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Necessary optimality conditions for an integro-differential Bolza problem via Dubovitskii-Milyutin method
A sufficient optimality condition for delayed state-linear optimal control problems
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal |
We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.
References:
[1] |
V. L. Bakke,
Optimal fields for problems with delays, J. Optim. Theory Appl., 33 (1981), 69-84.
doi: 10.1007/BF00935177. |
[2] |
H. T. Banks,
Necessary conditions for control problems with variable time lags, SIAM J. Control, 6 (1968), 9-47.
doi: 10.1137/0306002. |
[3] |
E. B. M. Bashier and K. C. Patidar,
Optimal control of an epidemiological model with multiple time delays, Appl. Math. Comput., 292 (2017), 47-56.
doi: 10.1016/j.amc.2016.07.009. |
[4] |
A. Boccia, P. Falugi, H. Maurer and R. Vinter, Free time optimal control problems with time delays, 52nd IEEE Conference on Decision and Control, (2013), 520-525.
doi: 10.1109/CDC.2013.6759934. |
[5] |
A. Boccia and R. B. Vinter,
The maximum principle for optimal control problems with time delays, SIAM J. Control Optim., 55 (2017), 2905-2935.
doi: 10.1137/16M1085474. |
[6] |
G. V. Bokov,
Pontryagin's maximum principle of optimal control problems with time-delay, J. Math. Sci. (N. Y.), 172 (2011), 623-634.
doi: 10.1007/s10958-011-0208-y. |
[7] |
F. Cacace, F. Conte, A. Germani and G. Palombo,
Optimal control of linear systems with large and variable input delays, Systems Control Lett., 89 (2016), 1-7.
doi: 10.1016/j.sysconle.2015.12.003. |
[8] |
W. L. Chan and S. P. Yung,
Sufficient conditions for variational problems with delayed argument, J. Optim. Theory Appl., 76 (1993), 131-144.
doi: 10.1007/BF00952825. |
[9] |
D. H. Chyung and E. B. Lee,
Linear optimal systems with time delays, SIAM J. Control Optim., 4 (1966), 548-575.
doi: 10.1137/0304042. |
[10] |
M. C. Delfour,
The linear-quadratic optimal control problem with delays in state and control variables: a state space approach, SIAM J. Control and Optim., 24 (1986), 835-883.
doi: 10.1137/0324053. |
[11] |
A. M. Elaiw and N. H. AlShamrani,
Stability of a general delay-distributed virus dynamics model with multi-staged infected progression and immune response, Math. Meth. Appl. Sci., 40 (2017), 699-719.
doi: 10.1002/mma.4002. |
[12] |
D. H. Eller, J. K. Aggarwal and H. T. Banks,
Optimal control of linear time-delay systems, IEEE Trans. Automat. Control, 14 (1969), 678-687.
doi: 10.1109/TAC.1969.1099301. |
[13] |
A. Friedman,
Optimal control for hereditary processes, Arch. Rational Mech. Anal., 15 (1964), 396-416.
doi: 10.1007/BF00256929. |
[14] |
D. M. Gay, The AMPL modeling language: An aid to formulating and solving optimization problems, in Numerical Analysis and Optimization, 95-116, Springer Proc. Math. Stat., 134, Springer, Cham, 2015.
doi: 10.1007/978-3-319-17689-5_5. |
[15] |
L. Göllmann, D. Kern and H. Maurer,
Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optim. Control Appl. Meth., 30 (2009), 341-365.
doi: 10.1002/oca.843. |
[16] |
L. Göllmann and H. Maurer,
Theory and applications of optimal control problems with multiple time-delays, J. Ind. Manag. Optim., 10 (2014), 413-441.
doi: 10.3934/jimo.2014.10.413. |
[17] |
T. Guinn,
Reduction of delayed optimal control problems to nondelayed problems, J. Optim. Theory Appl., 18 (1976), 371-377.
doi: 10.1007/BF00933818. |
[18] |
A. Halanay,
Optimal controls for systems with time lag, SIAM J. Control, 6 (1968), 215-234.
doi: 10.1137/0306016. |
[19] |
M. R. Hestenes,
On variational theory and optimal control theory, SIAM J. Control, 3 (1965), 23-48.
