On the existence and uniqueness of solutions of optimal control problems of linear distributed systems which are not solved with respect to the time derivative

M. V. Plekhanova   ; V. E. Fedorov

Izvestiya. Mathematics, Tome 75 (2011) no. 2, pp. 395-412

Résumé

We investigate optimal control problems for linear distributed systems which are not solved with respect to the time derivative and whose homogeneous part admits a degenerate strongly continuous solution semigroup. To this end, we first obtain theorems on the existence of a unique strong solution of the Cauchy problem. This enables us to formulate sufficient conditions for the solubility of the optimal control problems under consideration. In contrast to earlier papers on a similar topic, we substantially weaken the conditions on the quality functional with respect to the state function. The abstract results thus obtained are illustrated by an example of an optimal control problem for the linearized system of Navier–Stokes equations.

Détail

Keywords: optimal control problem, distributed system, equation of Sobolev type, degenerate operator semigroup, unique solubility.

Bibliographie

[1] R. E. Showalter, “Nonlinear degenerate evolution equations and partial differential equations of mixed type”, SIAM J. Math. Anal., 6:1 (1975), 25–42 | DOI | MR | Zbl

[2] G. A. Sviridyuk, V. E. Fëdorov, “On the identities of analytic semigroups of operators with kernels”, Siberian Math. J., 39:3 (1998), 522–533 | DOI | MR | Zbl

[3] V. E. Fedorov, “Degenerate strongly continuous semigroups of operators”, St. Petersburg Math. J., 12:3 (2001), 471–489 | MR | Zbl

[4] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev type equations and degenerate semigroups of operators, Inverse Ill-posed Probl. Ser., VSP, Utrecht, 2003 | MR | Zbl

[5] P. C. Müller, “Linear control design of linear descriptor systems”, 14th Triennial World Congress (Beijing), 1999, 31–36

[6] V. F. Chistyakov, A. A. Scheglova, Izbrannye glavy teorii algebro-differentsialnykh sistem, Nauka, Novosibirsk, 2003 | MR | Zbl

[7] G. A. Sviridyuk, A. A. Efremov, “Optimal control of Sobolev-type linear equations with relatively $p$-sectorial operators”, Differential Equations, 31:11 (1995), 1882–1890 | MR | Zbl

[8] G. A. Sviridyuk, A. A. Efremov, “Optimal control problem for a class of linear equations of Sobolev type”, Russian Math. (Iz. VUZ), 40:12 (1996), 60–71 | MR | Zbl

[9] G. A. Sviridyuk, A. A. Efremov, “Optimal control of a class of linear degenerate equations”, Dokl. Math., 59:1 (1999), 157–159 | MR | Zbl

[10] V. E. Fedorov, M. V. Plekhanova, “Optimal control of Sobolev type linear equations”, Differ. Equ., 40:11 (2004), 1627–1637 | DOI | MR | Zbl

[11] V. E. Fedorov, M. V. Plekhanova, “Issledovanie odnoi zadachi optimalnogo upravleniya”, Vestn. Chelyab. gos. ped. un-ta. Ser. 4. Estestv. nauki, 2005, no. 6, 23–31

[12] V. E. Fedorov, M. V. Plekhanova, “Slabye resheniya i problema kvadraticheskogo regulyatora dlya vyrozhdennogo differentsialnogo uravneniya v gilbertovom prostranstve”, Vychislitelnye tekhnologii, 9:2 (2004), 92–102 | Zbl

[13] M. V. Plekhanova, V. E. Fedorov, “Zadacha optimalnogo upravleniya dlya odnogo klassa vyrozhdennykh uravnenii”, Izv. RAN. Teoriya i sistemy upravleniya, 2004, no. 5, 40–44 | MR

[14] M. V. Plekhanova, V. E. Fedorov, “Zadachi startovogo upravleniya dlya lineinykh uravnenii sobolevskogo tipa”, Vestn. YuUrGU. Ser. Matem., fiz., khim., 2005, no. 6, 43–49

[15] A. V. Fursikov, Optimal control of distributed systems. Theory and applications, Transl. Math. Monogr., 187, Amer. Math. Soc., Providence, RI, 2000 | MR | Zbl | Zbl

[16] S. Mizohata, The theory of partial differential equations, Cambridge Univ. Press, London, 1973 | MR | Zbl

[17] G. A. Sviridyuk, G. A. Kuznetsov, “Relatively strongly $p$-sectorial linear operators”, Dokl. Math., 59:2 (1999), 298–300 | MR | Zbl