**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**32127

##### Stability of Stochastic Model Predictive Control for Schrödinger Equation with Finite Approximation

**Authors:**
Tomoaki Hashimoto

**Abstract:**

**Keywords:**
Optimal control,
stochastic systems,
quantum systems,
stabilization.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1128115

**References:**

[1] A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle and G. Gerber, Control of Chemical Reactions by Feedback-Optimized Phase-Shaped Femtosecond Laser Pulses, Science, Vol. 282, 1998, pp.919-922.

[2] T. Brixner, N.H. Damrauer, P. Niklaus and G. Gerber, Photoselective Adaptive Femtosecond Quantum Control in the Liquid Phase, Nature, Vol. 414, 2001, pp.57-60.

[3] T. Weinacht, J. Ahn and P. Bucksbaum, Controlling the Shape of a Quantum Wavefunction, Nature, Vol. 397, 1999, pp.233-235.

[4] H. Rabitz, R. de Vivie-Riedle, M. Motzkus and K. Kompa, Whither the Future of Controlling Quantum Phenomena?, Science, Vol. 288, 2000, pp.824-828.

[5] L. I. Schiff, Quantum Mechanics, Mcgraw-Hill College, 3rd Edition, 1968.

[6] D. J. Griffiths, Introduction to Quantum Mechanics, Prentice Hall, 2nd Edition, 2005.

[7] M. Mirrahimi, P. Rouchon and G. Turinici, Lyapunov control of bilinear Schr¨odinger equations, Automatica, Vol. 41, 2005, pp.1987-1994.

[8] K. Beauchard, J.-M. Coron, M. Mirrahimi and P. Rouchon, Implicit Lyapunov Control of Finite Dimensional Schr¨odinger Equations, Systems & Control Letters, Vol. 56, 2007, pp.388-395.

[9] M. Mirrahimi and R. Van Handel, Stabilizing Feedback Controls for Quantum Systems, SIAM Journal on Control and Optimization, Vol. 46, 2007, pp.445-467.

[10] X. Wang and S. G. Schirmer, Analysis of Effectiveness of Lyapunov Control for Non-Generic Quantum States, IEEE Transactions on Automatic Control, Vol. 55, 2010, pp.1406-1411.

[11] B.-Z. Guo and K.-Y. Yang, Output Feedback Stabilization of a One-Dimensional Schr¨odinger Equation by Boundary Observation With Time Delay, IEEE Transactions on Automatic Control, Vol. 55, 2010, pp.1226-1232.

[12] M. Krstic, B.-Z. Guo and A. Smyshlyaev, Boundary Controllers and Observers for the Linearized Schr¨odinger Equation, SIAM Journal on Control and Optimization, Vol. 49, 2011, pp.1479-1497.

[13] D. Alessandro and M. Dahleh, Optimal Control of Two-Level Quantum Systems, IEEE Transactions on Automatic Control, Vol. 46, 2001, pp.866-876.

[14] L. Baudouina and J. Salomonb, Constructive Solution of a Bilinear Optimal Control Problem for a Schr¨odinger Equation, Systems & Control Letters, Vol. 57, 2008, pp.453-464.

[15] S. Grivopoulos and B. Bamieh, Optimal Population Transfers in a Quantum System for Large Transfer Time, IEEE Transactions on Automatic Control, Vol. 53, 2008, pp.980-992.

[16] T. Hashimoto, Y. Yoshioka, T. Ohtsuka, Receding Horizon Control with Numerical Solution for Thermal Fluid Systems, Proceedings of SICE Annual Conference, pp. 1298-1303, 2012.

[17] T. Hashimoto, Y. Yoshioka, T. Ohtsuka, Receding Horizon Control with Numerical Solution for Spatiotemporal Dynamic Systems, Proceedings of IEEE Conference on Decision and Control, pp. 2920-2925, 2012.

[18] T. Hashimoto, Y. Yoshioka and T. Ohtsuka, Receding Horizon Control for Hot Strip Mill Cooling Systems, IEEE/ASME Transactions on Mechatronics, Vol. 18, No. 3, pp. 998-1005, 2013.

[19] T. Hashimoto, Y. Yoshioka and T. Ohtsuka, Receding Horizon Control With Numerical Solution for Nonlinear Parabolic Partial Differential Equations, IEEE Transactions on Automatic Control, Vol. 58, No. 3, pp. 725-730, 2013.

[20] T. Hashimoto, Y. Takiguchi and T. Ohtsuka, Receding Horizon Control for High-Dimensional Burgersf Equations with Boundary Control Inputs, Transactions of the Japan Society for Aeronautical and Space Sciences, Vol. 56, No.3, pp. 137-144, 2013.

