Advanced Search
Article Contents
Article Contents

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

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

Abstract Full Text(HTML) Figure(8) Related Papers Cited by
  • 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.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Functions $ \gamma = \psi\circ \ell^{-1} $ and $ \hat\gamma $ defined above

    Figure 2.  $ T $-periodic solutions $ x(\cdot,u^-,\bar x) $ and $ x(\cdot,u^+,\bar x) $

    Figure 3.  The solution $ \tilde x $ in thick line, $ x $ in thin line

    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)

  • [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. BittantiG. 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. BittantiA. 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. GuardabassiA. 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. EvansJ. 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. HatzimanikatisG. LyberatosS. 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.
  • 加载中



Article Metrics

HTML views(678) PDF downloads(293) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint