OJOp  Vol.4 No.3 , September 2015
A Maximum Principle for Smooth Infinite Horizon Optimal Control Problems with State Constraints and with Terminal Constraints at Infinity
Author(s) Atle Seierstad
ABSTRACT
Necessary conditions for optimality are proved for smooth infinite horizon optimal control problems with unilateral state constraints (pathwise constraints) and with terminal conditions on the states at the infinite horizon. The aim of the paper is to obtain strong necessary conditions including transversality conditions at infinity, which in many cases lead to a set of candidates for optimality containing only a few elements, similar to what is the case in finite horizon problems. However, strong growth conditions are needed for the results to hold.

Cite this paper
Seierstad, A. (2015) A Maximum Principle for Smooth Infinite Horizon Optimal Control Problems with State Constraints and with Terminal Constraints at Infinity. Open Journal of Optimization, 4, 100-130. doi: 10.4236/ojop.2015.43012.
References
[1]   Halkin, H. (1974) Necessary Conditions for Optimal Control Problems with Infinite Horizons. Econometrica, 42, 267-272.
http://dx.doi.org/10.2307/1911976

[2]   Michel, P. (1982) On the Transversality Condition in Infinite Horizon Problems. Econometrica, 50, 975-985.
http://dx.doi.org/10.2307/1912772

[3]   Seierstad, A. and Sydsaeter, K. (2009) Conditions Implying the Vanishing of the Hamiltonian at Infinity in Optimal Control Problems. Optimization Letters, 3, 507-512.
http://dx.doi.org/10.1007/s11590-009-0128-7

[4]   Aseev, S.M. and Veliov, V.M. (2011) Maximum Principles for Infinite-Horizon Optimal Control Problems with Dominating Discount. Research Report 2011-06 June, Operations Research and Control Systems, Institute of Mathematical Methods in Economics, Vienna University of Technology, Vienna.

[5]   Benveneniste, L. and Scheinkman, J. (1982) Duality Theory for Dynamic Optimization Models in Economics. Journal of Economic Theory, 27, 1-19.
http://dx.doi.org/10.1016/0022-0531(82)90012-6

[6]   Seierstad, A. and K. Sydsaeter (1987) Optimal Control Theory with Economic Applications. Amsterdam, The Netherland.

[7]   Seierstad, A. (1999) Necessary Conditions for Non-Smooth Infinite Horizon Control Problems. Journal of Optimization Theory and Applications, 103, 201-229.
http://dx.doi.org/10.1023/A:1021733719020

[8]   Pereira, F.L. and Silva, G.N. (2011) A Maximum Principle for Constrained Infinite Horizon Dynamic Control Systems. Preprints of the 18th IFAC World-Congress, Milano, 28 August-2 September 2011, 10207-10212.

[9]   de Oliveira, V.A. and Silva, G.N. (2009) Optimality Conditions for Infinite Horizon Control problems with State Constraints. Nonlinear Analysis, 71, e1788-e1795.

[10]   Aseev, S.M. and Veliov, V.M. (2012) Needle Variations in Infinite-Horizon Optimal Control. Research Report 2012-4, September, Operations Research and Control Systems, Institute of Mathematical Methods in Economics, Vienna University of Technology, Vienna.

[11]   Aseev, S.M. and Kryazhimskii, A.V. (2004) The Pontryagin Maximum Principle and Transversality Conditions for a Class of Optimal Control Problems with Infinite Time Horizons. SIAM Journal on Control and Optimization, 43, 1094-1119.

[12]   Weber, T.A. (2006) An Infinite-Horizon Maximum Principle with Bounds on the Adjoint Variable. Journal of Economic Dynamics and Control, 30, 229-241.
http://dx.doi.org/10.1016/j.jedc.2004.11.006

[13]   Arutyunov, A.V. and Aseev, S.M. (1977) Investigation of the Degeneracy Phenomenon of the Maximum Principle for Optimal Control with State Constraints. SIAM Journal on Control and Optimization, 35, 930-952.
http://dx.doi.org/10.1137/S036301299426996X

[14]   Vinter, R.B. and Ferreira, M.M.A. (1994) When Is the Maximum Principle for State Constrained Problems Nondegenerate? Journal of Mathematical Analysis and Applications, 187, 438-467.
http://dx.doi.org/10.1006/jmaa.1994.1366

[15]   Ferreira, M.M.A. and Fontes, F.A.C.C. (2004) Nondegeneracy and Normality in Necessary Conditions for Optimality: An Overview. Proceedings of the 6th Portuguese Conference on Automatic Control, CONTROLO, Faro, Portugal, 1-9 June 2004.

[16]   Vinter, R.B. (2000) Optimal Control. Birkhäuser, Boston.

[17]   Arutyunov, A.V., Karamzin, D.Y. and Pereira, F.L. (2011) The Maximum Principle for Optimal Control Problems with State Constraints by R.V. Gamkrelidze: Revisited. Journal of Optimization Theory and Applications, 149, 474-493.
http://dx.doi.org/10.1007/s10957-011-9807-5

[18]   Arutyunov, A.V., Aseev, S.M. and Blagodatskikh, V.I. (1994) First Order Necessary Conditions in the Problem of Optimal Control of a Differential Inclusion with Phase Constraints. Russian Academy of Sciences Sbornik Mathematics, 79, 117-139.
http://dx.doi.org/10.1070/sm1994v079n01abeh003493

[19]   Arutyunov, A.V. (2000) Optimality Conditions: Abnormal and Degenerate Problems. Kluwer Academic, Dortdrecht.
http://dx.doi.org/10.1007/978-94-015-9438-7

[20]   Arutyunov, A.V. and Aseev, S.M. (1995) State Constraints in Optimal Control: The Degeneracy Phenomenon. Systems & Control Letters, 26, 267-273.
http://dx.doi.org/10.1016/0167-6911(95)00021-Z

[21]   Rampazzo, F. and Vinter, R.B. (1999) A Theorem on the Existence of a Neighbouring Feasible Trajectory Satisfying a State Constraint, with Application to Optimal Control. IMA Journal of Mathematical Control and Information, 16, 335-351.
http://dx.doi.org/10.1093/imamci/16.4.335

[22]   Rampazzo, F. and Vinter, R.B. (2000) Degenerate Optimal Control Problems with State Constraints. SIAM Journal on Control and Optimization, 39, 989-1007.
http://dx.doi.org/10.1137/S0363012998340223

[23]   Bettiol, P. and Frankowska, H. (2007) Normality of the Maximum princIple for Nonconvex Constrained Bolza Problems. Journal of Differential Equations, 243, 2565-2569.
http://dx.doi.org/10.1016/j.jde.2007.05.005

[24]   Fontes, F.A.C.C. (2000) Normality in the Necessary Conditions of Optimality for Control Problems with State Constraints. Proceedings of the IASTED Conference on Control and Applications, Cancun, Mexico.

 
 
Top