doi: 10.1137/0303003. |
[20] |
D. K. Hughes,
Variational and optimal control problems with delayed argument, J. Optim. Theory Appl., 2 (1968), 1-14.
doi: 10.1007/BF00927159. |
[21] |
S. H. Hwang and Z. Bien,
Sufficient conditions for optimal time-delay systems with applications to functionally constrained control problems, Internat. J. Control, 38 (1983), 607-620.
doi: 10.1080/00207178308933097. |
[22] |
M. Q. Jacobs and T. Kao,
An optimum settling problem for time lag systems, J. Math. Anal. Appl., 40 (1972), 687-707.
doi: 10.1016/0022-247X(72)90013-3. |
[23] |
G. L. Kharatishvili,
The maximum principle in the theory of optimal processes involving delay, Soviet Math. Dokl., 2 (1961), 28-32.
|
[24] |
G. L. Kharatishvili, A maximum principle in extremal problems with delays, in Mathematical Theory of Control, 26-34, Academic Press, New York, 1967. |
[25] |
G. L. Kharatishvili and T. A. Tadumadze,
Nonlinear optimal control systems with variable lags, Mat. Sb. (N.S.), 107(149) (1978), 613-633.
|
[26] |
F. Khellat,
Optimal control of linear time-delayed systems by linear legendre multiwavelets, J. Optim. Theory Appl., 143 (2009), 107-121.
doi: 10.1007/s10957-009-9548-x. |
[27] |
J. Klamka, H. Maurer and A. Swierniak,
Local controllability and optimal control for a model of combined anticancer therapy with control delays, Math. Biosci. Eng., 14 (2016), 195-216.
doi: 10.3934/mbe.2017013. |
[28] |
R. W. Koepcke,
On the control of linear systems with pure time delay, J. Basic Eng, 87 (1965), 74-80.
doi: 10.1115/1.3650530. |
[29] |
H. N. Koivo and E. B. Lee,
Controller synthesis for linear systems with retarded state and control variables and quadratic cost, Automatica J. IFAC, 8 (1972), 203-208.
doi: 10.1016/0005-1098(72)90068-4. |
[30] |
C. H. Lee and S. P. Yung,
Sufficient conditions for optimal control problems with time delay, J. Optim. Theory Appl., 88 (1996), 157-176.
doi: 10.1007/BF02192027. |
[31] |
E. B. Lee,
Variational problems for systems having delay in the control action, IEEE Trans. Automat. Control, 13 (1968), 697-699.
doi: 10.1109/TAC.1968.1099029. |
[32] |
R. C. H. Lee and S. P. Yung,
Optimality conditions and duality for a non-linear time-delay control problem, Optimal Control Appl. Methods, 18 (1997), 327-340.
doi: 10.1002/(SICI)1099-1514(199709/10)18:5<327::AID-OCA614>3.0.CO;2-9. |
[33] |
E. B. Lee and L. Markus, Foundations of Optimal Control Theory, 2nd edition, Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1986. |
[34] |
A. P. Lemos-Paião, Introduction to Optimal Control Theory and Its Application to Diabetes, M.Sc. thesis, University of Aveiro, Aveiro, 2015. |
[35] |
M. N. Oǧuztöreli,
A time optimal control problem for systems described by differential difference equations, SIAM J. Control Optim., 1 (1963), 290-310.
doi: 10.1137/0301017. |
[36] |
M. N. Oǧuztöreli, Time-lag Control Systems, Academic Press, New York, 1966.
![]() ![]() |
[37] |
K. R. Palanisamy and R. G. Prasada,
Optimal control of linear systems with delays in state and control via Walsh functions, IEE Proceedings D - Control Theory and Applications, 130 (1983), 300-312.
doi: 10.1049/ip-d.1983.0051. |
[38] |
W. J. Palm and W. E. Schmitendorf,
Conjugate-point conditions for variational problems with delayed argument, J. Optim. Theory Appl., 14 (1974), 599-612.
doi: 10.1007/BF00932963. |
[39] |
H. Pirnay, R. López-Negrete and L. T. Biegler,
Optimal sensitivity based on IPOPT, Math. Program. Comput., 4 (2012), 307-331.
doi: 10.1007/s12532-012-0043-2. |
[40] |
L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, 2nd edition, Interscience, New York, 1962. |
[41] |
V. M. Popov and A. Halanay,
A problem in the theory of time delay optimum systems, Autom. Remote Control, 25 (1964), 1129-1134.
