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Infinite time optimal control and periodicity

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Abstract

For smooth nonlinear systems

$$\dot x(t) = X_0 (x(t)) + \sum\limits_{i = 1}^r {u_i (t)X_i (x(t)),}$$

the infinite time optimal control problems: maximize

$$\mathop {\lim \sup }\limits_{T \to \infty } \frac{1}{T}\int_0^T {g(x(t)), u(t)) dt}$$

(average yield criterion) or

$$\mathop {\lim }\limits_{T \to \infty } \int_0^T {e^{ - \delta l} g(x(t), u(t)) dt}$$

(discounted criterion) are considered, where the initial valuex(0) may be free or restricted. We study the existence of optimal periodic solutions for the above problems: if approximately optimal solutions have a limit point in the interior of some control set, then there exist approximately optimal periodic solutions. This result is applied to the growth of linear control semigroups and to a three-dimensional predator-prey harvesting model.

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References

  1. Arnold, L., and W. Kliemann. Qualitative theory of stochastic systems. In: Bharucha-Reid, A. T. (ed.), Probabilistic Analysis and Related Topics, Vol. III. Academic Press, New York (1983), pp. 1–79.

    Google Scholar 

  2. Arnold, L., and W. Kliemann. On unique ergodicity for degenerate diffusions. Stochastics,21 (1987), 41–61.

    Google Scholar 

  3. Arnold, L., W. Kliemann, and E. Oeljeklaus. Lyapunov exponents of linear stochastic systems. In: Arnold, L., and V. Wihstutz (eds.), Lyapunov Exponents. Lecture Notes in Mathematics, No. 1186. Springer-Verlag, New York (1986), pp. 85–125.

    Google Scholar 

  4. Aubin, J. P., and F. H. Clarke. Shadow prices and duality for a class of optimal control problems. SIAM J. Control Optim.,17 (1979), 567–585.

    Google Scholar 

  5. Cesari, L. Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, 3rd edn. Springer-Verlag, Berlin (1971).

    Google Scholar 

  6. Clark, C. W. Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York (1976).

    Google Scholar 

  7. Colonius, F. Optimal Periodic Control. Lecture Notes in Mathematics, No. 1313. Springer-Verlag, Berlin (1988).

    Google Scholar 

  8. Colonius, F. Asymptotic behavior of optimal control systems with high or low discount rates, Math. Oper. Res., to appear.

  9. Kliemann, W. Recurrence and invariant measures for degenerate diffusions. Ann. Probab.,15 (1986), 690–707.

    Google Scholar 

  10. Nemytskii, V. V., and V. V. Stepanov. Qualitative Theory of Differential Equations. Princeton University Press, Princeton (1960).

    Google Scholar 

  11. Rockafellar, R. T. Saddle points of Hamiltonian systems in convex problems of Lagrange. In: Cass, D., and K. Shell (eds.), The Hamiltonian Approach to Dynamic Economics. Academic Press, New York (1976).

    Google Scholar 

  12. Roxin, E. O. Limits sets in infinite horizon optimal control systems. In: Lakshmikantham, V. (ed.), Trends in the Theory and Practice of Non-Linear Analysis. Elsevier, Amsterdam (1985), pp. 401–407.

    Google Scholar 

  13. Sieveking, M. Dinamica de poblaciones por medio de ecuaciones differenciales ordinarias. Escuela Politecnica Nacional, Quito, Ecuador and Fachbereich Mathematik, Universität Frankfurt (1985).

  14. Sussmann, H. J. Orbits of families of vector fields and integrability of distributions. Trans. Amer. Math. Soc.,180 (1973), 171–188.

    Google Scholar 

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Authors and Affiliations

  1. Institute for Dynamical Systems, Universität Bremen, D-2800, Bremen 33, West Germany

    Fritz Colonius

  2. Department of Mathematics, Iowa State University, Ames, IA, USA

    Wolfgang Kliemann

Authors
  1. Fritz Colonius
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  2. Wolfgang Kliemann
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Additional information

This research was conducted while the authors were visiting the Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, USA. The research of F. Colonius was supported by Stiftung Volkswagenwerk.

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Colonius, F., Kliemann, W. Infinite time optimal control and periodicity. Appl Math Optim 20, 113–130 (1989). https://doi.org/10.1007/BF01447651

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