×

A unified numerical scheme for linear-quadratic optimal control problems with joint control and state constraints. (English) Zbl 1260.49064

Summary: We present a numerical scheme for solving the continuous-time convex linear-quadratic (LQ) optimal control problem with mixed polyhedral state and control constraints. Unifying a discretization of this optimal control problem as often employed in model predictive control and that obtained through time-stepping methods based on the differential variational inequality reformulation, the scheme solves a sequence of finite-dimensional convex quadratic programs (QPs) whose optimal solutions are employed to construct a sequence of discrete-time trajectories dependent on the time step. Under certain technical primal-dual assumptions primarily to deal with the algebraic constraints involving the state variable, we prove that such a numerical trajectory converges to an optimal trajectory of the continuous-time control problem as the time step goes to zero, with both the limiting optimal state and costate trajectories being absolutely continuous. This provides a constructive proof of the existence of a solution to the optimal control problem with such regularity properties. Additional properties of the optimal solutions to the LQ problem are also established that are analogous to those of the finite-dimensional convex QP. Our results are applicable to problems with convex but not necessarily strictly convex objective functions and with possibly unbounded mixed state-control constraints.

MSC:

49N10 Linear-quadratic optimal control problems
49M25 Discrete approximations in optimal control
65K05 Numerical mathematical programming methods
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/s10107-006-0041-0 · Zbl 1148.93003 · doi:10.1007/s10107-006-0041-0
[2] DOI: 10.1002/nme.1047 · Zbl 1075.70501 · doi:10.1002/nme.1047
[3] DOI: 10.1002/nme.512 · Zbl 1027.70001 · doi:10.1002/nme.512
[4] Archer U., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations 13 (1995)
[5] DOI: 10.1007/BF02238746 · Zbl 0769.65056 · doi:10.1007/BF02238746
[6] Bai Y., Z. Angewedte Math. Mech. 70 pp T624– (1990)
[7] DOI: 10.1007/s11228-006-0018-4 · Zbl 1126.34011 · doi:10.1007/s11228-006-0018-4
[8] DOI: 10.1007/s00245-007-9015-8 · Zbl 1140.49027 · doi:10.1007/s00245-007-9015-8
[9] DOI: 10.1016/j.jde.2007.05.005 · Zbl 1161.49021 · doi:10.1016/j.jde.2007.05.005
[10] DOI: 10.1137/080732614 · Zbl 1202.49020 · doi:10.1137/080732614
[11] DOI: 10.1137/1.9780898718577 · Zbl 1189.49001 · doi:10.1137/1.9780898718577
[12] DOI: 10.1137/1.9781611971224 · doi:10.1137/1.9781611971224
[13] Bressan A., Introduction to the Mathematical Theory of Control 2 (2007)
[14] DOI: 10.1016/j.na.2008.07.020 · Zbl 1165.49036 · doi:10.1016/j.na.2008.07.020
[15] DOI: 10.1016/S0167-6911(96)00057-6 · Zbl 0867.49025 · doi:10.1016/S0167-6911(96)00057-6
[16] Clarke F. F., Nonsmooth Analysis and Control Theory 178 (1998) · Zbl 1047.49500
[17] Cottle R. W., The Linear Complementarity Problem · Zbl 0186.23806 · doi:10.1007/0-306-48332-7_258
[18] DOI: 10.1137/0307003 · Zbl 0175.10502 · doi:10.1137/0307003
[19] DOI: 10.1007/BF00932851 · doi:10.1007/BF00932851
[20] DOI: 10.1137/0310048 · Zbl 0244.49018 · doi:10.1137/0310048
[21] DOI: 10.1090/S0025-5718-1969-0247746-7 · doi:10.1090/S0025-5718-1969-0247746-7
[22] DOI: 10.1007/BF00927675 · Zbl 0174.20704 · doi:10.1007/BF00927675
[23] Daniel J. W., The Approximate Minimization Of Functionals (1983)
[24] DOI: 10.1007/978-1-4613-8489-2_3 · doi:10.1007/978-1-4613-8489-2_3
[25] DOI: 10.1090/S0025-5718-00-01184-4 · Zbl 0987.49017 · doi:10.1090/S0025-5718-00-01184-4
[26] DOI: 10.1137/S0036142999351765 · Zbl 0968.49022 · doi:10.1137/S0036142999351765
[27] DOI: 10.1287/mnsc.17.11.698 · Zbl 0242.90040 · doi:10.1287/mnsc.17.11.698
