December  2018, 11(6): 1283-1316. doi: 10.3934/dcdss.2018072

Structure of approximate solutions of Bolza variational problems on large intervals

Department of Mathematics, The Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel

Received  March 2017 Revised  July 2017 Published  June 2018

In this paper we study the structure of approximate solutions of autonomous Bolza variational problems on large finite intervals. We show that approximate solutions are determined mainly by the integrand, and are essentially independent of the choice of time interval and data.

Citation: Alexander J. Zaslavski. Structure of approximate solutions of Bolza variational problems on large intervals. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1283-1316. doi: 10.3934/dcdss.2018072
References:
[1]

S. M. Aseev and V. M. Veliov, Maximum principle for infinite-horizon optimal control problems with dominating discount, Dynamics of Continuous, Discrete and Impulsive Systems, SERIES B, 19 (2012), 43-63.   Google Scholar

[2]

J. P. Aubin and I. Ekeland I, Applied Nonlinear Analysis, Wiley Interscience, New York, 1984.  Google Scholar

[3]

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

[4]

M. Bashir and J. Blot, Infinite dimensional infinite-horizon Pontryagin principles for discrete-time problems, Set-Valued and Variational Analysis, 23 (2015), 43-54.  doi: 10.1007/s11228-014-0302-7.  Google Scholar

[5]

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

[6]

J. Blot and N. Hayek, Infinite-Horizon Optimal Control in the Discrete-Time Framework, SpringerBriefs in Optimization, New York. 2014. doi: 10.1007/978-1-4614-9038-8.  Google Scholar

[7]

I. Bright, A reduction of topological infinite-horizon optimization to periodic optimization in a class of compact 2-manifolds, Journal of Mathematical Analysis and Applications, 394 (2012), 84-101.  doi: 10.1016/j.jmaa.2012.03.042.  Google Scholar

[8]

D. A. Carlson, A. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.  Google Scholar

[9]

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

[10]

T. DammL. GruneM. Stieler and K. Worthmann, An exponential turnpike theorem for dissipative discrete time optimal control problems, SIAM Journal on Control and Optimization, 52 (2014), 1935-1957.  doi: 10.1137/120888934.  Google Scholar

[11]

V. A. De Oliveira and G. N. Silva, Optimality conditions for infinite horizon control problems with state constraints, Nonlinear Analysis, 71 (2009), 1788-1795.  doi: 10.1016/j.na.2009.02.052.  Google Scholar

[12]

V. GaitsgoryL. Grune and N. Thatcher, Stabilization with discounted optimal control, Systems and Control Letters, 82 (2015), 91-98.  doi: 10.1016/j.sysconle.2015.05.010.  Google Scholar

[13]

N. Hayek, Infinite horizon multiobjective optimal control problems in the discrete time case, Optimization, 60 (2011), 509-529.  doi: 10.1080/02331930903480352.  Google Scholar

[14]

D. V. Khlopin, Necessary conditions of overtaking equilibrium for infinite horizon differential games, Mat.Teor. Igr Pril., 5 (2013), 105-136.   Google Scholar

[15]

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: 10.1007/BF00251430.  Google Scholar

[16]

V. LykinaS. 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: 10.1016/j.jmaa.2007.08.008.  Google Scholar

[17]

M. Mammadov, Turnpike theorem for an infinite horizon optimal control problem with time delay, SIAM Journal on Control and Optimization, 52 (2014), 420-438.  doi: 10.1137/130926808.  Google Scholar

[18]

M. Marcus and A. J. Zaslavski, The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincaré, Anal. non linéaire, 16 (1999), 593-629.  doi: 10.1016/S0294-1449(99)80029-8.  Google Scholar

[19]

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

[20]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions, Appl. Analysis, 90 (2011), 1075-1109.  doi: 10.1080/00036811003735840.  Google Scholar

[21]

P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule, Amer. Econom. Rev., 55 (1965), 486-496.   Google Scholar

[22]

E. Trelat and E. Zuazua, E The turnpike property in finite-dimensional nonlinear optimal control, Journal of Differential Equations, 218 (2015), 81-114.  doi: 10.1016/j.jde.2014.09.005.  Google Scholar

[23]

A. J. Zaslavski, Ground states in Frenkel-Kontorova model, Math. USSR Izvestiya, 29 (1987), 323-354.  doi: 10.1070/IM1987v029n02ABEH000972.  Google Scholar

[24]

A. J. Zaslavski, Dynamic properties of optimal solutions of variational problems, Nonlinear Analysis, Theory, Methods and Applications, 27 (1996), 895-931.  doi: 10.1016/0362-546X(95)00029-U.  Google Scholar

