An identification problem for first-order degenerate differential equations. (English) Zbl 1129.65044

The paper concerns the inverse problem for the degenerate differential equation of first order of the following form:
\[ \begin{aligned} {d\over{dt}}(Mu)+Lu=f(t)z&, \quad 0\leq t\leq\tau,\\ (Mu)(0)=Mu_0&,\\ \Phi[Mu(t)]=g(t)&, \quad 0\leq t\leq\tau, \end{aligned} \]
where \(M,L\) are closed linear operators in a reflexive Banach space \(X\), \(\Phi\in X^*\) and \(L\) is invertible. The considered identification problem consists in determining \(u\), and \(f\) under some regularity assumptions, satisfying the above differential equation with given \(u_0\) and \(g\) satisfying the compatibility condition \(\Phi[Mu_0]=g(0)\). A question of solvability of such a identification problem is investigated.
The main results concern the existence of a unique solution under assumptions on \(M\) and \(L\) less restrictive than those in recent publications [for example M. Al Horani, Matematiche 57, No. 2, 217–227 (2002; Zbl 1072.34055)]. Finally, the abstract results are applied to 3 examples of concrete identification problems for partial differential equations.


65J22 Numerical solution to inverse problems in abstract spaces
34G10 Linear differential equations in abstract spaces
34A55 Inverse problems involving ordinary differential equations
65L09 Numerical solution of inverse problems involving ordinary differential equations


Zbl 1072.34055
Full Text: DOI


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