April  2014, 10(2): 413-441. doi: 10.3934/jimo.2014.10.413

Theory and applications of optimal control problems with multiple time-delays

1. 

Department of Mechanical Engineering, Münster University of Applied Sciences, Stegerwaldstrasse 39, 48565 Steinfurt, Germany

2. 

Institute of Computational and Applied Mathematics, Westfälische Wilhelms-Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany

Received  November 2012 Revised  July 2013 Published  October 2013

In this paper we study optimal control problems with multiple time delays in control and state and mixed type control-state constraints. We derive necessary optimality conditions in the form of a Pontryagin type Minimum Principle. A discretization method is presented by which the delayed control problem is transformed into a nonlinear programming problem. It is shown that the associated Lagrange multipliers provide a consistent numerical approximation for the adjoint variables of the delayed optimal control problem. We illustrate the theory and numerical approach on an analytical example and an optimal control model from immunology.
Citation: Laurenz Göllmann, Helmut Maurer. Theory and applications of optimal control problems with multiple time-delays. Journal of Industrial & Management Optimization, 2014, 10 (2) : 413-441. doi: 10.3934/jimo.2014.10.413
References:
[1]

T. S. Angell and A. Kirsch, On the necessary conditions for optimal control of retarded systems,, Appl. Math. Optim., 22 (1990), 117. doi: 10.1007/BF01447323. Google Scholar

[2]

A. Asachenkov, G. Marchuk, R. Mohler and S. Zuev, Disease Dynamics,, Birkhäuser, (1994). Google Scholar

[3]

H. T. Banks, Necessary conditions for control problems with variable time lags,, SIAM J. Control, 6 (1968), 9. doi: 10.1137/0306002. Google Scholar

[4]

Q. Chai, R. Loxton, K. L. Teo and C. Yang, A class of optimal state-delay control problems,, Nonlinear Anal. Real World Appl., 14 (2013), 1536. doi: 10.1016/j.nonrwa.2012.10.017. Google Scholar

[5]

Q. Chai, R. Loxton, K. L. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems,, J. Ind. Manag. Optim., 9 (2013), 471. doi: 10.3934/jimo.2013.9.471. Google Scholar

[6]

W. L. Chan and S. P. Yung, Sufficient conditions for variational problems with delayed argument,, J. Optim. Theory Appl., 76 (1993), 131. doi: 10.1007/BF00952825. Google Scholar

[7]

F. Colonius and D. Hinrichsen, Optimal control of functional differential systems,, SIAM J. Control Optim., 16 (1978), 861. doi: 10.1137/0316060. Google Scholar

[8]

S. Dadebo and R. Luus, Optimal control of time-delay systems by dynamic programming,, Optimal Control Appl. Methods, 13 (1992), 29. doi: 10.1002/oca.4660130103. Google Scholar

[9]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, The Scientific Press, (1993). Google Scholar

[10]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints,, Optimal Control Appl. Methods, 30 (2009), 341. doi: 10.1002/oca.843. Google Scholar

[11]

T. Guinn, Reduction of delayed optimal control problems to nondelayed problems,, J. Optimization Theory Appl., 18 (1976), 371. doi: 10.1007/BF00933818. Google Scholar

[12]

W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numer. Math., 87 (2000), 247. doi: 10.1007/s002110000178. Google Scholar

[13]

A. Halanay, Optimal controls for systems with time lag,, SIAM J. Control, 6 (1968), 215. doi: 10.1137/0306016. Google Scholar

[14]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory,, John Wiley & Sons, (1966). Google Scholar

[15]

S.-C. Huang, Optimal control problems with retardations and restricted phase coordinates,, J. Optimization Theory Appl., 3 (1969), 316. doi: 10.1007/BF00931371. Google Scholar

[16]

G. L. Kharatishvili, Maximum principle in the theory of optimal time-delay processes,, Dokl. Akad. Nauk. USSR, 136 (1961), 39. Google Scholar

[17]

D. Kern, Notwendige Optimalitätsbedingungen und numerische Lösungsmethoden für optimale Steuerprozesse mit Retardierungen,, Diploma thesis, (2005). Google Scholar

[18]

R. Loxton, K. L. Teo and V. Rehbock, An optimization approach to state-delay identification,, IEEE Transactions on Automatic Control, 55 (2010), 2113. doi: 10.1109/TAC.2010.2050710. Google Scholar

[19]

R. M. May, Time-delay versus stability in population models with two and three tropic levels,, Ecology, 54 (1973), 315. doi: 10.2307/1934339. Google Scholar

[20]

A. A. Milyutin and N. P. Osmolovskii, Calculus of Variations and Optimal Control,, Translations of Mathematical Monographs, (1998). Google Scholar

