## The Solvability of the Inverse Problem for the Evolution Equation with a Superstable Semigroup

**Authors:**Tikhonov IV^{1}, Vu Nguyen S.T.^{2}**Affiliations:**- Lomonosov Moscow State University
- Moscow State University of Education

**Issue:**Vol 26, No 2 (2018)**Pages:**103-118**Section:**Mathematics**URL:**http://journals.rudn.ru/miph/article/view/18365**DOI:**https://doi.org/10.22363/2312-9735-2018-26-2-103-118- Cite item

# Abstract

For the evolution equation in a Banach space, the linear inverse source problem is studied. It is required to recover an unknown nonhomogeneous term by means of an additional nonlocal condition written in the form of a Riemann-Stieltjes integral. The main assumption is related to the superstability (quasinilpotency) of the evolution semigroup. More precisely, it is assumed that the evolutionary semigroup associated with the abstract differential equation has an infinite negative exponential type. Without other restrictions, a theorem on the solvability of the inverse problem is obtained. It is shown that the solution can be represented by a convergent Neumann series. Exact conditions under which an infinite series becomes a finite sum are found. Here, the algorithm for calculating the solution becomes finite. Model examples are considered, including an important example of the inverse problem with final overdetermination. The above results can be applicated in special parts of mathematical physics related to the theory of elasticity and the linear transport theory. As is customary, our research takes place in the general case with a choice of the complex scalar field, but the main facts are also true in the real case. The created theory allows transfer to nonlocal problems for evolution equations, when instead of the traditional initial condition special time averaging is used to find the solution.

# About the authors

### I V Tikhonov

Lomonosov Moscow State University
**Author for correspondence.**

Email: ivtikh@mail.ru

Leninskyie Gori, Moscow, 119991, Russian Federation

Doctor of Physical and Mathematical Sciences, professor of Department of Mathematical Physics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University

### Son Tung Vu Nguyen

Moscow State University of Education
Email: vnsontung@mail.ru

14 Krasnoprudnaya Str., Moscow, 107140, Russian Federation

Postgraduate Student of Mathematical Analysis Department, Moscow State University of Education

# References

- E. Hille, R. Phillips, Functional Analysis and Semigroups, IL, Moscow, 1962, in Russian.
- N. Dunford, J. Schwartz, Linear Operators. P. 1. General Theory, IL, Moscow, 1962, in Russian.
- S. G. Krein, Linear Differential Equations in Banach Space, Nauka, Moscow, 1967, in Russian.
- A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, N.Y., 1983.
- K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, N.Y., 2000.
- I. V. Tikhonov, Yu. S. Eidelman, Problems on Correctness of Ordinary and Inverse Problems for Evolutionary Equations of a Special Form, Math. Notes 56 (1994) 830–839, in Russian.
- A. I. Prilepko, I. V. Tikhonov, Recovery of the Nonhomogeneous Term in an Abstract Evolution Equation, Russian Acad. Sci. Izv. Math. 58 (1994) 167–188, in Russian.
- A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Basel, N.Y., 2000.
- I. V. Tikhonov, V. N. S. Tung, A Method of Solving the Inverse Problem for the Evolution Equation with a Superstable Semigroup, Journal Differential Equations and Control Processes (2) (2017) 51–58, in Russian.
- A. V. Balakrishnan, On Superstability of Semigroups, in: M. P. Polis, et al. (Eds.), Systems Modelling and Optimization, Proceedings of the 18th IFIP Conference on System Modelling and Optimization, CRC Research Notes in Mathematics, Chapman and Hall, 1999, pp. 12–19.
- A. V. Balakrishnan, Smart Structures and Super Stability, in: G. Lumer, L. Weis (Eds.), Evolution Equations and Their Applications in Physical and Life Sciences. Lecture Notes in Pure and Applied Mathematics, Vol. 215, 2001, pp. 43–53.
- A. V. Balakrishnan, Superstability of Systems, Applied Mathematics and Computation 164 (2005) 321–326.
- D. Creutz, M. Mazo, C. Preda, Superstability and Finite Time Extinction for C0– Semigroups, arXiv:0907.4812. Submitted (2013) 1–12.
- J.-H. Chen, W.-Y. Lu, Perturbation of Nilpotent Semigroups and Application to Heat Exchanger Equations, Applied Mathematics Letters 24 (2011) 1698–1701.
- I. Kmit, N. Lyul’ko, Perturbations of Superstable Linear Hyperbolic Systems, arXiv:1605.04703. Submitted (2017) 1–26.
- M. A. Krasnoselsky, G. M. Vainikko, P. P. Zabreiko, Y. B. Rutitsky, V. Y. Stetsenko, Approximate Solution of Operator Equations, Nauka, Moscow, 1969, in Russian.
- J. Malinen, O. Nevanlinna, J. Zem´anek, Microspectral analysis of quasinilpotent operators, arXiv:1211.4790v1. Submitted (2012) 1–25.
- R. Eskandari, F. Mirzapour, Hyperinvariant Subspaces and Quasinilpotent Operators, Bulletin of the Iranian Mathematical Society 41 (2015) 805–813.
- P. P. Zabreiko, On the Spectral Radius of Volterra Integral Operators, Lithuanian Math. Collection 7 (1967) 281–287, in Russian.
- E. S. Zhukovskii, On the Theory of Volterra Equations, Differential Equations 25 (1989) 1132–1137, in Russian.
- V. I. Sumin, A. V. Chernov, Operators in the Spaces of Measurable Functions: the Volterra Property and Quasinilpotency, Differential Equations 34 (1998) 1403–1411, in Russian.
- R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Mir, Moscow, 1998, in Russian.
- I. P. Natanson, Theory of Functions of a Real Variable, Nauka, Moscow, 1974, in Russian.