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September  2020, 10(3): 547-571. doi: 10.3934/mcrf.2020010

Optimal periodic control for scalar dynamics under integral constraint on the input

Térence Bayen 1, , Alain Rapaport 2, and  Fatima-Zahra Tani 2,

1. 

Avignon Université, Laboratoire de Mathématiques d'Avignon (EA 2151) F-84018 Avignon, France., MISTEA, Univ Montpellier, INRA, Montpellier SupAgro, France
2. 

MISTEA, Univ Montpellier, INRA, Montpellier SupAgro, France

Dedicated to Prof. Dr. Frédéric Bonnans on the occasion of his 60th birthday

Received  February 2018 Revised  February 2019 Published  September 2020 Early access  December 2019

This paper studies a periodic optimal control problem governed by a one-dimensional system, linear with respect to the control $ u $, under an integral constraint on $ u $. We give conditions for which the value of the cost function at steady state with a constant control $ \bar u $ can be improved by considering periodic control $ u $ with average value equal to $ \bar u $. This leads to the so-called "over-yielding" met in several applications. With the use of the Pontryagin Maximum Principle, we provide the optimal synthesis of periodic strategies under the integral constraint. The results are illustrated on a single population model in order to study the effect of periodic inputs on the utility of the stock of resource.

Keywords: Optimal control, Pontryagin maximum principle, periodic solutions, over-yielding.
Mathematics Subject Classification: 49J15, 49K15, 34C25, 49N20, 49J30.
Citation: Térence Bayen, Alain Rapaport, Fatima-Zahra Tani. Optimal periodic control for scalar dynamics under integral constraint on the input. Mathematical Control & Related Fields, 2020, 10 (3) : 547-571. doi: 10.3934/mcrf.2020010
References:
[1]

E.-M. Abulesz and G. Lyberatos, Periodic impulse-forcing of nonlinear systems: A new method, International Journal of Control, 48 (1988), 469-480.  doi: 10.1080/00207178808906191.  Google Scholar

[2]

E.-M. Abulesz and G. Lyberatos, Periodic optimization of microbial growth processes, Biotechnology and Bioengineering, 29 (1987), 1059-1067.  doi: 10.1002/bit.260290904.  Google Scholar

[3]

E. M. Abulesz and G. Lyberatos, Periodic operation of a continuous culture of Baker's yeast, Biotechnology and Bioengineering, 34 (1989), 741-749.  doi: 10.1002/bit.260340603.  Google Scholar

[4]

A. O. Belyakov and V. M. Veliov, Constant versus periodic fishing: Age structured optimal control approach, Math. Model. Nat. Phenom., 9 (2014), 20-37.  doi: 10.1051/mmnp/20149403.  Google Scholar

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D. S. Bernstein and E. G. Gilbert, Optimal periodic control: The $\pi$ test revisited, IEEE Transactions on Automatic Control, 25 (1980), 673-684.  doi: 10.1109/TAC.1980.1102394.  Google Scholar

[6]

S. Bittanti, G. Fronza and G. Guardabassi, Periodic control: A frequency domain approach, IEEE Transactions on Automatic Control, 18 (1973), 33-38.  doi: 10.1109/tac.1973.1100225.  Google Scholar

[7]

S. Bittanti, A. Locatelli and C. Maffezzoni, Second-variation methods in periodic optimization, J. Optimization Theory and Appl., 14 (1974), 31-49.  doi: 10.1007/BF00933173.  Google Scholar

[8]

G. Guardabassi, A. Locatelli and S. Rinaldi, Status of periodic optimization of dynamical systems, J. Optimization Theory and Appl., 14 (1974), 1-20.  doi: 10.1007/BF00933171.  Google Scholar

[9]

L. Cesari, Optimization-Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics (New York), 17. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[10]

C. W. Clark, Mathematical Bioeconomics: The Mathematics of Conservation, Third edition, Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2010.  Google Scholar

[11]

