## 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.

### MSC:

 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:

### References:

 [1] FAVINI, A., and LORENZI, A., Identification Problems in Singular Integro-Differential Equations of Parabolic Type I, Dynamics of Continuous, Discrete, and Impulsive Systems, Series A: Mathematical Analysis, Vol. 12, pp. 303–328, 2005. · Zbl 1081.45006 [2] AL HORANI, M. H., An Identification Problem for Some Degenerate Differential Equations, Le Matematiche, Vol. 57, pp. 217–227, 2002. · Zbl 1072.34055 [3] AL HORANI, M. H., and FAVINI, A., Degenerate First-Order Identification Problems in Banach Spaces, Differential Equations: Inverse and Direct Problems, Edited by A. Favini and A. Lorenzi, Taylor and Francis Group, pp. 1–15, New York, NY, 2006. · Zbl 1115.34012 [4] ASANOV, A., and ATAMANOV, E.R., Nonclassical and Inverse Problems for Pseudoparabolic Equations, VSP, Utrecht, Holland, 1997. · Zbl 0912.35155 [5] LORENZI, A., Introduction to Identification Problems via Functional Analysis, VSP, Utrecht, Holland, 2001. [6] PRILEPKO, A., ORLOVSKY, D. G., and VASIN, I., Methods for Solving Inverse Problems in Mathematical Physics, Dekker, New York, NY, 2000. · Zbl 0947.35173 [7] FAVINI, A., and LORENZI, A., Identification Problems for Singular Integro-Differential Equations of Parabolic Type II, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 56, pp. 879–904, 2004. · Zbl 1048.45009 [8] FAVINI, A., and LORENZI, A., Singular Integro-Differential Equations of Parabolic Type and Inverse Problems, Mathematical Models and Methods in Applied Sciences, Vol. 13, pp. 1745–1766, 2003. · Zbl 1059.45010 [9] AL HORANI, M. H., and FAVINI, A., Degenerate Second-Order Identification Problem in Banach Spaces, Journal of Optimization Theory and Applications. Vol. 120, pp. 305–326, 2004. · Zbl 1048.93020 [10] TRIEBEL, H., Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, Holland, 1978. · Zbl 0387.46032 [11] FAVINI, A., and YAGI, A., Degenerate Differential Equations in Banach Spaces, Dekker, New York, NY, 1999. · Zbl 0913.34001 [12] LUNARDI, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, Switzerland, 1995. · Zbl 0816.35001 [13] YOSIDA, K., Functional Analysis, Springer Verlag, Berlin, Germany, 1969. · Zbl 0126.11504 [14] FAVINI, A., LORENZI, A., TANABE, H., and YAGI, A., An L p -Approach to Singular Linear Parabolic Equations in Bounded Domains, Osaka Journal Mathematics, Vol. 42, pp. 385–406, 2005. · Zbl 1082.35073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.