In this paper, we consider the question of controllability of a class of linear and semilinear evolution equations on Hilbert space with measures as controls. We present necessary and sufficient conditions for weak and exact (strong) controllability of a linear system. Using this result we prove that exact controllability of the linear system implies exact controllability of a perturbed semilinear system. Controllability problem for the semilinear system is formulated as a fixed point problem on the space of vector measures and is concluded controllability from the existence of a fixed point. Our results cover impulsive controls as well as regular controls.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1052,
author = {N.U. Ahmed},
title = {Controllability of evolution equations and inclusions driven by vector measures},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
volume = {24},
year = {2004},
pages = {49-72},
zbl = {1076.93007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1052}
}
N.U. Ahmed. Controllability of evolution equations and inclusions driven by vector measures. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 24 (2004) pp. 49-72. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1052/
[000] [1] H.O. Fattorini, Some Remarks on Complete Controllability, SICON 4 (1966), 686-694. | Zbl 0168.34906
[001] [2] H.O. Fattorini, On Complete Controllability of Linear Systems, JDE 3 (1967), 391-402. | Zbl 0155.15903
[002] [3] H.O. Fattorini, Local Controllability of a Nonlinear Wave Equation, Mathematical Systems Theory 9 (1975), 3-45. | Zbl 0319.93009
[003] [4] H.O. Fattorini, Infinite Dimensional Optimization and Control Theory, Encyclopedia of mathematics and its applications, Cambridge University Press 1998.
[004] [5] R. Triggiani, A Note on the Lack of Exact Controllability and Optimization, SICON 15 (1977), 407-411. | Zbl 0354.93014
[005] [6] D.L. Russell, Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions, Siam Review 20 (1978), 639-739. | Zbl 0397.93001
[006] [7] N.U. Ahmed, Finite-Time Null Controllability for a Class of Linear Evolution Equations on a Banach Space with Control Constraints, J. Opt. Th. and Appl. 47 (2) (1985), 129-158. | Zbl 0549.49028
[007] [8] V. Lakshmikantham, D.D. Bainov and P.S. Simenov, Theory of Impulsive Differential Equations, World Scientific, 1989, Singapore, London.
[008] [9] N.U. Ahmed, Systems Governed by Impulsive Differential Inclusions on Hilbert Spaces, Nonlinear Analysis: TMA 45 (2001), 693-706. | Zbl 0995.34053
[009] [10] N.U. Ahmed, Necessary Conditions of Optimality for Impulsive Systems on Banach Spaces, Nonlinear Analysis: TMA 51 (2002), 409-424. | Zbl 1095.49513
[010] [11] N.U. Ahmed, Impulsive Perturbation of C₀ Semigroups and Evolution Inclusions, Nonlinear Funct. Anal. & Appl. 7 (4) (2002), 555-580. | Zbl 1037.35095
[011] [12] N.U. Ahmed, Existence of Optimal Controls for a General Class of Impulsive Systems on Banach Spaces, SIAM, Journal on Contr. and Optim. 42 (2) (2003), 665-685.
[012] [13] J. Diestel and J.J. Uhl, Jr., Vector Measures, AMS Mathematical Surveys 15 (1977), AMS, Providence, Rhode Island.
[013] [14] K. Yosida, Functional Analysis, (second edition), Springer-Verlag New York inc. 1968.
[014] [15] N.U. Ahmed, Semigroup Theory With Applications to Systems and Control, (1991), Pitman Research Notes in Mathematics Series, 246, Longman Scientific and Technical, U.K, Co-published with John Wiley, New York, USA.
[015] [16] S. Hu and N.S. Papageorgiou, Hand Book of Multivalued Analysis, Vol.1, Theory, Kluwer Academic Publishers, Dordrecht, Boston, London 1997.
[016] [17] M. Kisielewicz, M. Michta and J. Motyl, Set Valued Approach to Stochastic Control (I): Existence and Regularity Properties, Dynamic Systems and Applications 12 (2003), 405-432. | Zbl 1063.93047
[017] [18] M. Kisielewicz, M. Michta and J. Motyl, Set Valued Approach to Stochastic Control (II): Viability and Semimartingale Issues, Dynamic Systems and Applications 12 (2003), 433-466. | Zbl 1064.93042