Identification of a Distributed Parameter (Conductivity) in a System Governed by a Parabolic Equation: Uniqueness and Stability Results

The parameter identification problem considered herewith consists of identifying the measurable, bounded and strictly positive position - dependent conductivity from interior potential measurements in one spatial dimension. At least one solution is assumed to exist. Since conductivity satisfies a first order ordinary differential equation (ODE), uniqueness follows from stating either a regular or a singular initial value problem. Regular problems arise when a Cauchy datum is available at a point where the coefficient of a
_x
does not vanish. Singular problems are met when the potential is stationary, according to the conditions originally given by Kitamura and Nakagiri [SIAM J. Control and Optimiz., 15, (1977), pp 785 - 802] or when the {potential; source term} pair satisfies a non - local property known as self-identifiability. Potentials shall be piecewise differentiable in all cases. When stability is examined, potentials are chosen in a subset of W
₂
^2,1
; the starting point is an ODE for the conductivity difference, the defect equation. Two distinct integration procedures are applied to the latter, according to whether uniqueness is due to a regular or a singular problem. Standard stability theory applies to the former and leads to L
¹
-estimates, provided Gronwall - Bellman's inequality is extended to measurable functions. Singular problems yield at most L
¹
- stability estimates. Some examples and counterexamples are provided. The results help in interpreting the performance of some known distributed parameter identification methods.