Geodesic
Controllability of evolution equations and inclusions driven by vector measures
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 24 (2004) no. 1, pp. 49-72

Voir la notice de l'article provenant de la source Library of Science

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.
Keywords: controlability, impulsive systems, differential inclusions, Hilbert spaces, vector valued measures, C₀ semigroups 
Ahmed, N.U. Controllability of evolution equations and inclusions driven by vector measures. Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 24 (2004) no. 1, pp. 49-72. http://geodesic.mathdoc.fr/item/DMDICO_2004_24_1_a4/
@article{DMDICO_2004_24_1_a4,
     author = {Ahmed, N.U.},
     title = {Controllability of evolution equations and inclusions driven by vector measures},
     journal = {Discussiones Mathematicae. Differential Inclusions, Control and Optimization},
     pages = {49--72},
     year = {2004},
     volume = {24},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMDICO_2004_24_1_a4/}
}
TY  - JOUR
AU  - Ahmed, N.U.
TI  - Controllability of evolution equations and inclusions driven by vector measures
JO  - Discussiones Mathematicae. Differential Inclusions, Control and Optimization
PY  - 2004
SP  - 49
EP  - 72
VL  - 24
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/DMDICO_2004_24_1_a4/
LA  - en
ID  - DMDICO_2004_24_1_a4
ER  - 
%0 Journal Article
%A Ahmed, N.U.
%T Controllability of evolution equations and inclusions driven by vector measures
%J Discussiones Mathematicae. Differential Inclusions, Control and Optimization
%D 2004
%P 49-72
%V 24
%N 1
%U http://geodesic.mathdoc.fr/item/DMDICO_2004_24_1_a4/
%G en
%F DMDICO_2004_24_1_a4

[1] H.O. Fattorini, Some Remarks on Complete Controllability, SICON 4 (1966), 686-694.

[2] H.O. Fattorini, On Complete Controllability of Linear Systems, JDE 3 (1967), 391-402.

[3] H.O. Fattorini, Local Controllability of a Nonlinear Wave Equation, Mathematical Systems Theory 9 (1975), 3-45.

[4] H.O. Fattorini, Infinite Dimensional Optimization and Control Theory, Encyclopedia of mathematics and its applications, Cambridge University Press 1998.

[5] R. Triggiani, A Note on the Lack of Exact Controllability and Optimization, SICON 15 (1977), 407-411.

[6] D.L. Russell, Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions, Siam Review 20 (1978), 639-739.

[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.

[8] V. Lakshmikantham, D.D. Bainov and P.S. Simenov, Theory of Impulsive Differential Equations, World Scientific, 1989, Singapore, London.

[9] N.U. Ahmed, Systems Governed by Impulsive Differential Inclusions on Hilbert Spaces, Nonlinear Analysis: TMA 45 (2001), 693-706.

[10] N.U. Ahmed, Necessary Conditions of Optimality for Impulsive Systems on Banach Spaces, Nonlinear Analysis: TMA 51 (2002), 409-424.

[11] N.U. Ahmed, Impulsive Perturbation of C₀ Semigroups and Evolution Inclusions, Nonlinear Funct. Anal. Appl. 7 (4) (2002), 555-580.

[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.

[13] J. Diestel and J.J. Uhl, Jr., Vector Measures, AMS Mathematical Surveys 15 (1977), AMS, Providence, Rhode Island.

[14] K. Yosida, Functional Analysis, (second edition), Springer-Verlag New York inc. 1968.

[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.

[16] S. Hu and N.S. Papageorgiou, Hand Book of Multivalued Analysis, Vol.1, Theory, Kluwer Academic Publishers, Dordrecht, Boston, London 1997.

[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.

[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.