[28] Facchinei F., Finite-Dimensional Variational Inequalities and Complementarity Problems (2003) · Zbl 1062.90002
[29] DOI: 10.3166/ejc.9.190-206 · Zbl 1293.93288 · doi:10.3166/ejc.9.190-206
[30] DOI: 10.1002/nav.3800030109 · doi:10.1002/nav.3800030109
[31] Frankowska H., J. Convex Anal. 13 pp 299– (2006)
[32] DOI: 10.1137/S0363012902404711 · Zbl 1048.49026 · doi:10.1137/S0363012902404711
[33] DOI: 10.1016/0005-1098(89)90002-2 · Zbl 0685.93029 · doi:10.1016/0005-1098(89)90002-2
[34] DOI: 10.1007/s10589-009-9291-0 · Zbl 1226.49026 · doi:10.1007/s10589-009-9291-0
[35] DOI: 10.1016/j.automatica.2010.06.048 · Zbl 1219.49028 · doi:10.1016/j.automatica.2010.06.048
[36] DOI: 10.1137/060675745 · Zbl 1162.70003 · doi:10.1137/060675745
[37] DOI: 10.1137/S0363012902411581 · Zbl 1072.49025 · doi:10.1137/S0363012902411581
[38] DOI: 10.1109/TAC.2007.895915 · Zbl 1366.90161 · doi:10.1109/TAC.2007.895915
[39] DOI: 10.1109/TAC.2008.927799 · Zbl 1367.90109 · doi:10.1109/TAC.2008.927799
[40] DOI: 10.1137/0712063 · Zbl 0322.49021 · doi:10.1137/0712063
[41] DOI: 10.1137/0317026 · Zbl 0426.90083 · doi:10.1137/0317026
[42] DOI: 10.1007/s002110000178 · Zbl 0991.49020 · doi:10.1007/s002110000178
[43] DOI: 10.1137/0322027 · Zbl 0555.49022 · doi:10.1137/0322027
[44] DOI: 10.1137/080725258 · Zbl 1203.65123 · doi:10.1137/080725258
[45] DOI: 10.1137/1037043 · Zbl 0832.49013 · doi:10.1137/1037043
[46] DOI: 10.1080/002071798222208 · Zbl 0932.49028 · doi:10.1080/002071798222208
[47] Hoffman A. J., J. Res. Nat. Bureau Standards 49 pp 263– (1952) · doi:10.6028/jres.049.027
[48] DOI: 10.1007/s10589-007-9098-9 · Zbl 1219.49029 · doi:10.1007/s10589-007-9098-9
[49] DOI: 10.1137/S0895479892224768 · Zbl 0799.65063 · doi:10.1137/S0895479892224768
[50] Malanowski K. M., Arch. Automatyki I Telemech. 23 pp 227– (1978)
[51] DOI: 10.1137/0718039 · Zbl 0471.65033 · doi:10.1137/0718039
[52] DOI: 10.1016/0305-0548(69)90004-5 · Zbl 0606.90102 · doi:10.1016/0305-0548(69)90004-5
[53] DOI: 10.1016/0167-6377(88)90047-8 · Zbl 0653.90055 · doi:10.1016/0167-6377(88)90047-8
[54] DOI: 10.1016/S0005-1098(99)00214-9 · Zbl 0949.93003 · doi:10.1016/S0005-1098(99)00214-9
[55] DOI: 10.1016/S0098-1354(98)00301-9 · doi:10.1016/S0098-1354(98)00301-9
[56] DOI: 10.1007/s10107-006-0052-x · Zbl 1139.58011 · doi:10.1007/s10107-006-0052-x
[57] DOI: 10.1007/s10107-007-0117-5 · Zbl 1194.49033 · doi:10.1007/s10107-007-0117-5
[58] DOI: 10.1137/0311042 · Zbl 0238.49020 · doi:10.1137/0311042
[59] DOI: 10.1137/1015071 · Zbl 0274.49001 · doi:10.1137/1015071
[60] DOI: 10.1007/BF01585175 · Zbl 0805.90107 · doi:10.1007/BF01585175
[61] DOI: 10.1002/nme.1582 · Zbl 1110.70303 · doi:10.1002/nme.1582
[62] DOI: 10.1016/S0005-1098(00)00004-2 · Zbl 0955.93504 · doi:10.1016/S0005-1098(00)00004-2
[63] DOI: 10.1016/S0967-0661(02)00186-7 · doi:10.1016/S0967-0661(02)00186-7
[64] DOI: 10.1007/BFb0120929 · Zbl 0449.90090 · doi:10.1007/BFb0120929
[65] DOI: 10.1137/0325045 · Zbl 0617.49010 · doi:10.1137/0325045
[66] DOI: 10.1137/0327054 · Zbl 0682.49019 · doi:10.1137/0327054
[67] DOI: 10.1007/s10107-004-0544-5 · Zbl 1076.90060 · doi:10.1007/s10107-004-0544-5
[68] Sethi S. P., Optimal Control Theory: Applications to Management Science and Economics, 2. ed. (2000) · Zbl 0998.49002
[69] DOI: 10.1016/j.na.2005.07.041 · Zbl 1093.49024 · doi:10.1016/j.na.2005.07.041
[70] Sontag, E. D. 1998. ”Mathematical Control Theory: Deterministic Finite-Dimensional Systems”. In , 2, New York: Springer-Verlag. · Zbl 0945.93001 · doi:10.1007/978-1-4612-0577-7
[71] Stöver, R. 1999. ”Collocation methods for solving linear differential-algebraic boundary value problems”. Bremen: Zentrum für Technomathematik, Universität. Report 99-08
[72] Vinter R., Optimal Control (2000)
[73] DOI: 10.1137/S0363012992240503 · Zbl 0876.49026 · doi:10.1137/S0363012992240503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.