Identification for degenerate problems of hyperbolic type

Abstract

In a Hilbert space X, we consider the degenerate problem (Mu) (0) = Mu ₀, where M and L are closed linear operators in X and M is not necessarily invertible. Given the additional information Φ [Mu(t)] = g(t), with Φ ∈ X*, g ∈ C ¹([0, τ]; ℝ) and z ∈ X, we are concerned with the determination of the conditions under which we can identify f ∈ C([0, τ]; ℝ) such that u be a strict solution to the above problem, i.e., Mu ∈ C ¹([0, τ]; X), Lu ∈ C([0, τ]; X). A similar identification problem is solved for an equation of second order in time. Examples for both situations are given.

Keywords:

• degenerate hyperbolic equations
• identification problems
• evolution equations

AMS Subject Classifications:

• 35L80
• 35R30
• 65N21

Acknowledgements

This work has been supported by the project PRIN 2008 ‘Analisi matematica nei problemi inversi per le applicazioni’ financed by MIUR. GM has been supported by the Grant CNCSIS PCCE-55/2008, ‘Sisteme diferentiale in analiza neliniara si aplicatii’.