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September  2018, 8(3&4): 535-555. doi: 10.3934/mcrf.2018022

Necessary conditions for infinite horizon optimal control problems with state constraints

Vincenzo Basco 1,2, , Piermarco Cannarsa 1, and  Hélène Frankowska 2,

1. 

Dipartimento di Matematica, Università degli Studi di Roma "Tor Vergata", Via della Ricerca Scientifica, 1 - 00133 Roma, Italy
2. 

IMJ-PRG, UMR 7586 CNRS, Sorbonne Université, case 247, 4 place Jussieu, 75252 Paris, France

Received  November 2017 Revised  January 2018 Published  September 2018

Fund Project: The research of third author benefited from the support of the FMJH Program Gaspard Monge in optimization and operation research, and from the support to this program from EDF under the grant PGMO 2015-2832H.

Partial and full sensitivity relations are obtained for nonauto-nomous optimal control problems with infinite horizon subject to state constraints, assuming the associated value function to be locally Lipschitz in the state. Sufficient structural conditions are given to ensure such a Lipschitz regularity in presence of a positive discount factor, as it is typical of macroeconomics models.

Keywords: Infinite horizon optimal control, state constraints, value function, maximrinciple, sensitivity relations.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022
References:
[1]

K. Arrow and M. Kurz, Optimal growth with irreversible investment in a Ramsey model, Econometrica, 38 (1970), 331-344.  doi: 10.2307/1913014.  Google Scholar

[2]

S. M. Aseev, On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems, Tr. Inst. Mat. Mekh., 19 (2013), 15-24.   Google Scholar

[3]

S. M. Aseev and V. M. Veliov, Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions, Tr. Inst. Mat. Mekh., 20 (2014), 41-57.   Google Scholar

[4]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[5]

V. Basco and H. Frankowska, Lipschitz continuity of the value function for the infinite horizon optimal control problem under state constraints, (submitted). Google Scholar

[6]

J. P. Bénassy, Macroeconomic Theory, Oxford University Press, 2010. Google Scholar

[7]

L. M. Benveniste and J. A. Scheinkman, Duality theory for dynamic optimization models of economics: the continuous time case, J. Econom. Theory, 27 (1982), 1-19.  doi: 10.1016/0022-0531(82)90012-6.  Google Scholar

[8]

P. Bettiol, H. Frankowska and R. B. Vinter, Improved sensitivity relations in state constrained optimal control, Appl. Math. Optim., 71 (2015), 353-377.  doi: 10.1007/s00245-014-9260-6.  Google Scholar

[9]

O. J. Blanchard and S. Fischer, Lectures on Macroeconomics, MIT press, 1989. Google Scholar

[10]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkhäuser Boston, Inc., Boston, MA, 2004.  Google Scholar

[11]

A. Cernea and H. Frankowska, A connection between the maximum principle and dynamic programming for constrained control problems, SIAM J. Control Optim., 44 (2005), 673-703.  doi: 10.1137/S0363012903430585.  Google Scholar

[12]

H. Frankowska and M. Mazzola, On relations of the adjoint state to the value function for optimal control problems with state constraints, Nonlinear Differential Equations Appl., 20 (2013), 361-383.  doi: 10.1007/s00030-012-0183-0.  Google Scholar

[13]

P. Loreti and M. E. Tessitore, Approximation and regularity results on constrained viscosity solutions of Hamilton-Jacobi-Bellman equations, J. Math. Systems Estim. Control, 4 (1994), 467-483.   Google Scholar

[14]

F. P. Ramsey, A mathematical theory of saving, The Economic Journal, 38 (1928), 543-559.  doi: 10.2307/2224098.  Google Scholar

[15]

R. T. Rockafellar and R. B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[16]

A. Seierstad, Necessary conditions for nonsmooth, infinite-horizon, optimal control problems, J. Optim. Theory Appl., 103 (1999), 201-229.  doi: 10.1023/A:1021733719020.  Google Scholar

