# On exact controllability and complete stabilizability for linear systems

### Abstract

In this paper we consider linear systems with control described by the equation $\dot x = \mathcal A x +\mathcal B u$ where functions u and x take values in U and X respectively. For such object, a short review of results concerning relations between exact controllability and complete stabilizability (stabilizability with arbitrary decay rate) is given. The analysis is done in various situations: bounded or unbounded state and control operators $\mathcal A$ and $\mathcal B$, Banach or Hilbert spaces U and X. The well known equivalence between complete controllability and pole assignment in the situation of finite dimensional spaces is no longer true in general in infinite dimensional spaces. Exact controllability is not sufficient for complete stabilizability if U and X are Banach spaces. In Hilbert space setting this implication holds true. The converse also is not so simple: in some situations, complete stabilizability implies exact controllability (Banach space setting with bounded operators), in other situation, it is not true. The corresponding results are given with some ideas for the proofs. Complete technical development are indicated in the cited literature. Several examples are given. Special attention is paid to the case of infinite dimensional systems generated by delay systems of neutral type in some general form (distributed delays). The question of the relation between exact null controllability and complete stabilizability is more precisely investigated. In general there is no equivalence between the two notions. However for some classes of neutral type equations there is an equivalence. The question how the equivalence occurs for more general systems is still open. This is a short and non exhaustive review of some research on control theory for infinite dimensional spaces. Our works in this area were initiated by V. I. Korobov during the 70th of the past century in Kharkov State University.### Downloads

### References

A. D. Andrew, W. M. Patterson. Range inclusion and factorization of operators on classical Banach spaces. J. Math. Anal. Appl., - 1991. - 1. V. 156. - P. 40-43. DOI: 10.1016/0022-247X(91)90380-I.

P. Barkhayev, R. Rabah, G. Sklyar. Conditions of exact null controllability and the problem of complete stabilizability for time-delay systems. In Stabilization of Distributed Parameter Systems: Design Methods and Applications. Cham, G. Sklyar, A. Zuyev, eds., Springer Intern. Publ. -2021. - P. 1-15. DOI: 10.1007/978-3-030-61742-4_1.

R. Datko. Uniform asymptotic stability of evolutionary processes in a Banach space. SIAM J. Math. Anal., - 1972. - 3. V. 3. - P. 428-445. DOI: 10.1137/0503042.

R. G. Douglas. On majorization, factorization, and range inclusion of operators on Hilbert space. Proc. Amer. Math. Soc., - 1966. - 2. V. 17. - P. 413-415. DOI: 10.1090/S0002-9939-1966-0203464-1.

G. Eckstein. Exact controllability and spectrum assignment. In Topics in modern operator theory (Timi¸soara/Herculane, 1980), volume 2 of Operator Theory: Adv. Appl., - 1981. Birkh¨auser, Basel-Boston, Mass. - P. 81-94. DOI: 10.1007/978-3-0348-5456-6_7.

M. R. Embry. Factorization of operators on Banach space. Proc. Amer. Math. Soc., - 1973. - 3. V. 38. - P. 587-590. DOI: 10.2307/2038955.

M. A. Kaashoek, Cornelis V. M. van der Mee, Leiba Rodman. Analytic operator functions with compact spectrum. III: Hilbert space case: Inverse problem and applications. J. Oper. Theory, - 1983. -2. V. 10. - P. 219-250.

V. E. Khartovski˘ı, A. T. Pavlovskaya. Complete controllability and controllability for linear autonomous systems of neutral type. Automation and Remote Control, - 2013. - 5. V. 74. - P. 769-784. DOI: 10.1134/S0005117913050032.

V. Komornik. Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim., - 1997. - 5. V. 35. - P. 1591-1613. DOI: 10.1137/S0363012996301609.

V. I. Korobov, R. Rabah. Exact controllability in Banach space. Differential Equations, -1980. -12. V. 15. - P. 1531-1537. Translated from russian: Differentsialnye Uravnenia, -1980. -12. V. 15. - P. 2142-2150.

J.-C. Louis, D. Wexler. On exact controllability in Hilbert spaces. Journal of Differential Equations, - 1983. -2. V. 49. - P. 258-269. DOI: 10.1016/0022-0396(83)90014-1.

E. Makarov, M. Niezabitowski, S. Popova, V. Zaitsev, M. Zhuravleva. On Assignment of the Upper Bohl Exponent for Linear Discrete-Time Systems in Infinite-Dimensional Spaces. Proceedings of 25th International Conference on Methods and Models in Automation and Robotics (MMAR), - 2021. - P. 239-244. DOI: 10.1109/MMAR49549.2021.9528496.

A. S. Markus, V. R. Olshevsky. Complete controllability of spectrum assignment in infinite dimensional spaces. Integral Equations Oper. Theory, - 1993. - 1. V. 17. - P. 107-122, 1993. Translation of the paper published in russian: Matematicheskiye issledovaniya, Kishinev,- 1989, V. 106. - P. 97-113. DOI: 10.1007/bf01322549.

