Abstract

We consider the optimal control problem of feeding in minimal time a tank where several species compete for a single resource, with the objective being to reach a given level of the resource. We allow controls to be bounded measurable functions of time plus possible impulses. For the one-species case, we show that the immediate one-impulse strategy (filling the whole reactor with one single impulse at the initial time) is optimal when the growth function is monotonic. For nonmonotonic growth functions with one maximum, we show that a particular singular arc strategy (precisely defined in section 3) is optimal. These results extend and improve former ones obtained for the class of measurable controls only. For the two-species case with monotonic growth functions, we give conditions under which the immediate one-impulse strategy is optimal. We also give optimality conditions for the singular arc strategy (at a level that depends on the initial condition) to be optimal. The possibility for the immediate one-impulse strategy to be nonoptimal while both growth functions are monotonic is a surprising result and is illustrated with the help of numerical simulations.

MSC codes

  1. 49J15
  2. 49N25

Keywords

  1. minimal time problem
  2. chemostat
  3. Hamilton–Jacobi–Bellman equation
  4. Pontryagin's minimum principle
  5. impulse control

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
G. D'Ans, D. Gottlieb, and P. Kokotovic, Optimal control of bacterial growth, Automatica, 8 (1972), pp. 729–736.
2.
M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhaüser, Cambridge, MA, 1997.
3.
D. Berovic and R. Vinter, The application of dynamic programming to optimal inventory control, IEEE Trans. Automat. Control, 49 (2004), pp. 676–685.
4.
F. Bonnans and P. Rouchon, Commande et optimisation de systèmes dynamiques, Éditions de l'École Polytechnique, Montreal, 2005.
5.
A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory Appl., 71 (1991), pp. 67–83.
6.
A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), pp. 435–457.
7.
E. Crück, Problèmes de cible sous contraintes d'état pour des systèmes non linéaires avec sauts d'état, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), pp. 403–408.
8.
V. A. Dykhta and O. N. Samsonyuk, Optimal Impulse Control with Applications (in Russian), Fizmatlit, Moscow, 2000.
9.
C. Gao, K. Li, E. Feng, and Z. Xiu, Nonlinear impulsive system of fed-batch culture in fermentative production and its properties, Chaos Solitons Fractals, 28 (2006), pp. 271–277.
10.
J. Hong, Optimal substrate feeding policy for fed batch fermentation with substrate and product inhibition kinetics, Biotechnol. Bioengng., 28 (1986), pp. 1421–1431.
11.
R. L. Irvine and L. H. Ketchum, Sequencing batch reactors for biological wastewater treatment, Crit. Rev. Environ. Control, 18 (1989), pp. 255–294.
12.
H. C. Lim, Y. J. Tayeb, J. M. Modak, and P. Bonte, Computational algorithms for optimal feed rates for a class of fed-batch fermentation: Numerical results for penicillin and cell production, Biotechnol. Bioengng., 28 (1986), pp. 1408–1420.
13.
A. Miele, Extremization of linear integrals by Green's Theorem, Optimization Techniques, Academic Press, New York, 1962, pp. 69–98.
14.
B. M. Miller, Conditions for optimality in generalized control problems. I. Necessary conditions for optimality, Automat. Remote Control, 53 (1992), pp. 362–370.
15.
B. M. Miller, Conditions for optimality in generalized control problems. II. Sufficient conditions for optimality, Automat. Remote Control, 53 (1992), pp. 505–513.
16.
B. M. Miller and E. J. Rubinovich, Optimal impulse control problem with constrained number of impulses, Math. Comput. Simulation, 34 (1992), pp. 23–49.
17.
B. Miller, The generalized solutions of ordinary differential equations in the impulse control problems, J. Math. Systems Estim. Control, 6 (1994), pp. 415–435.
18.
B. Miller, The generalized solutions of nonlinear optimization problems with impulse control, SIAM J. Control Optim., 34 (1996), pp. 1420–1440.
19.
B. M. Miller and E. Y. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems. Kluwer/Plenum Publishers, New York, 2003.
20.
J. Moreno, Optimal control of bioreactors for the wastewater treatment, Optimal Control Appl. Methods, 20 (1999), pp. 145–164.
21.
M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differ. Integral Equ., 8 (1995), pp. 269–288.
22.
M. Motta and F. Rampazzo, Nonlinear systems with unbounded controls and state constraints: A problem of proper extension, NoDEA Nonlinear Differential Equations Appl., 3 (1996), pp. 191–216.
23.
M. Motta and F. Rampazzo, Dynamic programming for nonlinear systems driven by ordinary and impulsive controls, SIAM J. Control Optim., 34 (1996), pp. 199–225.
24.
M. Motta and C. Sartori, Minimum time with bounded energy, minimum energy with bounded time, SIAM J. Control Optim., 42 (2003), pp. 789–809.
25.
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated from Russian by K. N. Trirogoff; L. W. Neustadt, ed., Interscience Publishers, John Wiley & Sons, Inc., New York, London, 1962.
26.
F. Rampazzo and C. Sartori, The minimum time function with unbounded controls, J. Math. Systems Estim. Control, 8 (1998), p. 34.
27.
R. W. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures, J. Soc. Ind. Appl. Math. Ser. A Control, 3 (1965), pp. 191–205.
28.
H. L. Smith, Monotone Dynamical Systems. AMS, Providence, RI, 1995.
29.
H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, London, 1995.
30.
R. G. Tsoneva, T. D. Patarinska, and I. P. Popchev, Augmented Lagrange decomposition method for optimal control calculation of batch fermentation processes, Bioprocess Engrg., 18 (1998), pp. 143-153.
31.
R. Vinter, Optimal Control, Birkhäuser, Cambridge, MA, 2000.
32.
H. Wang, E. Feng, and Z. Xiu, Optimality condition of the nonlinear impulsive system in fed-batch fermentation, Nonlinear Anal., 68 (2008), pp. 12–23.
33.
P. Wolenski and S. Zabic, A differential solution concept for impulsive systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal., 13 (2006), pp. 199-210.
34.
P. Wolenski and S. Zabic, A sampling method and approximation results for impulsive systems, SIAM J. Control Optim., 46 (2007), pp. 983-998.
35.
S. T. Zavalishchin and A. N. Sesekin, Dynamic impulse systems: Theory and applications, Math. Appl. 394. Kluwer Academic Publishers Group, Dordrecht, 1997.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 2827 - 2856
ISSN (online): 1095-7138

History

Submitted: 22 June 2007
Accepted: 6 July 2008
Published online: 19 November 2008

MSC codes

  1. 49J15
  2. 49N25

Keywords

  1. minimal time problem
  2. chemostat
  3. Hamilton–Jacobi–Bellman equation
  4. Pontryagin's minimum principle
  5. impulse control

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media