[25]

A. J. Zaslavski, Existence and uniform boundedness of optimal solutions of variational problems, Abstract and Applied Analysis, 3 (1998), 265-292.  doi: 10.1155/S1085337598000566.  Google Scholar

[26]

A. J. Zaslavski, Turnpike property for extremals of variational problems with vector-valued functions, Transactions of the AMS, 351 (1999), 211-231.  doi: 10.1090/S0002-9947-99-02132-7.  Google Scholar

[27]

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

[28]

A. J. Zaslavski, A nonintersection property for extremals of variational problems with vector-valued functions, Ann. Inst. H. Poincare, Anal.non lineare, 23 (2006), 929-948.  doi: 10.1016/j.anihpc.2006.01.002.  Google Scholar

[29]

A. J. Zaslavski, Turnpike properties of approximate solutions of autonomous variational problems, Control and Cybernetics, 37 (2008), 491-512.   Google Scholar

[30]

A. J. Zaslavski, Convergence of approximate solutions of variational problems, Control and Cybernetics, 38 (2009), 1607-1629.   Google Scholar

[31]

A. J. Zaslavski, Turnpike Phenomenon and Infinite Horizon Optimal Control, Springer Optimization and Its Applications, New York, 2014. doi: 10.1007/978-3-319-08828-0.  Google Scholar

[32]

A. J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications, Springer, Cham-Heidelberg-New York-Dordrecht-London, 2015. doi: 10.1007/978-3-319-19141-6.  Google Scholar

[33]

A. J. Zaslavski, Structure of extremals of variational problems in the regions close to the endpoints, Calculus of Variations and PDE's, 53 (2015), 847-878.  doi: 10.1007/s00526-014-0769-y.  Google Scholar

[34]

A. J. Zaslavski, Structure of solutions of optimal control problems on large intervals: A survey of recent results, Pure and Applied Functional Analysis, 1 (2016), 123-158.   Google Scholar

[35]

A. J. Zaslavski, Linear control systems with nonconvex integrands on large intervals, Pure and Applied Functional Analysis, 1 (2016), 441-474.   Google Scholar

[36]

A. J. Zaslavski, Structure of approximate solutions of autonomous variational problems, Applied Analysis and Optimization, 1 (2017), 113-151.   Google Scholar

show all references

References:
[1]

S. M. Aseev and V. M. Veliov, Maximum principle for infinite-horizon optimal control problems with dominating discount, Dynamics of Continuous, Discrete and Impulsive Systems, SERIES B, 19 (2012), 43-63.   Google Scholar

[2]

J. P. Aubin and I. Ekeland I, Applied Nonlinear Analysis, Wiley Interscience, New York, 1984.  Google Scholar

[3]

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

[4]

M. Bashir and J. Blot, Infinite dimensional infinite-horizon Pontryagin principles for discrete-time problems, Set-Valued and Variational Analysis, 23 (2015), 43-54.  doi: 10.1007/s11228-014-0302-7.  Google Scholar

[5]

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

[6]

J. Blot and N. Hayek, Infinite-Horizon Optimal Control in the Discrete-Time Framework, SpringerBriefs in Optimization, New York. 2014. doi: 10.1007/978-1-4614-9038-8.  Google Scholar

[7]

I. Bright, A reduction of topological infinite-horizon optimization to periodic optimization in a class of compact 2-manifolds, Journal of Mathematical Analysis and Applications, 394 (2012), 84-101.  doi: 10.1016/j.jmaa.2012.03.042.  Google Scholar

[8]

D. A. Carlson, A. Haurie and A. Leizarowitz, Infinite Horizon Optimal Control, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-76755-5.  Google Scholar

[9]

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

[10]

T. DammL. GruneM. Stieler and K. Worthmann, An exponential turnpike theorem for dissipative discrete time optimal control problems, SIAM Journal on Control and Optimization, 52 (2014), 1935-1957.  doi: 10.1137/120888934.  Google Scholar

[11]

V. A. De Oliveira and G. N. Silva, Optimality conditions for infinite horizon control problems with state constraints, Nonlinear Analysis, 71 (2009), 1788-1795.  doi: 10.1016/j.na.2009.02.052.  Google Scholar

[12]

V. GaitsgoryL. Grune and N. Thatcher, Stabilization with discounted optimal control, Systems and Control Letters, 82 (2015), 91-98.  doi: 10.1016/j.sysconle.2015.05.010.  Google Scholar

[13]

N. Hayek, Infinite horizon multiobjective optimal control problems in the discrete time case, Optimization, 60 (2011), 509-529.  doi: 10.1080/02331930903480352.  Google Scholar

[14]

D. V. Khlopin, Necessary conditions of overtaking equilibrium for infinite horizon differential games, Mat.Teor. Igr Pril., 5 (2013), 105-136.   Google Scholar

[15]

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: 10.1007/BF00251430.  Google Scholar

[16]

V. LykinaS. 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: 10.1016/j.jmaa.2007.08.008.  Google Scholar

[17]

M. Mammadov, Turnpike theorem for an infinite horizon optimal control problem with time delay, SIAM Journal on Control and Optimization, 52 (2014), 420-438.  doi: 10.1137/130926808.  Google Scholar

[18]

M. Marcus and A. J. Zaslavski, The structure of extremals of a class of second order variational problems, Ann. Inst. H. Poincaré, Anal. non linéaire, 16 (1999), 593-629.  doi: 10.1016/S0294-1449(99)80029-8.  Google Scholar

[19]

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

[20]

B. S. Mordukhovich, Optimal control and feedback design of state-constrained parabolic systems in uncertainly conditions, Appl. Analysis, 90 (2011), 1075-1109.  doi: 10.1080/00036811003735840.  Google Scholar

[21]

P. A. Samuelson, A catenary turnpike theorem involving consumption and the golden rule, Amer. Econom. Rev., 55 (1965), 486-496.   Google Scholar

[22]

E. Trelat and E. Zuazua, E The turnpike property in finite-dimensional nonlinear optimal control, Journal of Differential Equations, 218 (2015), 81-114.  doi: 10.1016/j.jde.2014.09.005.  Google Scholar

[23]

A. J. Zaslavski, Ground states in Frenkel-Kontorova model, Math. USSR Izvestiya, 29 (1987), 323-354.  doi: 10.1070/IM1987v029n02ABEH000972.  Google Scholar

[24]

A. J. Zaslavski, Dynamic properties of optimal solutions of variational problems, Nonlinear Analysis, Theory, Methods and Applications, 27 (1996), 895-931.  doi: 10.1016/0362-546X(95)00029-U.  Google Scholar

[25]

A. J. Zaslavski, Existence and uniform boundedness of optimal solutions of variational problems, Abstract and Applied Analysis, 3 (1998), 265-292.  doi: 10.1155/S1085337598000566.  Google Scholar

[26]

A. J. Zaslavski, Turnpike property for extremals of variational problems with vector-valued functions, Transactions of the AMS, 351 (1999), 211-231.  doi: 10.1090/S0002-9947-99-02132-7.  Google Scholar

[27]

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

[28]

A. J. Zaslavski, A nonintersection property for extremals of variational problems with vector-valued functions, Ann. Inst. H. Poincare, Anal.non lineare, 23 (2006), 929-948.  doi: 10.1016/j.anihpc.2006.01.002.  Google Scholar

[29]

A. J. Zaslavski, Turnpike properties of approximate solutions of autonomous variational problems, Control and Cybernetics, 37 (2008), 491-512.   Google Scholar

[30]

A. J. Zaslavski, Convergence of approximate solutions of variational problems, Control and Cybernetics, 38 (2009), 1607-1629.   Google Scholar

[31]

A. J. Zaslavski, Turnpike Phenomenon and Infinite Horizon Optimal Control, Springer Optimization and Its Applications, New York, 2014. doi: 10.1007/978-3-319-08828-0.  Google Scholar

[32]

A. J. Zaslavski, Turnpike Theory of Continuous-Time Linear Optimal Control Problems, Springer Optimization and Its Applications, Springer, Cham-Heidelberg-New York-Dordrecht-London, 2015. doi: 10.1007/978-3-319-19141-6.  Google Scholar

[33]

A. J. Zaslavski, Structure of extremals of variational problems in the regions close to the endpoints, Calculus of Variations and PDE's, 53 (2015), 847-878.  doi: 10.1007/s00526-014-0769-y.  Google Scholar

[34]

A. J. Zaslavski, Structure of solutions of optimal control problems on large intervals: A survey of recent results, Pure and Applied Functional Analysis, 1 (2016), 123-158.   Google Scholar

[35]

A. J. Zaslavski, Linear control systems with nonconvex integrands on large intervals, Pure and Applied Functional Analysis, 1 (2016), 441-474.   Google Scholar

[36]

A. J. Zaslavski, Structure of approximate solutions of autonomous variational problems, Applied Analysis and Optimization, 1 (2017), 113-151.   Google Scholar

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