M. Megan, V. Hiri¸s. On the space of linear controllable systems in Hilbert spaces. Glasnik Mat. Ser. III, - 1975. - 1. V. 10(30). - P. 161-167.

A. V. Metel'skii, S. A. Minyuk. Criteria for the constructive identifiability and complete controllability of linear time-independent systems of neutral type. Izv. Ross. Akad. Nauk Teor. Sist. Upr., - 2006. - 5. - P. 15-23.

D. A. O’Connor, T. J. Tarn. On stabilization by state feedback for neutral differential equations. IEEE Trans. Automat. Control, - 1983. - 5. V. 28. - P. 615-618. DOI: 10.1109/TAC.1983.1103286.

A. W. Olbrot, L. Pandolfi. Null controllability of a class of functional differential systems. Int. J. Control, - 1988. - 1 V. 47. - P. 193-208. DOI: 10.1080/00207178808906006.

L. Pandolfi. Stabilization of neutral functional differential equations. J. Optimization Theory Appl., - 1976. - 2. V. 20. - P. 191-204. DOI: 10.1007/BF01767451.

R. Rabah. Commandabilit´e des syst`emes lin´eaires `a retard constant dans les espaces de Banach (Controllability of linear systems with constant delay in Banach spaces). RAIRO Automat.-Prod. Inform. Ind., -1986. - 6. V. 20. - P. 529-539.

R. Rabah, G. M. Sklyar, P. Yu. Barkhayev. On exact controllability of neutral time-delay systems. Ukrainian Math. Journal, - 2016. - 6. V. 68. - P. 800-815. DOI: 10.1007/s11253-016-1265-7.

R. Rabah, G. M. Sklyar, A. V. Rezounenko. Stability analysis of neutral type systems in Hilbert space. J. Differential Equations, - 2005. - 2. V. 214. - P. 391-428. DOI: 10.1016/j.jde.2004.08.001.

R. Rabah, G. M. Sklyar, A.V. Rezounenko. On strong regular stabilizability for linear neutral type systems. J. Differential Equations, - 2008. - 3. V. 245. - P. 569-593. DOI: 10.1016/j.jde.2008.02.041.

R. Rabah, J. Karrakchou. On Exact Controllability and Complete Stabilizability for Linear Systems in Hilbert Spaces. Applied Mathematics Letters, - 1997. - 1. V. 10. - P. 35-40. DOI: 10.1016/S0893-9659(96)00107-3.

R. Rabah, G. Sklyar, P. Barkhayev. Exact null controllability, complete stabilizability and continuous final observability of neutral type systems. Int. J. Appl. Math. Comput. Sci., - 2017. - 3. V. 27. - P. 489-499. DOI: 10.1515/amcs-2017-0034.

R. Rabah, G. M. Sklyar. The analysis of exact controllability of neutral-type systems by the moment problem approach. SIAM J. Control Optim., - 2007. - 6. V. 46. - P. 2148-2181. DOI: 10.1137/060650246.

R. Rabah, G. M. Sklyar, P. Yu. Barkhayev. Stability and stabilizability of mixed retarded-neutral type systems. ESAIM Control Optim. Calc. Var., - 2012. - 3. V. 18. - P. 656-692. DOI: 10.1051/cocv/2011166.

W. Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Inc., New York, 2d edition. 1991. 424 p.

G. M. Sklyar. The problem of the perturbation of an element of a Banach algebra by a right ideal and its application to the question of the stabilization of linear systems in Banach spaces. Vestnik Kharkiv University, - 1982. - 230. - P. 32-35. In Russian.

M. Slemrod. A note on complete controllability and stabilizability for linear control systems in Hilbert space. SIAM J. Control, - 1974. - 3. V. 12. - P. 500-508. DOI: 10.1137/0312038.

R. Triggiani. A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control Optim., - 1977. - 3. 15. - P. 407-411, DOI: 10.1137/0315028.

R. Triggiani. Addendum: “A note on the lack of exact controllability for mild solutions in Banach spaces” [SIAM J. Control Optim. 15 (1977), no. 3, 407–411; MR 55 #8942]. SIAM J. Control Optim., - 1980. - 1. V. 18. - P. 98-99, DOI: 10.1137/0318007.

J. van Neerven. The asymptotic behaviour of semigroups of linear operators. Operator theory advances and application. V. 88. Birkh¨auser, Basel. 1996. 234 p.

W. M. Wonham. Linear multivariable control: a geometric approach. Springer, New York, 3rd edition, 1985. 334 p.

Jerzy Zabczyk. Mathematical control theory: an introduction. Systems & Control: Foundations & Applications. 1992. Birkh¨auser Boston, Boston. 260 p.

Yi. Zeng, Z. Xie, F. Guo. On exact controllability and complete stabilizability for linear systems. Appl. Math. Lett., - 2013. - 7. 26. - P. 766-768, DOI: 10.1016/j.aml.2013.02.008.

*Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics*,

*94*, 4-23. https://doi.org/10.26565/2221-5646-2021-94-01

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