Geodesic
Boundary control and a matrix inverse problem for the equation $u_{tt}-u_{xx}+V(x)u=0$
Sbornik. Mathematics, Tome 72 (1992) no. 2, pp. 287-310 Cet article a éte moissonné depuis la source Math-Net.Ru

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The authors solve the problem of recovering the matrix-valued potential $V(x)$, $x>0$, from the given reaction operator $R\colon u(0,t)\mapsto u_x(0,t)$, $t>0$. They show the connections between this problem and the theory of boundary control, which allows them to obtain analogues of the classical Gel'fand–Levitan–Krein equations. They establish the basis property for a family of vector-valued exponentials; this property is connected with the spectral characteristics of the boundary value problem. They prove the controllability of the corresponding system under a boundary control $u(0,t)=f(t)$. 
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     title = {Boundary control and a~matrix inverse problem for the equation $u_{tt}-u_{xx}+V(x)u=0$},
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S. A. Avdonin; M. I. Belishev; S. A. Ivanov. Boundary control and a matrix inverse problem for the equation $u_{tt}-u_{xx}+V(x)u=0$. Sbornik. Mathematics, Tome 72 (1992) no. 2, pp. 287-310. http://geodesic.mathdoc.fr/item/SM_1992_72_2_a0/

[1] Iskenderov A. D., “O variatsionnykh postanovkakh mnogomernykh obratnykh zadach matematicheskoi fiziki”, DAN SSSR, 274:3 (1984), 531–533 | MR | Zbl

[2] Anikonov Yu. E., Bubnov B. A., “Voprosy upravleniya i obratnye zadachi”, DAN SSSR, 304:2 (1989), 309–312 | MR | Zbl

[3] Belishev M. I., “Ob odnom podkhode k mnogomernym obratnym zadacham dlya volnovogo uravneniya”, DAN SSSR, 297:3 (1987), 524–527 | MR

[4] Belishev M. I., “Uravneniya tipa Gelfanda - Levitana v mnogomernoi obratnoi zadache dlya volnovogo uravneniya”, Zap. nauch. seminarov LOMI, 165:17 (1987), 15–20 | Zbl

[5] Belishev M. I., Blagoveschenskii A. S., “Pryamoi metod nestatsionarnoi obratnoi zadachi dlya mnogomernogo volnovogo uravneniya”, Uslovno-korrektnye zadachi matematicheskoi fiziki i analiza, Izd-vo Krasnoyarsk. un-ta, Krasnoyarsk, 1988, 42–49

[6] Belishev M. I., “Volnovye bazisy v mnogomernykh obratnykh zadachakh”, Matem. sb., 180 (222) (1989), 584–602 | MR

[7] Blagoveschenskii A. S., “O nesamosopryazhennoi obratnoi kraevoi zadache dlya giperbolicheskogo differentsialnogo uravneniya”, Probl. matem. fiziki, no. 5, 1971, 38–62

[8] Avdonin S. A., Ivanov S. A., “Porozhdayuschaya matritsa-funktsiya v zadachakh upravleniya kolebaniyami svyazannykh strun”, DAN SSSR, 307:5 (1989), 1033–1037 | Zbl

[9] Avdonin S. A., Ivanov S. A., Upravlyaemost sistem s raspredelennymi parametrami i semeistva eksponent, UMK VO, Kiev, 1989

[10] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1971 | MR | Zbl

[11] Levitan B. M., Obratnye zadachi Shturma-Liuvillya, Nauka, M., 1984 | MR

[12] Lions Zh.-L., Optimalnoe uravnenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Mir, M., 1972 | MR

[13] Butkovskii A. G., Metody upravleniya sistemami s raspredelennymi parametrami, Nauka, M., 1975 | Zbl

[14] Russell D. L., “Controllability and stabilizability theory for linear partial differential equations”, SIAM Review, 20:4 (1978), 639–739 | DOI | MR | Zbl

[15] Gelfand I. M., Levitan B. M., “Ob opredelenii differentsialnogo uravneniya po ego spektralnoi funktsii”, Izv. AN SSSR. Ser. matem., 15:4 (1951), 309–360 | MR | Zbl

[16] Atkinson F. V., Diskretnye i nepreryvnye granichnye zadachi, Mir, M., 1968 | MR | Zbl

[17] Gokhberg I. Ts., Krein M. G., Teoriya volterrovykh operatorov v gilbertovom prostranstve i ee prilozheniya, Nauka, M., 1967 | MR

[18] Smirnov V. I., Kurs vysshei matematiki, Gostekhizdat, M., 1947

[19] Nikolskii N. K., Lektsii ob operatore sdviga, Nauka, M., 1980 | MR

[20] Pavlov B. S., “Spektralnyi analiz differentsialnogo operatora s razmazannym granichnym usloviem”, Probl. matem. fiziki, no. 6, L., 1973, 101–119 | Zbl

[21] Pavlov B. S., “Bazisnost sistemy eksponent i uslovie Makenkhoupta”, DAN SSSR, 247:1 (1979), 37–40 | MR | Zbl

[22] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR