986), 155-171. doi: doi:10.1007/BF01442233.  Google Scholar

[8]

A. Leizarowitz and V. J. Mizel, One dimensional infinite horizon variational problems arising in continuum mechanics, Arch. Rational Mech. Anal., 106 (1989), 161-194. doi: doi:10.1007/BF00251430.  Google Scholar

[9]

V. Lykina, S. Pickenhain and M. Wagner, Different interpretations of the improper integral objective in an infinite horizon control problem, J. Math. Anal. Appl, 340 (2008), 498-510. doi: doi:10.1016/j.jmaa.2007.08.008.  Google Scholar

[10]

V. L. Makarov and A. M. Rubinov, "Mathematical Theory of Economic Dynamics and Equilibria,'' Springer-Verlag, New York, 1977.  Google Scholar

[11]

M. A. Mamedov and S. Pehlivan, Statistical convergence of optimal paths, Math. Japon., 52 (2000), 51-55.  Google Scholar

[12]

M. A. Mamedov and S. Pehlivan, Statistical cluster points and turnpike theorem in nonconvex problems, J. Math. Anal. Appl., 256 (2001), 686-693. doi: doi:10.1006/jmaa.2000.7061.  Google Scholar

[13]

L. W. McKenzie, Turnpike theory, Econometrica 44 (1976), 841-866. doi: doi:10.2307/1911532.  Google Scholar

[14]

B. Mordukhovich, Minimax design for a class of distributed parameter systems, Automat. Remote Control, 50 (1990), 1333-1340. Google Scholar

[15]

B. Mordukhovich and I. Shvartsman, Optimization and feedback control of constrained parabolic systems under uncertain perturbations, in "Optimal Control, Stabilization and Nonsmooth Analysis,'' Lecture Notes Control Inform. Sci. Springer, 2004, 121-132.  Google Scholar

[16]

J. Moser, Minimal solutions of variational problems on a torus, Ann. Inst. H. Poincare, Analyse Nonlineare, 3 (1986), 229-272.  Google Scholar

[17]

S. Pickenhain, V. Lykina and M. Wagner, On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems, Control Cybernet. 37 (2008), 451-468.  Google Scholar

[18]

A. M. Rubinov, Economic dynamics, J. Soviet Math., 26 (1984), 1975-2012. doi: doi:10.1007/BF01084444.  Google Scholar

[19]

P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule, American Economic Review, 55 (1965), 486-496. Google Scholar

[20]

A. J. Zaslavski, Optimal programs on infinite horizon 1, SIAM Journal on Control and Optimization, 33 (1995), 1643-1660. doi: doi:10.1137/S036301299325726X.  Google Scholar

[21]

A. J. Zaslavski, Optimal programs on infinite horizon 2, SIAM Journal on Control and Optimization, 33 (1995), 1661-1686. doi: doi:10.1137/S0363012993257271.  Google Scholar

[22]

A. J. Zaslavski, "Turnpike Properties in the Calculus of Variations and Optimal Control,'' Springer, New York, 2006.  Google Scholar

[23]

A. J. Zaslavski, Turnpike results for a discrete-time optimal control system arising in economic dynamics, Nonlinear Analysis, 67 (2007), 2024-2049. doi: doi:10.1016/j.na.2006.08.029.  Google Scholar

[24]

A. J. Zaslavski, Two turnpike results for a discrete-time optimal control system, Nonlinear Analysis, 71 (2009), 902-909. doi: doi:10.1016/j.na.2008.12.053.  Google Scholar

[25]

A. J .Zaslavski, Stability of a turnpike phenomenon for a discrete-time optimal control system, J. Optim. Theory Appl., 145 (2010), 597-612. doi: doi:10.1007/s10957-010-9677-2.  Google Scholar

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