|
[42] |
D. Rocha, C. J. Silva and D. F. M. Torres,
Stability and optimal control of a delayed HIV model, Math. Meth. Appl. Sci., 41 (2018), 2251-2260.
doi: 10.1002/mma.4207. |
[43] |
L. D. Sabbagh,
Variational problems with lags, J. Optim. Theory Appl., 3 (1969), 34-51.
doi: 10.1007/BF00929540. |
[44] |
S. P. M. Santos, N. Martins and D. F. M. Torres,
Higher-order variational problems of Herglotz type with time delay, Pure Appl. Funct. Anal., 1 (2016), 291-307.
|
[45] |
W. E. Schmitendorf,
A sufficient condition for optimal control problems with time delays, Automatica J. IFAC, 9 (1973), 633-637.
doi: 10.1016/0005-1098(73)90048-4. |
[46] |
C. J. Silva, H. Maurer and D. F. M. Torres,
Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337.
doi: 10.3934/mbe.2017021. |
[47] |
M. A. Soliman,
A new necessary condition for optimality systems with time delay, J. Optim. Theory Appl., 11 (1973), 249-254.
doi: 10.1007/BF00935193. |
[48] |
E. Stumpf,
Local stability analysis of differential equations with state-dependent delay, Discrete Contin. Dyn. Syst., 36 (2016), 3445-3461.
doi: 10.3934/dcds.2016.36.3445. |
[49] |
Y. Xia, M. Fu and P. Shi, Analysis and Synthesis of Dynamical Systems with Time-Delays, Lecture Notes in Control and Information Sciences, 387. Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-02696-6. |
[50] |
J. Xu, Y. Geng and Y. Zhou,
Global stability of a multi-group model with distributed delay and vaccination, Math. Meth. Appl. Sci., 40 (2017), 1475-1486.
doi: 10.1002/mma.4068. |
[51] |
R. Xu, S. Zhang and F. Zhang,
Global dynamics of a delayed SEIS infectious disease model with logistic growth and saturation incidence, Math. Meth. Appl. Sci., 39 (2016), 3294-3308.
doi: 10.1002/mma.3774. |
show all references
References:
[1] |
V. L. Bakke,
Optimal fields for problems with delays, J. Optim. Theory Appl., 33 (1981), 69-84.
doi: 10.1007/BF00935177. |
[2] |
H. T. Banks,
Necessary conditions for control problems with variable time lags, SIAM J. Control, 6 (1968), 9-47.
doi: 10.1137/0306002. |
[3] |
E. B. M. Bashier and K. C. Patidar,
Optimal control of an epidemiological model with multiple time delays, Appl. Math. Comput., 292 (2017), 47-56.
doi: 10.1016/j.amc.2016.07.009. |
[4] |
A. Boccia, P. Falugi, H. Maurer and R. Vinter, Free time optimal control problems with time delays, 52nd IEEE Conference on Decision and Control, (2013), 520-525.
doi: 10.1109/CDC.2013.6759934. |
[5] |
A. Boccia and R. B. Vinter,
The maximum principle for optimal control problems with time delays, SIAM J. Control Optim., 55 (2017), 2905-2935.
doi: 10.1137/16M1085474. |
[6] |
G. V. Bokov,
Pontryagin's maximum principle of optimal control problems with time-delay, J. Math. Sci. (N. Y.), 172 (2011), 623-634.
doi: 10.1007/s10958-011-0208-y. |
[7] |
F. Cacace, F. Conte, A. Germani and G. Palombo,
Optimal control of linear systems with large and variable input delays, Systems Control Lett., 89 (2016), 1-7.
doi: 10.1016/j.sysconle.2015.12.003. |
[8] |
W. L. Chan and S. P. Yung,
Sufficient conditions for variational problems with delayed argument, J. Optim. Theory Appl., 76 (1993), 131-144.
doi: 10.1007/BF00952825. |
[9] |
D. H. Chyung and E. B. Lee,
Linear optimal systems with time delays, SIAM J. Control Optim., 4 (1966), 548-575.
doi: 10.1137/0304042. |
[10] |
M. C. Delfour,
The linear-quadratic optimal control problem with delays in state and control variables: a state space approach, SIAM J. Control and Optim., 24 (1986), 835-883.