[21] T. Hashimoto, Y. Takiguchi and T. Ohtsuka, Output Feedback Receding Horizon Control for Spatiotemporal Dynamic Systems, Proceedings of Asian Control Conference, 2013.

[22] R. Satoh, T. Hashimoto and T. Ohtsuka, Receding Horizon Control for Mass Transport Phenomena in Thermal Fluid Systems, Proceedings of Australian Control Conference, pp. 273-278, 2014.

[23] T. Hashimoto, Receding Horizon Control for a Class of Discrete-time Nonlinear Implicit Systems, Proceedings of IEEE Conference on Decision and Control, pp. 5089-5094, 2014.

[24] T. Hashimoto, Optimal Feedback Control Method Using Magnetic Force for Crystal Growth Dynamics, International Journal of Science and Engineering Investigations, Vol. 4, Issue 45, pp. 1-6, 2015.

[25] T. Hashimoto, R. Satoh and T. Ohtsuka, Receding Horizon Control for Spatiotemporal Dynamic Systems, Mechanical Engineering Journal, Vol. 3, No. 2, 15-00345, 2016.

[26] T. Hashimoto, I. Yoshimoto, T. Ohtsuka, Probabilistic Constrained Model Predictive Control for Schr¨odinger Equation with Finite Approximation, Proceedings of SICE Annual Conference, pp. 1613-1618, 2012.

[27] T. Hashimoto, Probabilistic Constrained Model Predictive Control for Linear Discrete-time Systems with Additive Stochastic Disturbances, Proceedings of IEEE Conference on Decision and Control, pp. 6434-6439, 2013.

[28] T. Hashimoto, Computational Simulations on Stability of Model Predictive Control for Linear Discrete-time Stochastic Systems, International Journal of Computer, Electrical, Automation, Control and Information Engineering, Vol. 9, No. 8, pp. 1385-1390, 2015.

[29] T. Hashimoto, Conservativeness of Probabilistic Constrained Optimal Control Method for Unknown Probability Distribution, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, Vol. 9, No. 9, pp. 11-15, 2015.

[30] T. Hashimoto, A Method for Solving Optimal Control Problems subject to Probabilistic Affine State Constraints for Linear Discrete-time Uncertain Systems, International Journal of Mechanical and Production Engineering, Vol. 3, Issue 12, pp. 6-10, 2015.

[31] T. Hashimoto, Solutions to Probabilistic Constrained Optimal Control Problems Using Concentration Inequalities, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, Vol. 10, No. 10, pp. 441-446, 2016.

[32] S. Boucheron, G. Lugosi and P. Massart Concentration Inequalities: A Nonasymptotic Thepry of Independence, Oxford University Press, 2013.

[33] J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Advances in Computational Mathematics, Vol. 6, 1996, pp. 207-226.

[34] B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, Springer, 6th edition, 2010.

[35] J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operation Research and Financial Engineering, Springer, 2006.

[36] L. E. Ghaoui and S. I. Niculescu, Advances in Linear Matrix Inequality Methods in Control, Society for Industrial and Applied Mathematics, 1987.

[37] T. Hashimoto, T. Amemiya and H. A. Fujii, Stabilization of Linear Uncertain Delay Systems with Antisymmetric Stepwise Configurations, Journal of Dynamical and Control Systems, Vol. 14, No. 1, pp. 1-31, 2008.

[38] T. Hashimoto, T. Amemiya and H. A. Fujii, Output Feedback Stabilization of Linear Time-varying Uncertain Delay Systems, Mathematical Problems in Engineering, Vol. 2009, Article ID. 457468, 2009.

[39] T. Hashimoto and T. Amemiya, Stabilization of Linear Time-varying Uncertain Delay Systems with Double Triangular Configuration, WSEAS Transactions on Systems and Control, Vol. 4, No.9, pp.465-475, 2009.

[40] T. Hashimoto, Stabilization of Abstract Delay Systems on Banach Lattices using Nonnegative Semigroups, Proceedings of the 50th IEEE Conference on Decision and Control, pp. 1872-1877, 2011.

[41] T. Hashimoto, A Variable Transformation Method for Stabilizing Abstract Delay Systems on Banach Lattices, Journal of Mathematics Research, Vol. 4, No. 2, pp.2-9, 2012.

[42] T. Hashimoto, An Optimization Algorithm for Designing a Stabilizing Controller for Linear Time-varying Uncertain Systems with State Delays, Computational Mathematics and Modeling, Vol.24, No.1, pp.90-102, 2013.