Abstract
The concept of local controllability is investigated for non-relativistic quantum systems. Sufficient conditions will be sought such that the solution of the controlled Schrodinger equation can be guided, over a short time interval, to any chosen point in a suitably prescribed neighborhood of the solution in the absence of control. Evolution equations which are linear in the controls but nonlinear in the quantum state Ψ are considered. Our formulation and analysis will (for the most part) run parallel to those of Hermes.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Tarn, T.J., G.M. Huang and J.W. Clark: Modelling of quantum mechanical control systems, Mathematical Modelling, 1 (1980), 109–121.
van der Schaft, Aryan J.: Hamiltonian and quantum mechanical control systems, in: Proceedings of the 4th International Seminar on Mathematical Theory of Dynamical Systems and Microphysics, Udine (Ed. A. Blaquiere, S. Diner and G. Lochak), Springer-Verlag, Wien-New York 1987.
Clark, J.W. and T.J. Tarn: Quantum nondemolition filtering, in: Proceedings of the 4th International Seminar on Mathematical Theory of Dynamical Systems and Microphysics, Udine (Ed. A. Blaquiere, S. Diner and G. Lochak), Springer-Verlag, Wien-New York 1987.
Tarn, T.J., J.W. Clark, C.K. Ong and G.M. Huang: Continuous-time quantum mechanical filter, in: Proceedings of the Joint Workshop on Feedback and Synthesis of Linear and Nonlinear Systems, Bielefeld and Rome (Ed. D. Hinrichsen and A. Isidori), Springer-Verlag, Berlin 1982.
Clark, J.W., C.K. Ong, T.J. Tarn and G.M. Huang: Quantum nondemolition filters, Mathematical Systems Theory, 18(1985), 33–35.
Ong, C.K., G.M. Huang, T.J. Tarn and J.W. Clark: Invertibility of quantum-mechanical control systems, Mathematical Systems Theory, 17(1984), 335–350.
Belavkin, Viacheslav: Non-demolition measurement and control in quantum dynamical systems, in: Proceedings of the 4th International Seminar on Mathematical Theory of Dynamical Systems and Microphysics, Udine (Ed. A. Blaquiere, S. Diner and G. Lochak), Springer-Verlag, Wien-New York 1987.
Peirce, A.P., M.A. Dahleh and H. Rabitz: Optimal control of quantum mechanical systems: existence, numerical approximations, and applications, to appear in: Proceedings of the IEEE International Conference: Control 88, University of Oxford, UK, April (1988).
Butkovskiy, A.G. and Yu. I. Samoilenko, Control of quantum systems, automation and remote control, No. 4, April (1979), 485–502; Control of quantum systems, automation and remote control, No. 5 May (1979), 629–645.
Butkovskiy, A.G. and Ye. I. Pustil'nykova: The method of seeking finite control for quantum mechanical processes, in: Proceedings of the 4th International Seminar on Mathematical Theory of Dynamical Systems and Microphysics, Udine (Ed. A. Blaquiere, S. Diner and G. Lochak), Springer-Verlag, Wien-New York 1987.
Butkovskiy, A.G. and Yu. I. Samoilenko: Controllability of quantum mechanical systems, Dokl, Akad. Nauk SSSR 250, 51 (1980) [Sov, Phys, Dokl, 25, 22 (1980)].
Huang, G.M., T.J. Tarn and J.W. Clark: On the controllability of quantum mechanical systems, J. Math. Phys. 24 (1983) 2608–2618.
Sussmann, H. and V. Jurdjevic: Controllability of non-linear systems, Journal of Differential Equations, 12 (1962), 95–116.
Jurdjevic, V. and H. Sussmann: Control systems on lie groups, Journal of Differential Equations, 12 (1972), 313–329.
Krener, Arthur J.: A Generalization of chow's theorem and the bang-bang theorem to nonlinear control problems, SIAM Journal of Control, Vol. 12, No. 1, Feb. (1974).
Brockett, Roger W.: Nonlinear systems and differential geometry: Proceedings of IEEE, Vol. 64, No. 1, Jan (1976).
Kunita, Hiroshi: On the controllability of nonlinear systems, with applications of polynomial systems, Applied Mathematics and Optimization (1976), 89–99.
Hermes, H.: Local controllability of obserables in finite and infinite dimensional nonlinear control systems, Applied Mathematics and Optimization, 5, (1979), 117–125.
Messiah, A.: Quantum mechanics, Vols. I and II, Wiley, New York (1961).
Beals, R. and C. Feffermann: On Local solvability of linear partial differential equations, Annals of Mathematics, 97 (1973), 483–498.
Brown, G.E.: Unified theory of nuclear models and forces, North-Holland, Amsterdam, (1971).
Santilli, R.M.: Need of subjecting to an experimental verification the validity within a hadron of Einstein's special relativity and Pauli's exclusion principle, Hadronic Journal, 1, (1978), 574–901.
Abraham, R., and J.E. Marsden: Foundations of mechanics, 2nd ed. Benjamin, Reading, (1978).
Wigner, E.P.: The Scientist speculates. I.J. Good, Ed. W. Heinemann, London, (1961).
d'Espagnat, B.: Conceptual foundations of quantum mechanics, Benjamin, Reading, (1976).
Hermes, H.: Controllability of nonlinear delay differential equations, Nonlinear Analysis, Theory, Methods and Applications, 3 (1979).
Kailath, T.: Linear systems, Prentice-Hall, Inc., Englewood Cliffs, (1980).
Helstrom, C.W.: Quantum detection and estimation theory, Academic Press, New York, (1976).
Ilic, D.: D. Sc. Dissertation, Washington University (1978), unpublished.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this paper
Cite this paper
Tarn, T.J., Clark, J.W., Huang, G.M. (1989). Local controllability of generalized quantum mechanical systems. In: Blaquiére, A. (eds) Modeling and Control of Systems. Lecture Notes in Control and Information Sciences, vol 121. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0041193
Download citation
DOI: https://doi.org/10.1007/BFb0041193
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50790-1
Online ISBN: 978-3-540-46087-9
eBook Packages: Springer Book Archive