[21]

B. S. Mordukhovich and R. Trubnik, Stability of discrete approximations and necessary optimality conditions for delay-differential inclusions,, Ann. Oper. Res., 101 (2001), 149. doi: 10.1023/A:1010968423112. Google Scholar

[22]

L. W. Neustadt, Optimization. A Theory of Necessary Conditions,, Princeton University Press, (1976). Google Scholar

[23]

M. N. Oǧuztöreli, Time-Lag Control Systems,, Mathematics in Science and Engineering, 24 (1966). Google Scholar

[24]

S. H. Oh and R. Luus, Optimal feedback control of time-delay systems,, AIChE J., 22 (1976), 140. doi: 10.1002/aic.690220117. Google Scholar

[25]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Translated from the Russian by K. N. Trirogoff, (1962). Google Scholar

[26]

W. H. Ray and M. A. Soliman, The optimal control of processes containing pure time delays-I, necessary conditions for an optimum,, Chemical Engin. Science, 25 (1970), 1911. doi: 10.1016/0009-2509(70)87009-9. Google Scholar

[27]

M. A. Soliman and W. H. Ray, Optimal control of multivariable systems with pure time delays,, Automatica, 7 (1971), 681. doi: 10.1016/0005-1098(71)90006-9. Google Scholar

[28]

M. A. Soliman and W. H. Ray, On the optimal control of systems having pure time delays and singular arcs. I. Some necessary conditions for optimality,, Int. J. Control (1), 16 (1972), 963. doi: 10.1080/00207177208932327. Google Scholar

[29]

R. F. Stengel, R. Ghigliazza, N. Kulkarni and O. Laplace, Optimal control of innate immune response,, Optimal Control Appl. Methods, 23 (2002), 91. doi: 10.1002/oca.704. Google Scholar

[30]

R. F. Stengel and R. Ghigliazza, Stochastic optimal therapy for enhanced immune response,, Mathematical Biosciences, 191 (2004), 123. doi: 10.1016/j.mbs.2004.06.004. Google Scholar

[31]

R. J. Vanderbei, LOQO: An interior point code for quadratic programming,, Optim. Methods Softw., 11/12 (1999), 451. doi: 10.1080/10556789908805759. Google Scholar

[32]

R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming,, Comput. Optim. Appl., 13 (1999), 231. doi: 10.1023/A:1008677427361. Google Scholar

[33]

A. Wächter, An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering,, Ph.D thesis, (2002). Google Scholar

[34]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Math. Program., 106 (2006), 25. doi: 10.1007/s10107-004-0559-y. Google Scholar

[35]

J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). Google Scholar

show all references

References:
[1]

T. S. Angell and A. Kirsch, On the necessary conditions for optimal control of retarded systems,, Appl. Math. Optim., 22 (1990), 117. doi: 10.1007/BF01447323. Google Scholar

[2]

A. Asachenkov, G. Marchuk, R. Mohler and S. Zuev, Disease Dynamics,, Birkhäuser, (1994). Google Scholar

[3]

H. T. Banks, Necessary conditions for control problems with variable time lags,, SIAM J. Control, 6 (1968), 9. doi: 10.1137/0306002. Google Scholar

[4]

Q. Chai, R. Loxton, K. L. Teo and C. Yang, A class of optimal state-delay control problems,, Nonlinear Anal. Real World Appl., 14 (2013), 1536. doi: 10.1016/j.nonrwa.2012.10.017. Google Scholar

[5]

Q. Chai, R. Loxton, K. L. Teo and C. Yang, A unified parameter identification method for nonlinear time-delay systems,, J. Ind. Manag. Optim., 9 (2013), 471. doi: 10.3934/jimo.2013.9.471. Google Scholar

[6]

W. L. Chan and S. P. Yung, Sufficient conditions for variational problems with delayed argument,, J. Optim. Theory Appl., 76 (1993), 131. doi: 10.1007/BF00952825. Google Scholar

[7]

F. Colonius and D. Hinrichsen, Optimal control of functional differential systems,, SIAM J. Control Optim., 16 (1978), 861. doi: 10.1137/0316060. Google Scholar

[8]

S. Dadebo and R. Luus, Optimal control of time-delay systems by dynamic programming,, Optimal Control Appl. Methods, 13 (1992), 29. doi: 10.1002/oca.4660130103. Google Scholar

[9]

R. Fourer, D. M. Gay and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming,, The Scientific Press, (1993). Google Scholar

[10]

L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints,, Optimal Control Appl. Methods, 30 (2009), 341. doi: 10.1002/oca.843. Google Scholar

[11]

