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May  2019, 24(5): 2293-2313. doi: 10.3934/dcdsb.2019096

A sufficient optimality condition for delayed state-linear optimal control problems

Ana P. Lemos-Paião , Cristiana J. Silva and  Delfim F. M. Torres ,

Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

*Corresponding author: Delfim F. M. Torres (delfim@ua.pt)

Received  December 2017 Revised  January 2019 Published  May 2019 Early access  March 2019

Fund Project: This work is part of first author's Ph.D., which is carried out at the University of Aveiro.

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.

Keywords: Delayed optimal control problems, delayed state-linear control systems, time delays in state and control variables, sufficient optimality condition, augmented problem.
Mathematics Subject Classification: Primary: 49K15; Secondary: 34H99.
Citation: Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096
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.

Figure 1.  Scheme of new state variables.
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Figure 2.  Optimal control: green line-initial data; blue line-analytical solution; red dashed line-numerical solution.
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Figure 3.  Optimal state: green line-initial data; blue line-analytical solution; red dashed line-numerical solution.
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