On the lower semicontinuity of functionals involving Lebesgue or improper Riemann integrals in infinite horizon optimal control problems

• Autorzy: Pickenhain S. , Lykina V. , Wagner M.
• Języki publikacji: EN

Abstrakty

EN

This paper deals with infinite horizon optimal control problems, which are formulated in weighted Sobolev spaces ... [wzór] and weighted Lp-spaces ... [wzór]. We ask for the consequences of the interpretation of the integral within the objective as a Lebesgue or an improper Riemann integral. In order to justify the use of both types of integrals, various applications of infinite horizon problems are presented. We provide examples showing that lower semicontinuity may fail for objectives involving Lebesgue as well as improper Riemann integrals. Further we prove a lower semicontinuity theorem for an objective with Lebesgue integral under more restrictive growth conditions on the integrand.

Słowa kluczowe

EN

optimal control; infinite horizon; weighted Sobolev spaces; lower semicontinuity; Lebesgue integral; improper Riemann integral

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2008
• Tom: Vol. 37, no 2
• Strony: 451--468
• Opis fizyczny: Bibliogr. 24 poz.

Twórcy

autor

Pickenhain S.

autor

Lykina V.

autor

Wagner M.

• Cottbus University of Technology, Institute of Mathematics, P. O. B. 10 13 44, D-03013 Cottbus, Germany,

Bibliografia

• ADAMS, R.A. and FOURNIER, J.J.F. (2007) Sobolev Spaces. 2nd ed. Academic Press / Elsevier, Amsterdam.
• ANTOCI, F. (2003) Some necessary and some sufficient conditions for the compactness of the embedding of weighted Sobolev spaces. Ricerche Mat. 52, 55-71.
• BENVENISTE, L.M. and SCHEINKMAN, J.A. (1982) Duality theory for dynamic optimization models of economics: the continuous time case. J. of Economic Theory 27, 1-19.
• BLOT, J. and HAYEK, N. (1996) Second-order necessary conditions for the infinite-horizon variational problems. Mathematics of Operations Research 21, 979-990.
• BLOT, J. and HAYEK, N. (2000) Sufficient conditions for infinite-horizon calculus of variations problems. ESAIM Control Optim. Calc. Var. 5, 279-292.
• BLOT, J. and MICHEL, P. (1996) First-order necessary conditions for infinite-horizon variational problems. J. Optim. Theory Appl. 88, 339-364.
• CARLSON, D.A., HAURIE, A.B. and LEIZAROWITZ, A. (1991) Infinite Horizon Optimal Control. 2nd ed. Springer, Berlin.
• COLONIUS, F. and KLIEMANN, W. (1989) Infinite time optimal control and periodicity. Appl. Math. Optim. 20, 113-130.
• DMITRUK, A.V. and KUZ’KINA, N. V. (2005) Existence theorem in the optimal control problem on an infinite time interval. Mathematical Notes 78, 466-480.
• DUNFORD, N. and SCHWARTZ, J.T. (1988) Linear Operators. Part I: General Theory. Wiley-Interscience, New York.
• ELSTRODT, J. (1996) Mass- und Integrationstheorie. Springer, Berlin.
• FEICHTINGER, G. and HARTL, R.F. (1986) Optimale Kontrolle okonomischer Prozesse. De Gruyter, Berlin - New York.
• FICHTENHOLZ, G.M. (1990) Differential- und Integralrechnung. Band II. 10th ed. VEB Deutscher Verlag der Wissenschaften, Berlin.
• GOH, B.S., LEITMANN, G. and VINCENT, T.L. (1974) Optimal control of a prey-predator system. Mathematical Biosciences 19, 263-286.
• HALKIN, H. (1974) Necessary conditions for optimal control problems with infinite horizons. Econometrica 42, 267-272.
• KLÖTZLER, R. (1979) On a general conception of duality in optimal control. In: Fábera, J., ed., Equadiff IV. Proceedings of the Czechoslovak Conference on Differential Equations and their Applications held in Prague, August 22-26, 1977. Springer, Berlin, Lecture Notes in Mathematics 703, 189-196.
• KUFNER, A. (1985) Weighted Sobolev Spaces. John Wiley & Sons, Chichester.
• LEIZAROWITZ, A. and MIZEL, V.J. (1989) One-dimensional infinite-horizon variational problems arising in continuum mechanics. Arch. Rat. Mech. Anal. 106, 161-194.
• MAGILL, M.J.P. (1982) Pricing infinite horizon programs. J. Math. Anal. Appl. 88, 398-421.
• PICKENHAIN, S. and LYKINA, V. (2006) Sufficiency conditions for infinite horizon optimal control problems. In: Seeger, A., ed., Recent Advances in Optimization. Springer, Berlin, Lecture Notes in Economics and Mathematical Systems 563, 217-232.
• SETHI, S.P. and THOMPSON, G.L. (2000) Optimal Control Theory. Applications to Management Science and Economics. 2nd ed. Kluwer, Boston-Dordrecht-London .
• VAINBERG, M.M. (1964) Variational Methods for the Study of Nonlinear Operators. Holden-Day, San Francisco-London-Amsterdam.
• YOSIDA, K. (1980) Functional Analysis. 6th ed. Springer, Berlin-Heidelberg-New York.
• ZASLAVSKI, A.J. (1995) The existence of periodic minimal energy configurations for one-dimensional infinite horizon variational problems arising in continuum mechanics. J. Math. Anal. Appl. 194, 459-476.