Cite this article as:

Vu Nguyen Son Tung .. . Special Examples of Superstable Semigroups and Their Application in the Inverse Problems Theory. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2018, vol. 18, iss. 3, pp. 252-262. DOI: https://doi.org/10.18500/1816-9791-2018-18-3-252-262

# Special Examples of Superstable Semigroups and Their Application in the Inverse Problems Theory

Special examples of superstable (quasinilpotent) semigroups and their application in the theory of linear inverse problems for evolutionary equations are studied. The term “semigroup” means here the semigroup of bounded linear operators of class C 0 . The standard research scheme is used. The linear inverse problem with the final overdetermination in a Banach space for the evolution equation is considered. A special assumption is introduced, related to the superstability of the main evolutionary semigroup. For the inverse problem we establish the existence and uniqueness theorem of the solution. It is noted that the solution of the problem can be represented by the convergent Neumann series. To illustrate the general theory, we consider special examples of superstable semigroups that are generated by a one-dimensional streaming operator with absorption in the weighted Banachs pace of function son the ray .It is shown tha the reare many possibilities for choosing the absorption coefficient and the weight function, under which the superstability of the corresponding semigroup is guaranteed. The established results allow applying to a particular inverse problem for the transport equation with absorption on the ray. The applied approach can be extended to themultidimensional transport equation in an unbounded domain without the collision integral.

1. Prilepko A. I., Orlovsky D. G., Vasin I. A. Methods for Solving Inverse Problems in Mathematical Physics. New York ; Basel, Marcel Dekker, 2000. 723 p.

2. Prilepko A. I., Tikhonov I. V. Recovery of the nonhomogeneous term in an abstract evolution equation. Russian Acad. Sci. Izv. Math., 1995, vol. 44, iss. 2, pp. 373–394. DOI: http://dx.doi.org/10.1070/IM1995v044n02ABEH001602

3. Tikhonov I. V., Eidelman Yu. S. Problems on correctness of ordinary and inverse problems for evolutionary equations of a special form. Math. Notes, 1994, vol. 56, iss. 2, pp. 830– 839. DOI: https://doi.org/10.1007/BF02110743

4. Tikhonov I. V., Vu Nguyen Son Tung. The solvability of the inverse problem for the evolution equation with a superstable semigroup. RUDN Journal of MIPh, 2018, vol. 26, iss. 2, pp. 103–118 (in Russian). DOI: http://dx.doi.org/10.22363/2312-9735-2018-26-2-103-118

5. Balakrishnan A. V. On superstability of semigroups. Systems modelling and optimization: Proceedings of the 18th IFIP Conference on System Modelling and Optimization. Ser. Chapman & Hall/CRC Research Notes in Mathematics, CRC Press, 1999, pp. 12–19.

6. Balakrishnan A. V. Superstability of systems. Appl. Math. and Comput., 2005, vol. 164, iss. 2, pp. 321–326. DOI: https://doi.org/10.1016/j.amc.2004.06.052

7. Jian-Hua Chen, Wen-Ying Lu. Perturbation of nilpotent semigroups and application to heat exchanger equations. Appl. Math. Letters, 2011, vol. 24, pp. 1698–1701. DOI: https://doi.org/10.1016/j.aml.2011.04.023

8. Creutz D., Mazo M., Preda C. Superstability and finite time extinction for C 0 - Semigroups. arXiv:0907.4812v4 [math.FA], 12 p. Available at: https://arxiv.org/pdf/0907.4812.pdf (accessed 24 December 2013).

9. Kmit I., Lyul’ko N. Perturbations of superstable linear hyperbolic systems. arX-iv:1605.04703v3 [math.AP], 29 p. Available at: https://arxiv.org/pdf/1605.04703.pdf (ac- cessed 9 January 2018).

10. Krein S. G. Lineinye differentsial’nye uravneniia v banakhovom prostranstve [Linear differential equations in Banach space]. Moscow, Nauka, 1967, 464 p. (in Russian).

11. Pazy A. Semigroups of linear operators and applications to partial differential equations. New York, Springer, 1983. 279 p.

12. Engel K.-J., Nagel R. One-parameter semigroups for linear evolution equations. New York, Springer, 2000. 586 p.

13. Iskenderov A. D., Tagiev R. G. The inverse problem on the determination of right sides of evolution equations in Banach space. Nauchn. Tr., Azerb. Gos. Univ., 1979, iss 1, pp. 51–56 (in Russian).

14. Rundell W. Determination of an unknown non-homogeneous term in a linear partial differential equation from overspecified boundary data. Appl. Anal., 1980, vol. 10, iss. 3, pp. 231–242. DOI: https://doi.org/10.1080/00036818008839304

15. Eidelman. Yu. S. Krayevyye zadachi dlya differentsial’nykh uravneniy s parametrami. Avtoref. dis. kand. fiz.-mat. nauk [Boundary value problems for differential equations with parameters : Thesis Diss. Cand. Sci. (Phys. and Math.)]. Voronez, 1984. 16 p. (in Russian).

16. Orlovsky D. G. On a problem of determining the parameter of an evolution equation. Differ. Equ., 1990, vol. 26, iss. 9, pp. 1201–1207.

17. Tikhonov I. V., Vu Nguyen Son Tung. Formulas for an explicit solution of the model nonlocal problem associated with the ordinary transport equation. Yakutian Math. J., 2017, vol. 24, no. 1, pp. 57–73. DOI: https://doi.org/10.25587/SVFU.2017.1.8437