On an infinite dimensional linear-quadratic problem with fixed endpoints: The continuity question

K.Maciej Przyłuski

doi:10.2478/amcs-2014-0053

.blurhash-client-img { display: none !important; }

On an infinite dimensional linear-quadratic problem with fixed endpoints: The continuity question

International Journal of Applied Mathematics and Computer Science

Volume 24 (2014): Issue 4 (December 2014)

By: K.Maciej Przyłuski

Open Access

|Dec 2014

Abstract

In a Hilbert space setting, necessary and sufficient conditions for the minimum norm solution u to the equation Su = Rz to be continuously dependent on z are given. These conditions are used to study the continuity of minimum energy and linear-quadratic control problems for infinite dimensional linear systems with fixed endpoints.

References

Athans, M. (1971). The role and use of the stochastic linear-quadratic-Gaussian problem in control system design, IEEE Transactions on Automatic Control, AC-16(6): 529-552.10.1109/TAC.1971.1099818
Search in Google Scholar
Aubin, J.-P. (2000). Applied Functional Analysis, Wiley, New York, NY.10.1002/9781118032725
Search in Google Scholar
Balakrishnan, A.V. (1981). Applied Functional Analysis, Springer, New York, NY.
Search in Google Scholar
Brockett, R.F. (1970). Finite-Dimensional Linear Systems, Wiley, New York, NY.
Search in Google Scholar
Chow, G.P. (1976). Analysis and Control of Dynamic Economic Systems, Wiley, New York, NY.
Search in Google Scholar
Corless, M.J. and Frazho, A.E. (2003). Linear Systems and Control.10.1201/9780203911372
Search in Google Scholar
An Operator Perspective, Marcel Dekker, New York, NY.
Search in Google Scholar
Curtain, R.F. (1984). Linear-quadratic control problem with fixed endpoints in infinite dimensions, Journal of Optimization Theory and Its Applications 44(1): 55-74.10.1007/BF00934894
Search in Google Scholar
Curtain, R.F. and Pritchard A.J. (1978). Infinite-Dimensional Linear Systems Theory, Springer, Berlin.10.1007/BFb0006761
Search in Google Scholar
Curtain, R.F. and Zwart H. (1995). An Introduction to Infinite- Dimensional Linear Systems Theory, Springer, New York, NY.10.1007/978-1-4612-4224-6
Search in Google Scholar
Douglas, R.G. (1966). On majorization, factorization, and range inclusion of operators on Hilbert space, Proceedings of the American Mathematical Society 18(2): 413-415.10.1090/S0002-9939-1966-0203464-1
Search in Google Scholar
Emirsajłow, Z. (1989). Feedback control in LQCP with a terminal inequality constraint, Journal of Optimization Theory and Applications 62(3): 387-403.10.1007/BF00939813
Search in Google Scholar
Evans, L.C. (2010). Partial Differential Equations, American Mathematical Society, Providence RI.
Search in Google Scholar
Federico, S. (2011). A stochastic control problem with delay arising in a pension fund model, Finance and Stochastics 15(3): 421-459.10.1007/s00780-010-0146-4
Search in Google Scholar
Fuhrmann, P.A. (1972). On weak and strong reachability and controllability of infinite-dimensional linear systems, Journal of Optimization Theory and Its Applications 9(2): 77-89.10.1007/BF00932345
Search in Google Scholar
Kandilakis, D. and Papageorgiou, N.S. (1992). Evolution inclusions of the subdifferential type depending on a parameter, Commentationes Mathematicae Universitatis Carolinae 33(3): 437-449.
Search in Google Scholar
Kendrick, D.A. (1981). Stochastic Control for Economic Models, McGraw-Hill, New York, NY.
Search in Google Scholar
Kobayashi, T. (1978). Some remarks on controllability for distributed parameter systems, SIAM Journal on Control and Optimization 16(5): 733-742.10.1137/0316049
Search in Google Scholar
Laurent, P.-J. (1972). Approximation et Optimisation, Hermann, Paris.
Search in Google Scholar
Ljungqvist, L. and Sargent, T.J. (2004). Recursive Macroeconomic Theory, MIT Press, Cambridge, MA.
Search in Google Scholar
Luenberger, D.G. (1969). Optimization by Vector Space Methods, Wiley, New York, NY.
Search in Google Scholar
Papageorgiou, N.S. (1991). On the dependence of the solutions and optimal solutions of control problems on the control constraint set, Journal of Mathematical Analysis and Applications 158(2): 427-447.10.1016/0022-247X(91)90247-W
Search in Google Scholar
Porter, W.A. (1966). Modern Foundations of System Engineering, Macmillan, New York, NY.
Search in Google Scholar
Przyłuski, K.M. (1981). Remarks on continuous dependence of an optimal control on parameters, in O. Moeschlin and D. Pallaschke (Eds.), Game Theory and Mathematical Economics, North-Holland, Amsterdam, pp. 333-337.
Search in Google Scholar
Rolewicz, S. (1987). Functional Analysis and Control Theory. Linear Systems, PWN, Warsaw, (in Polish).10.1007/978-94-015-7758-8
Search in Google Scholar
Sent, E.-M. (1998). The Evolving Rationality of Rational Expectations: An Assessment of Thomas Sargent’s Achievements, Cambridge University Press, Cambridge.10.1017/CBO9780511528514
Search in Google Scholar
Triggiani, R. (1975a). A note on the lack of exact controllability for mild solutions in Banach spaces, SIAMJournal on Control and Optimization 15(3): 407-411.10.1137/0315028
Search in Google Scholar
Triggiani, R. (1975b). On the lack of exact controllability for mild solutions in Banach spaces, Journal of Mathematical Analysis and Applications 50(2): 438-446.10.1016/0022-247X(75)90033-5
Search in Google Scholar
Triggiani, R. (1976). Extensions of rank conditions for controllability and observability to Banach spaces and unbounded operators, SIAM Journal on Control and Optimization 14(2): 313-338. 10.1137/0314022
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/amcs-2014-0053 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X

Journal RSS Feed

Language: English

Page range: 723 - 733

Submitted on: Sep 30, 2013

Published on: Dec 20, 2014

Published by: University of Zielona Góra

In partnership with: Paradigm Publishing Services

Publication frequency: 4 issues per year

Keywords:

minimum norm problem,

linear-quadratic control,

linear-quadratic economies,

controllability,

continuity of optimal control

Related subjects:

Mathematics,

Applied mathematics

© 2014 K.Maciej Przyłuski, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 24 (2014): Issue 4 (December 2014)