doi: 10.1137/0324053. |
[11] |
A. M. Elaiw and N. H. AlShamrani,
Stability of a general delay-distributed virus dynamics model with multi-staged infected progression and immune response, Math. Meth. Appl. Sci., 40 (2017), 699-719.
doi: 10.1002/mma.4002. |
[12] |
D. H. Eller, J. K. Aggarwal and H. T. Banks,
Optimal control of linear time-delay systems, IEEE Trans. Automat. Control, 14 (1969), 678-687.
doi: 10.1109/TAC.1969.1099301. |
[13] |
A. Friedman,
Optimal control for hereditary processes, Arch. Rational Mech. Anal., 15 (1964), 396-416.
doi: 10.1007/BF00256929. |
[14] |
D. M. Gay, The AMPL modeling language: An aid to formulating and solving optimization problems, in Numerical Analysis and Optimization, 95-116, Springer Proc. Math. Stat., 134, Springer, Cham, 2015.
doi: 10.1007/978-3-319-17689-5_5. |
[15] |
L. Göllmann, D. Kern and H. Maurer,
Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optim. Control Appl. Meth., 30 (2009), 341-365.
doi: 10.1002/oca.843. |
[16] |
L. Göllmann and H. Maurer,
Theory and applications of optimal control problems with multiple time-delays, J. Ind. Manag. Optim., 10 (2014), 413-441.
doi: 10.3934/jimo.2014.10.413. |
[17] |
T. Guinn,
Reduction of delayed optimal control problems to nondelayed problems, J. Optim. Theory Appl., 18 (1976), 371-377.
doi: 10.1007/BF00933818. |
[18] |
A. Halanay,
Optimal controls for systems with time lag, SIAM J. Control, 6 (1968), 215-234.
doi: 10.1137/0306016. |
[19] |
M. R. Hestenes,
On variational theory and optimal control theory, SIAM J. Control, 3 (1965), 23-48.
doi: 10.1137/0303003. |
[20] |
D. K. Hughes,
Variational and optimal control problems with delayed argument, J. Optim. Theory Appl., 2 (1968), 1-14.
doi: 10.1007/BF00927159. |
[21] |
S. H. Hwang and Z. Bien,
Sufficient conditions for optimal time-delay systems with applications to functionally constrained control problems, Internat. J. Control, 38 (1983), 607-620.
doi: 10.1080/00207178308933097. |
[22] |
M. Q. Jacobs and T. Kao,
An optimum settling problem for time lag systems, J. Math. Anal. Appl., 40 (1972), 687-707.
doi: 10.1016/0022-247X(72)90013-3. |
[23] |
G. L. Kharatishvili,
The maximum principle in the theory of optimal processes involving delay, Soviet Math. Dokl., 2 (1961), 28-32.
|
[24] |
G. L. Kharatishvili, A maximum principle in extremal problems with delays, in Mathematical Theory of Control, 26-34, Academic Press, New York, 1967. |
[25] |
G. L. Kharatishvili and T. A. Tadumadze,
Nonlinear optimal control systems with variable lags, Mat. Sb. (N.S.), 107(149) (1978), 613-633.
|
[26] |
F. Khellat,
Optimal control of linear time-delayed systems by linear legendre multiwavelets, J. Optim. Theory Appl., 143 (2009), 107-121.
doi: 10.1007/s10957-009-9548-x. |
[27] |
J. Klamka, H. Maurer and A. Swierniak,
Local controllability and optimal control for a model of combined anticancer therapy with control delays, Math. Biosci. Eng., 14 (2016), 195-216.
doi: 10.3934/mbe.2017013. |
[28] |
R. W. Koepcke,
On the control of linear systems with pure time delay, J. Basic Eng, 87 (1965), 74-80.
doi: 10.1115/1.3650530. |
[29] |
H. N. Koivo and E. B. Lee,
Controller synthesis for linear systems with retarded state and control variables and quadratic cost, Automatica J. IFAC, 8 (1972), 203-208.
doi: 10.1016/0005-1098(72)90068-4. |
[30] |
C. H. Lee and S. P. Yung,
Sufficient conditions for optimal control problems with time delay, J. Optim. Theory Appl., 88 (1996), 157-176.
doi: 10.1007/BF02192027. |
[31] |
E. B. Lee,
Variational problems for systems having delay in the control action, IEEE Trans. Automat. Control, 13 (1968), 697-699.
