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Abstract
We consider a system of differential equations on a Banach space X given by: x'(t) = Ax(t) + u(t)f(t, x(t)), x(0) = x0, where A is an infinitesimal generator of a C0-semigroup, f : R0+ × X → X is a locally Lipschitz function, and u ∈ Lp([0, T], R) is a control defined on [0, T] with 1 < p ≤ ∞. Using the Compactness Principle and the generalization of Gronwalls Lemma, the system is shown to be controllable for a γ-bounded function f. Another result of this study is the local existence and the uniqueness of the solution of the system for locally bounded function f through weighted ω-norm.
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References
- Ball, J. M., Marsden, J. E., and Slemrod, M., Controllability for Distributed Bilinear System, SIAM J. Control and Optimization, 20, 1982
- Boussa¨ıd, N., Caponigro, M., and Chambrion, T., ”On the Ball-Marsden-Slemrod Obstruction for Bilinear Control System”, IEEE 58th Conference on Decision and Control (CDC), France, 2009.
- Boussa¨ıd, N., Caponigro, M., and Chambrion, T., ”Impulsive Control of Bilinear Schr¨odinger Equation: Propagators and Attainable Sets”, IEEE 58th Conference on Decision and Control (CDC), France, 2009.
- Thomas, C., and Laurent, T., Obstruction to The Bilinear Control of The Gross-Pitaevskii Equation: An Example with an Unbounded Potential, IFAC PapersOnLine, 52-16, 204-209, 2019.
- Gunther, D., Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture, arXiv:2111.10460v2, 2021.
- Jonas, L., A Remark on The Attainable Set of Schr¨odinger Equation, arXiv:1904.00591v2, 2019.
- Younes, L., Houssine El, M. E., and Noureddine, E., A New Generalization of Lemma Gronwall-Bellman, Applied Mathematical Sciences, 6, 621-628, 2012.
- Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
- Zeidler, E., Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems, Springer, New York, 1986
References
Ball, J. M., Marsden, J. E., and Slemrod, M., Controllability for Distributed Bilinear System, SIAM J. Control and Optimization, 20, 1982
Boussa¨ıd, N., Caponigro, M., and Chambrion, T., ”On the Ball-Marsden-Slemrod Obstruction for Bilinear Control System”, IEEE 58th Conference on Decision and Control (CDC), France, 2009.
Boussa¨ıd, N., Caponigro, M., and Chambrion, T., ”Impulsive Control of Bilinear Schr¨odinger Equation: Propagators and Attainable Sets”, IEEE 58th Conference on Decision and Control (CDC), France, 2009.
Thomas, C., and Laurent, T., Obstruction to The Bilinear Control of The Gross-Pitaevskii Equation: An Example with an Unbounded Potential, IFAC PapersOnLine, 52-16, 204-209, 2019.
Gunther, D., Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture, arXiv:2111.10460v2, 2021.
Jonas, L., A Remark on The Attainable Set of Schr¨odinger Equation, arXiv:1904.00591v2, 2019.
Younes, L., Houssine El, M. E., and Noureddine, E., A New Generalization of Lemma Gronwall-Bellman, Applied Mathematical Sciences, 6, 621-628, 2012.
Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
Zeidler, E., Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems, Springer, New York, 1986