R. T. Evans, J. L. Speyer and C.-H. Chuang, Solution of a periodic optimal control problem by asymptotic series, J. Optimization Theory and Appl., 52 (1987), 343-364.  doi: 10.1007/BF00938212.  Google Scholar

[12]

E. G. Gilbert, Optimal periodic control: A general theory of necessary conditions, SIAM J. Control Optim., 15 (1977), 717-746.  doi: 10.1137/0315046.  Google Scholar

[13]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganisms Cultures, ISTE, London, John Wiley & Sons, Inc., Hoboken, NJ, 2017.  Google Scholar

[14]

V. Hatzimanikatis, G. Lyberatos, S. Pavlou and S. A. Svoronos, A method for pulsed periodic optimization of chemical reaction systems, Chemical Engineering Science, 48 (1993), 789-797.  doi: 10.1016/0009-2509(93)80144-F.  Google Scholar

[15]

L. Idels, Stability analysis of periodic Fox production models, Can. Appl. Math. Q., 14 (2006), 331-341.   Google Scholar

[16]

L. Idels and M. Wang, Harvesting strategies with modified effort function, Intern. J. of Modelling, Identification and Control, Special Issue "Modeling Complex Systems" (IJMIC), 3 (2008), 83-87.   Google Scholar

[17]

C. Maffezzoni, Hamilton-Jacobi theory for periodic control problems, J. Optimization Theory and Appl., 14 (1974), 21-29.  doi: 10.1007/BF00933172.  Google Scholar

[18]

L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar

[19]

L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical Theory of Optimal Processes, Pergamon Press Book, The Macmillan Co., New York, 1964.  Google Scholar

[20]

J. L. Speyer and R. T. Evans, A second variation theory for optimal periodic processes, IEEE Transactions on Automatic Control, 29 (1984), 138-148.  doi: 10.1109/TAC.1984.1103482.  Google Scholar

[21]

Q. H. Wang and J. L. Speyer, Necessary and sufficient conditions for local optimality of a periodic process, SIAM J. Control Optim., 28 (1990), 482-497.  doi: 10.1137/0328027.  Google Scholar

show all references

References:
[1]

E.-M. Abulesz and G. Lyberatos, Periodic impulse-forcing of nonlinear systems: A new method, International Journal of Control, 48 (1988), 469-480.  doi: 10.1080/00207178808906191.  Google Scholar

[2]

E.-M. Abulesz and G. Lyberatos, Periodic optimization of microbial growth processes, Biotechnology and Bioengineering, 29 (1987), 1059-1067.  doi: 10.1002/bit.260290904.  Google Scholar

[3]

E. M. Abulesz and G. Lyberatos, Periodic operation of a continuous culture of Baker's yeast, Biotechnology and Bioengineering, 34 (1989), 741-749.  doi: 10.1002/bit.260340603.  Google Scholar

[4]

A. O. Belyakov and V. M. Veliov, Constant versus periodic fishing: Age structured optimal control approach, Math. Model. Nat. Phenom., 9 (2014), 20-37.  doi: 10.1051/mmnp/20149403.  Google Scholar

[5]

D. S. Bernstein and E. G. Gilbert, Optimal periodic control: The $\pi$ test revisited, IEEE Transactions on Automatic Control, 25 (1980), 673-684.  doi: 10.1109/TAC.1980.1102394.  Google Scholar

[6]

S. Bittanti, G. Fronza and G. Guardabassi, Periodic control: A frequency domain approach, IEEE Transactions on Automatic Control, 18 (1973), 33-38.  doi: 10.1109/tac.1973.1100225.  Google Scholar

[7]

S. Bittanti, A. Locatelli and C. Maffezzoni, Second-variation methods in periodic optimization, J. Optimization Theory and Appl., 14 (1974), 31-49.  doi: 10.1007/BF00933173.  Google Scholar

[8]

G. Guardabassi, A. Locatelli and S. Rinaldi, Status of periodic optimization of dynamical systems, J. Optimization Theory and Appl., 14 (1974), 1-20.  doi: 10.1007/BF00933171.  Google Scholar