doi: 10.1109/TAC.1968.1099029. |
[32] |
R. C. H. Lee and S. P. Yung,
Optimality conditions and duality for a non-linear time-delay control problem, Optimal Control Appl. Methods, 18 (1997), 327-340.
doi: 10.1002/(SICI)1099-1514(199709/10)18:5<327::AID-OCA614>3.0.CO;2-9. |
[33] |
E. B. Lee and L. Markus, Foundations of Optimal Control Theory, 2nd edition, Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1986. |
[34] |
A. P. Lemos-Paião, Introduction to Optimal Control Theory and Its Application to Diabetes, M.Sc. thesis, University of Aveiro, Aveiro, 2015. |
[35] |
M. N. Oǧuztöreli,
A time optimal control problem for systems described by differential difference equations, SIAM J. Control Optim., 1 (1963), 290-310.
doi: 10.1137/0301017. |
[36] |
M. N. Oǧuztöreli, Time-lag Control Systems, Academic Press, New York, 1966.
![]() ![]() |
[37] |
K. R. Palanisamy and R. G. Prasada,
Optimal control of linear systems with delays in state and control via Walsh functions, IEE Proceedings D - Control Theory and Applications, 130 (1983), 300-312.
doi: 10.1049/ip-d.1983.0051. |
[38] |
W. J. Palm and W. E. Schmitendorf,
Conjugate-point conditions for variational problems with delayed argument, J. Optim. Theory Appl., 14 (1974), 599-612.
doi: 10.1007/BF00932963. |
[39] |
H. Pirnay, R. López-Negrete and L. T. Biegler,
Optimal sensitivity based on IPOPT, Math. Program. Comput., 4 (2012), 307-331.
doi: 10.1007/s12532-012-0043-2. |
[40] |
L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, 2nd edition, Interscience, New York, 1962. |
[41] |
V. M. Popov and A. Halanay,
A problem in the theory of time delay optimum systems, Autom. Remote Control, 25 (1964), 1129-1134.
|
[42] |
D. Rocha, C. J. Silva and D. F. M. Torres,
Stability and optimal control of a delayed HIV model, Math. Meth. Appl. Sci., 41 (2018), 2251-2260.
doi: 10.1002/mma.4207. |
[43] |
L. D. Sabbagh,
Variational problems with lags, J. Optim. Theory Appl., 3 (1969), 34-51.
doi: 10.1007/BF00929540. |
[44] |
S. P. M. Santos, N. Martins and D. F. M. Torres,
Higher-order variational problems of Herglotz type with time delay, Pure Appl. Funct. Anal., 1 (2016), 291-307.
|
[45] |
W. E. Schmitendorf,
A sufficient condition for optimal control problems with time delays, Automatica J. IFAC, 9 (1973), 633-637.
doi: 10.1016/0005-1098(73)90048-4. |
[46] |
C. J. Silva, H. Maurer and D. F. M. Torres,
Optimal control of a tuberculosis model with state and control delays, Math. Biosci. Eng., 14 (2017), 321-337.
doi: 10.3934/mbe.2017021. |
[47] |
M. A. Soliman,
A new necessary condition for optimality systems with time delay, J. Optim. Theory Appl., 11 (1973), 249-254.
doi: 10.1007/BF00935193. |
[48] |
E. Stumpf,
Local stability analysis of differential equations with state-dependent delay, Discrete Contin. Dyn. Syst., 36 (2016), 3445-3461.
doi: 10.3934/dcds.2016.36.3445. |
[49] |
Y. Xia, M. Fu and P. Shi, Analysis and Synthesis of Dynamical Systems with Time-Delays, Lecture Notes in Control and Information Sciences, 387. Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-02696-6. |
[50] |
J. Xu, Y. Geng and Y. Zhou,
Global stability of a multi-group model with distributed delay and vaccination, Math. Meth. Appl. Sci., 40 (2017), 1475-1486.
doi: 10.1002/mma.4068. |
[51] |
R. Xu, S. Zhang and F. Zhang,
Global dynamics of a delayed SEIS infectious disease model with logistic growth and saturation incidence, Math. Meth. Appl. Sci., 39 (2016), 3294-3308.
doi: 10.1002/mma.3774. |



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