Optimal control of systems determined by strongly nonlinear operator valued measures

N.U. Ahmed

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2008)

  • Volume: 28, Issue: 1, page 165-189
  • ISSN: 1509-9407

Abstract

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In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪ H ↪ V* with continuous and dense embeddings where H is a separable Hilbert space and V is a reflexive Banach space with dual V*. The system is given by dx + A(dt,x) = f(t,x)γ(dt) + B(t)u(dt), x(0) = ξ, t ∈ I ≡ [0,T] where A is a strongly nonlinear operator valued measure mapping Σ × V to V* with Σ denoting the sigma algebra of subsets of the set I and f is a nonlinear operator mapping I × H to H, γ is a countably additive bounded positive measure and the control u is a suitable vector measure. We present existence, uniqueness and regularity properties of weak solutions and then prove the existence of optimal controls (vector valued measures) for a class of control problems.

How to cite

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N.U. Ahmed. "Optimal control of systems determined by strongly nonlinear operator valued measures." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 28.1 (2008): 165-189. <http://eudml.org/doc/271157>.

@article{N2008,
abstract = { In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪ H ↪ V* with continuous and dense embeddings where H is a separable Hilbert space and V is a reflexive Banach space with dual V*. The system is given by dx + A(dt,x) = f(t,x)γ(dt) + B(t)u(dt), x(0) = ξ, t ∈ I ≡ [0,T] where A is a strongly nonlinear operator valued measure mapping Σ × V to V* with Σ denoting the sigma algebra of subsets of the set I and f is a nonlinear operator mapping I × H to H, γ is a countably additive bounded positive measure and the control u is a suitable vector measure. We present existence, uniqueness and regularity properties of weak solutions and then prove the existence of optimal controls (vector valued measures) for a class of control problems. },
author = {N.U. Ahmed},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {evolution equations; strongly nonlinear operator valued measures; existence of solutions; regularity properties; optimal control; controlled systems in abstract spaces; measure valued syste},
language = {eng},
number = {1},
pages = {165-189},
title = {Optimal control of systems determined by strongly nonlinear operator valued measures},
url = {http://eudml.org/doc/271157},
volume = {28},
year = {2008},
}

TY - JOUR
AU - N.U. Ahmed
TI - Optimal control of systems determined by strongly nonlinear operator valued measures
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2008
VL - 28
IS - 1
SP - 165
EP - 189
AB - In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪ H ↪ V* with continuous and dense embeddings where H is a separable Hilbert space and V is a reflexive Banach space with dual V*. The system is given by dx + A(dt,x) = f(t,x)γ(dt) + B(t)u(dt), x(0) = ξ, t ∈ I ≡ [0,T] where A is a strongly nonlinear operator valued measure mapping Σ × V to V* with Σ denoting the sigma algebra of subsets of the set I and f is a nonlinear operator mapping I × H to H, γ is a countably additive bounded positive measure and the control u is a suitable vector measure. We present existence, uniqueness and regularity properties of weak solutions and then prove the existence of optimal controls (vector valued measures) for a class of control problems.
LA - eng
KW - evolution equations; strongly nonlinear operator valued measures; existence of solutions; regularity properties; optimal control; controlled systems in abstract spaces; measure valued syste
UR - http://eudml.org/doc/271157
ER -

References

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  1. [1] N.U. Ahmed, Differential inclusions, operator valued measures and optimal control, Special Issue of Dynamic Systems and Applications, Set-Valued Methods in Dynamic Systems, Guest Editors: M. Michta and J. Motyl 16 (2007), 13-36. 
  2. [2] N.U. Ahmed, Evolution equations determined by operator valued measures and optimal control, Nonlinear Analysis 67 (2007), 3199-3216. Zbl1119.49004
  3. [3] N.U. Ahmed, Vector and operator valued measures as controls for infinite dimensional systems: optimal control, Discuss. Math. Differential Inclusions Control and Optimization 28 (2008), 95-131. Zbl1181.28013
  4. [4] N.U. Ahmed, A class of semilinear parabolic and hyperbolic systems determined by operator valued measures, DCDIS 14 (4) (2007). 
  5. [5] N.U. Ahmed, Parabolic systems determined by strongly nonlinear operator valued measures, Nonlinear Analysis, Special Issue (Felicitation of Professor V. Lakshmikantham on his 85th birth date). 
  6. [6] N.U. Ahmed, Optimization and Identification of Systems Governed by Evolution Equations on Banach Spaces, Pitman Research Notes in Mathematics Series 184, Longman Scientific and Technical, U.K. and co-publisher John Wiley, New York, 1988. 
  7. [7] N.U. Ahmed, K.L. Teo and S.H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis 54 (2003), 907-925. Zbl1030.34056
  8. [8] N.U. Ahmed, Some remarks on the dynamics of impulsive systems in Banach spaces, DCDIS 8 (2001), 261-274. Zbl0995.34050
  9. [9] N.U. Ahmed, Impulsive perturbation of C₀-semigroups by operator valued measures, Nonlinear Functional Analysis & Applications 9 (1) (2004), 127-147. Zbl1053.34055
  10. [10] N.U. Ahmed, Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control. Optim. 42 (2) (2003), 669-685. Zbl1037.49036
  11. [11] J. Diestel and J.J. Uhl, Jr., Vector Measures, Mathematical Surveys, no. 15, American Mathematical Society, Providence, Rhode Island, 1977. 
  12. [12] N. Dunford and J.T. Schwartz, Linear Operators, Part 1: General Theory, Interscience Publishers, Inc., New York, London, 1958, 1964. Zbl0084.10402
  13. [13] H.O. Fattorini, Infinite Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications, Vol. 62, Cambridge University, Cambridge, U.K., 1999. Zbl0931.49001

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