[9]

L. Cesari, Optimization-Theory and Applications. Problems with Ordinary Differential Equations, Applications of Mathematics (New York), 17. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8165-5.  Google Scholar

[10]

C. W. Clark, Mathematical Bioeconomics: The Mathematics of Conservation, Third edition, Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2010.  Google Scholar

[11]

R. T. Evans, J. L. Speyer and C.-H. Chuang, Solution of a periodic optimal control problem by asymptotic series, J. Optimization Theory and Appl., 52 (1987), 343-364.  doi: 10.1007/BF00938212.  Google Scholar

[12]

E. G. Gilbert, Optimal periodic control: A general theory of necessary conditions, SIAM J. Control Optim., 15 (1977), 717-746.  doi: 10.1137/0315046.  Google Scholar

[13]

J. Harmand, C. Lobry, A. Rapaport and T. Sari, The Chemostat: Mathematical Theory of Microorganisms Cultures, ISTE, London, John Wiley & Sons, Inc., Hoboken, NJ, 2017.  Google Scholar

[14]

V. Hatzimanikatis, G. Lyberatos, S. Pavlou and S. A. Svoronos, A method for pulsed periodic optimization of chemical reaction systems, Chemical Engineering Science, 48 (1993), 789-797.  doi: 10.1016/0009-2509(93)80144-F.  Google Scholar

[15]

L. Idels, Stability analysis of periodic Fox production models, Can. Appl. Math. Q., 14 (2006), 331-341.   Google Scholar

[16]

L. Idels and M. Wang, Harvesting strategies with modified effort function, Intern. J. of Modelling, Identification and Control, Special Issue "Modeling Complex Systems" (IJMIC), 3 (2008), 83-87.   Google Scholar

[17]

C. Maffezzoni, Hamilton-Jacobi theory for periodic control problems, J. Optimization Theory and Appl., 14 (1974), 21-29.  doi: 10.1007/BF00933172.  Google Scholar

[18]

L. Perko, Differential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar

[19]

L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishchenko, Mathematical Theory of Optimal Processes, Pergamon Press Book, The Macmillan Co., New York, 1964.  Google Scholar

[20]

J. L. Speyer and R. T. Evans, A second variation theory for optimal periodic processes, IEEE Transactions on Automatic Control, 29 (1984), 138-148.  doi: 10.1109/TAC.1984.1103482.  Google Scholar

[21]

Q. H. Wang and J. L. Speyer, Necessary and sufficient conditions for local optimality of a periodic process, SIAM J. Control Optim., 28 (1990), 482-497.  doi: 10.1137/0328027.  Google Scholar

Figure 1.  Functions $ \gamma = \psi\circ \ell^{-1} $ and $ \hat\gamma $ defined above
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Figure 2.  $ T $-periodic solutions $ x(\cdot,u^-,\bar x) $ and $ x(\cdot,u^+,\bar x) $
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Figure 3.  The solution $ \tilde x $ in thick line, $ x $ in thin line
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Figure 4.  Optimal criterion $ J_{T}(\hat u_{T}) $ (left) and $ x_m $, $ x_M $ (right) as functions of the period $ T $ for the logistic growth
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Figure 5.  Graphs of the functions $ h $ (left) and $ \psi $ (right) for $ r = 0.3 $, $ K = 5 $, $ \alpha = 2.5 $, $ E_{max} = 0.5893 $, $ E^\star = 0.6235 $
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Figure 6.  Optimal criterion $ J_{T}(\hat u_{T}) $ (left) and $ x_m $, $ x_M $ (right) as functions of the period $ T $ for the depensation model (case 1)
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Figure 7.  Plot of the function $ F $ defined by (22) (left), and $ x_m $, $ x_M $, $ x_T^- $, $ x_T^+ $ (right) as functions of the period $ T $ $ (T<6) $ for the depensation model (case 2)
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Figure 8.  Optimal criterion $ J_{T}(\hat u_{T}) $ for the depensation model (case 2)
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