T. Guinn, Reduction of delayed optimal control problems to nondelayed problems,, J. Optimization Theory Appl., 18 (1976), 371. doi: 10.1007/BF00933818. Google Scholar

[12]

W. Hager, Runge-Kutta methods in optimal control and the transformed adjoint system,, Numer. Math., 87 (2000), 247. doi: 10.1007/s002110000178. Google Scholar

[13]

A. Halanay, Optimal controls for systems with time lag,, SIAM J. Control, 6 (1968), 215. doi: 10.1137/0306016. Google Scholar

[14]

M. R. Hestenes, Calculus of Variations and Optimal Control Theory,, John Wiley & Sons, (1966). Google Scholar

[15]

S.-C. Huang, Optimal control problems with retardations and restricted phase coordinates,, J. Optimization Theory Appl., 3 (1969), 316. doi: 10.1007/BF00931371. Google Scholar

[16]

G. L. Kharatishvili, Maximum principle in the theory of optimal time-delay processes,, Dokl. Akad. Nauk. USSR, 136 (1961), 39. Google Scholar

[17]

D. Kern, Notwendige Optimalitätsbedingungen und numerische Lösungsmethoden für optimale Steuerprozesse mit Retardierungen,, Diploma thesis, (2005). Google Scholar

[18]

R. Loxton, K. L. Teo and V. Rehbock, An optimization approach to state-delay identification,, IEEE Transactions on Automatic Control, 55 (2010), 2113. doi: 10.1109/TAC.2010.2050710. Google Scholar

[19]

R. M. May, Time-delay versus stability in population models with two and three tropic levels,, Ecology, 54 (1973), 315. doi: 10.2307/1934339. Google Scholar

[20]

A. A. Milyutin and N. P. Osmolovskii, Calculus of Variations and Optimal Control,, Translations of Mathematical Monographs, (1998). Google Scholar

[21]

B. S. Mordukhovich and R. Trubnik, Stability of discrete approximations and necessary optimality conditions for delay-differential inclusions,, Ann. Oper. Res., 101 (2001), 149. doi: 10.1023/A:1010968423112. Google Scholar

[22]

L. W. Neustadt, Optimization. A Theory of Necessary Conditions,, Princeton University Press, (1976). Google Scholar

[23]

M. N. Oǧuztöreli, Time-Lag Control Systems,, Mathematics in Science and Engineering, 24 (1966). Google Scholar

[24]

S. H. Oh and R. Luus, Optimal feedback control of time-delay systems,, AIChE J., 22 (1976), 140. doi: 10.1002/aic.690220117. Google Scholar

[25]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Translated from the Russian by K. N. Trirogoff, (1962). Google Scholar

[26]

W. H. Ray and M. A. Soliman, The optimal control of processes containing pure time delays-I, necessary conditions for an optimum,, Chemical Engin. Science, 25 (1970), 1911. doi: 10.1016/0009-2509(70)87009-9. Google Scholar

[27]

M. A. Soliman and W. H. Ray, Optimal control of multivariable systems with pure time delays,, Automatica, 7 (1971), 681. doi: 10.1016/0005-1098(71)90006-9. Google Scholar

[28]

M. A. Soliman and W. H. Ray, On the optimal control of systems having pure time delays and singular arcs. I. Some necessary conditions for optimality,, Int. J. Control (1), 16 (1972), 963. doi: 10.1080/00207177208932327. Google Scholar

[29]

R. F. Stengel, R. Ghigliazza, N. Kulkarni and O. Laplace, Optimal control of innate immune response,, Optimal Control Appl. Methods, 23 (2002), 91. doi: 10.1002/oca.704. Google Scholar

[30]

R. F. Stengel and R. Ghigliazza, Stochastic optimal therapy for enhanced immune response,, Mathematical Biosciences, 191 (2004), 123. doi: 10.1016/j.mbs.2004.06.004. Google Scholar

[31]

R. J. Vanderbei, LOQO: An interior point code for quadratic programming,, Optim. Methods Softw., 11/12 (1999), 451. doi: 10.1080/10556789908805759. Google Scholar

[32]

R. J. Vanderbei and D. F. Shanno, An interior-point algorithm for nonconvex nonlinear programming,, Comput. Optim. Appl., 13 (1999), 231. doi: 10.1023/A:1008677427361. Google Scholar

[33]

A. Wächter, An Interior Point Algorithm for Large-Scale Nonlinear Optimization with Applications in Process Engineering,, Ph.D thesis, (2002). Google Scholar

[34]

A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,, Math. Program., 106 (2006), 25. doi: 10.1007/s10107-004-0559-y. Google Scholar

[35]

J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). Google Scholar

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