Alemdar Hasanov 2009 Inverse Problems 25 115015 doi:10.1088/0266-5611/25/11/115015
Alemdar Hasanov
Show affiliationsInverse problems of determining the unknown source term F(x, t) in the cantilevered beam equation utt = (EI(x)uxx)xx + F(x, t) from the measured data μ(x) := u(x, T) or ν(x) := ut(x, T) at the final time t = T are considered. In view of weak solution approach, explicit formulae for the Fréchet gradients of the cost functionals J1(F) = ||u(x, T; w) − μ(x)||20 and J2(F) = ||ut(x, T; w) − ν(x)||20 are derived via the solutions of corresponding adjoint (backward beam) problems. The Lipschitz continuity of the gradients is proved. Based on these results the gradient-type monotone iteration process is constructed. Uniqueness and ill-conditionedness of the considered inverse problems are analyzed.
46.40.-f Vibrations and mechanical waves
46.70.De Beams, plates and shells
02.60.Lj Ordinary and partial differential equations; boundary value problems
35R30 Inverse problems (undetermined coefficients, etc.) for PDE
35L20 Boundary value problems for second-order, hyperbolic equations
74B15 Equations linearized about a deformed state (small deformations superposed on large)
Issue 11 (November 2009)
Received 27 June 2009, in final form 13 September 2009
Published 29 October 2009
Alemdar Hasanov 2009 Inverse Problems 25 115015
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