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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
| 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 |
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.
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 and Related Fields,
2020, 10
(3)
: 547-571.
doi: 10.3934/mcrf.2020010
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
C. W. Clark, Mathematical Bioeconomics: The Mathematics of Conservation, Third edition, Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2010. |
| [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. |
| [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. |
| [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. |
| [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. |
| [15] |
L. Idels,
Stability analysis of periodic Fox production models, Can. Appl. Math. Q., 14 (2006), 331-341.
|
| [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.
|
| [17] |
C. Maffezzoni,
Hamilton-Jacobi theory for periodic control problems, J. Optimization Theory and Appl., 14 (1974), 21-29.
doi: 10.1007/BF00933172. |
| [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. |
| [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. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [10] |
C. W. Clark, Mathematical Bioeconomics: The Mathematics of Conservation, Third edition, Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2010. |
| [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. |
| [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. |
| [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. |
| [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. |
| [15] |
L. Idels,
Stability analysis of periodic Fox production models, Can. Appl. Math. Q., 14 (2006), 331-341.
|
| [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.
|
| [17] |
C. Maffezzoni,
Hamilton-Jacobi theory for periodic control problems, J. Optimization Theory and Appl., 14 (1974), 21-29.
doi: 10.1007/BF00933172. |
| [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. |
| [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. |
| [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. |
| [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. |
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
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 $
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)
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)
Figure 8.
Optimal criterion $ J_{T}(\hat u_{T}) $ for the depensation model (case 2)
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