Absolute stability criteria for nonlinear operator equations

A. L. Likhtarnikov

Izvestiya. Mathematics , Tome 11 (1977) no. 5, pp. 1011-1029.

Résumé

Conditions are obtained for the stability in the large of solutions of nonlinear equations of the form \begin{equation} \frac{dx}{dt}=Ax+bu+f,\qquad u=\varphi(y,t),\quad y=Cx. \end{equation} Here $A$ is the infinitesimal generator of a semigroup of class $C_0$, the maps $b\colon U\to X$ and $C\colon X\to Y$ are bounded linear operators, and $U,X$ and $Y$ are (generally different) Hilbert spaces. The equations (1) describe a wide class of distributed parameter control systems. The results obtained have the following features: a) The stability conditions pertain not to an individual system but to classes of systems; the stability holds uniformly in a certain sense for all systems of a particular class (“absolute stability in a given class of nonlinearities”). b) For some classes of nonlinearities, the conditions are not only sufficient but necessary. Bibliography: 15 titles.

Bibliographie

[1] Popov V. M., Giperustoichivost avtomaticheskikh sistem, Nauka, M., 1970 | MR

[2] Yakubovich V. A., Metody issledovaniya nelineinykh sistem avtomaticheskogo upravleniya, gl. II, III, ed. R. P. Nelepin, Nauka, M., 1975, 74–180 | MR

[3] Yakubovich V. A., “Chastotnaya teorema v teorii upravleniya”, Sib. matem. zh., 14:2 (1973), 384–419

[4] Yakubovich V. A., “Neobkhodimye i dostatochnye usloviya absolyutnoi ustoichivosti pri nekotorykh tipakh kvadratichnykh ogranichenii”, Dokl. AN SSSR, 209:2 (1973), 312–315 | Zbl

[5] Yakubovich V. A., “Minimizatsiya kvadratichnykh funktsionalov pri kvadratichnykh ogranicheniyakh i neobkhodimost chastotnogo usloviya v kvadratichnom kriterii absolyutnoi ustoichivosti nelineinykh sistem upravleniya”, Dokl. AN SSSR, 209:5 (1973), 1039–1042 | Zbl

[6] Yakubovich V. A., “Chastotnaya teorema dlya sluchaya, kogda prostranstva sostoyanii i upravlenii – gilbertovy, i ee primenenie v nekotorykh zadachakh sinteza optimalnogo upravleniya, I”, Sib. matem. zh., 15:3 (1974), 639–668 ; “II”, Сиб. матем. ж., 16:5 (1975), 1081–1102 | Zbl | MR | Zbl

[7] Likhtarnikov A. L., Yakubovich V. A., “Chastotnaya teorema dlya silno nepreryvnykh odnoparametricheskikh polugrupp”, Izv. AN SSSR. Ser. matem., 41 (1977), 895–910

[8] Hsu C., “Stability analysis of reactor systems via Lyapunov's second method”, Trans. ASME, J. Basic Eng., ser. D, 89 (1967), 307–310

[9] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1963 | MR

[10] Daletskii Yu. P., Krein M. G., Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1969 | MR

[11] Vang P. K. S., “Usloviya ustoichivosti diffuzionnoi sistemy s nelineinymi obratnymi svyazyami”, Avtomatika i telemekhanika, 1968, no. 5, 10–23

[12] Smirnova V. B., Ustoichivost nekotorykh klassov sistem avtomaticheskogo regulirovaniya s raspredelennymi parametrami, Kand. dissertatsiya, LGU, 1975

[13] Agranovich M. S., Vishik M. I., “Ellipticheskie zadachi s parametrom i parabolicheskie zadachi obschego vida”, Uspekhi matem. nauk, XIX:3 (1964), 53–161

[14] Borisovich Yu. G., Turbabin A. S., “K zadache Koshi dlya lineinykh neodnorodnykh differentsialnykh uravnenii s zapazdyvayuschim argumentom”, Dokl. AN SSSR, 185:4 (1969), 741–744 | Zbl

[15] Milshtein G. N., “Eksponentsialnaya ustoichivost polozhitelnykh polugrupp v lineinom topologicheskom prostranstve, II”, Izv. VUZov. Ser. matem., 1975, no. 10, 51–61