[17]

A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications, North-Holland Publishing Co., Amsterdam, 1987.  Google Scholar

[18]

G. Sorger, On the long-run distribution of capital in the Ramsey model, J. Econom. Theory, 105 (2002), 226-243.  doi: 10.1006/jeth.2001.2841.  Google Scholar

[19]

R. Vinter, Optimal Control, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-8086-2.  Google Scholar

show all references

References:
[1]

K. Arrow and M. Kurz, Optimal growth with irreversible investment in a Ramsey model, Econometrica, 38 (1970), 331-344.  doi: 10.2307/1913014.  Google Scholar

[2]

S. M. Aseev, On some properties of the adjoint variable in the relations of the Pontryagin maximum principle for optimal economic growth problems, Tr. Inst. Mat. Mekh., 19 (2013), 15-24.   Google Scholar

[3]

S. M. Aseev and V. M. Veliov, Maximum principle for infinite-horizon optimal control problems under weak regularity assumptions, Tr. Inst. Mat. Mekh., 20 (2014), 41-57.   Google Scholar

[4]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009. doi: 10.1007/978-0-8176-4848-0.  Google Scholar

[5]

V. Basco and H. Frankowska, Lipschitz continuity of the value function for the infinite horizon optimal control problem under state constraints, (submitted). Google Scholar

[6]

J. P. Bénassy, Macroeconomic Theory, Oxford University Press, 2010. Google Scholar

[7]

L. M. Benveniste and J. A. Scheinkman, Duality theory for dynamic optimization models of economics: the continuous time case, J. Econom. Theory, 27 (1982), 1-19.  doi: 10.1016/0022-0531(82)90012-6.  Google Scholar

[8]

P. Bettiol, H. Frankowska and R. B. Vinter, Improved sensitivity relations in state constrained optimal control, Appl. Math. Optim., 71 (2015), 353-377.  doi: 10.1007/s00245-014-9260-6.  Google Scholar

[9]

O. J. Blanchard and S. Fischer, Lectures on Macroeconomics, MIT press, 1989. Google Scholar

[10]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Birkhäuser Boston, Inc., Boston, MA, 2004.  Google Scholar

[11]

A. Cernea and H. Frankowska, A connection between the maximum principle and dynamic programming for constrained control problems, SIAM J. Control Optim., 44 (2005), 673-703.  doi: 10.1137/S0363012903430585.  Google Scholar

[12]

H. Frankowska and M. Mazzola, On relations of the adjoint state to the value function for optimal control problems with state constraints, Nonlinear Differential Equations Appl., 20 (2013), 361-383.  doi: 10.1007/s00030-012-0183-0.  Google Scholar

[13]

P. Loreti and M. E. Tessitore, Approximation and regularity results on constrained viscosity solutions of Hamilton-Jacobi-Bellman equations, J. Math. Systems Estim. Control, 4 (1994), 467-483.   Google Scholar

[14]

F. P. Ramsey, A mathematical theory of saving, The Economic Journal, 38 (1928), 543-559.  doi: 10.2307/2224098.  Google Scholar

[15]

R. T. Rockafellar and R. B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-02431-3.  Google Scholar

[16]

A. Seierstad, Necessary conditions for nonsmooth, infinite-horizon, optimal control problems, J. Optim. Theory Appl., 103 (1999), 201-229.  doi: 10.1023/A:1021733719020.  Google Scholar

[17]

A. Seierstad and K. Sydsæter, Optimal Control Theory with Economic Applications, North-Holland Publishing Co., Amsterdam, 1987.  Google Scholar

[18]

G. Sorger, On the long-run distribution of capital in the Ramsey model, J. Econom. Theory, 105 (2002), 226-243.  doi: 10.1006/jeth.2001.2841.  Google Scholar

[19]

R. Vinter, Optimal Control, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-8086-2.  Google Scholar

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