# Controllability of evolution equations and inclusions driven by vector measures

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2004)

- Volume: 24, Issue: 1, page 49-72
- ISSN: 1509-9407

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topN.U. Ahmed. "Controllability of evolution equations and inclusions driven by vector measures." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 24.1 (2004): 49-72. <http://eudml.org/doc/271479>.

@article{N2004,

abstract = {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.},

author = {N.U. Ahmed},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {controlability; impulsive systems; differential inclusions; Hilbert spaces; vector valued measures; C₀ semigroups; controllability; semilinear control systems; infinite-dimensional control systems; time-varying systems; countably additive nonnegative finite measure; Banach fixed point theorem},

language = {eng},

number = {1},

pages = {49-72},

title = {Controllability of evolution equations and inclusions driven by vector measures},

url = {http://eudml.org/doc/271479},

volume = {24},

year = {2004},

}

TY - JOUR

AU - N.U. Ahmed

TI - Controllability of evolution equations and inclusions driven by vector measures

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2004

VL - 24

IS - 1

SP - 49

EP - 72

AB - 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.

LA - eng

KW - controlability; impulsive systems; differential inclusions; Hilbert spaces; vector valued measures; C₀ semigroups; controllability; semilinear control systems; infinite-dimensional control systems; time-varying systems; countably additive nonnegative finite measure; Banach fixed point theorem

UR - http://eudml.org/doc/271479

ER -

## References

top- [1] H.O. Fattorini, Some Remarks on Complete Controllability, SICON 4 (1966), 686-694. Zbl0168.34906
- [2] H.O. Fattorini, On Complete Controllability of Linear Systems, JDE 3 (1967), 391-402. Zbl0155.15903
- [3] H.O. Fattorini, Local Controllability of a Nonlinear Wave Equation, Mathematical Systems Theory 9 (1975), 3-45. Zbl0319.93009
- [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. Zbl0354.93014
- [6] D.L. Russell, Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions, Siam Review 20 (1978), 639-739. Zbl0397.93001
- [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. Zbl0549.49028
- [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. Zbl0995.34053
- [10] N.U. Ahmed, Necessary Conditions of Optimality for Impulsive Systems on Banach Spaces, Nonlinear Analysis: TMA 51 (2002), 409-424. Zbl1095.49513
- [11] N.U. Ahmed, Impulsive Perturbation of C₀ Semigroups and Evolution Inclusions, Nonlinear Funct. Anal. & Appl. 7 (4) (2002), 555-580. Zbl1037.35095
- [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. Zbl1063.93047
- [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. Zbl1064.93042

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