Geodesic
On the optimal control problem governed by the equations of von Kármán. III. The case of an arbitrary large perpendicular load
Applications of Mathematics, Tome 32 (1987) no. 4, pp. 315-331
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We shall deal with an optimal control problem for the deffection of a thin elastic plate. We consider the perpendicular load on the plate as the control variable. In contrast to the papers [1], [2], arbitrarily large loads are edmitted. As the unicity of a solution of the state equation is not guaranteed, we consider the cost functional defined on the set of admissible controls and states, and the state equation plays the role of the constraint. The existence of an optimal couple (i.e., control and state) is verified. By using Lagrange multipliers, some necessary optimality conditions are derived. A control problem with the cost functional involving all possible solutions of the state equation for arbitrary perpendicular load-control is investigated in the last part. The optimal control problem is solved via a sequence of penalized optimal control problems.
We shall deal with an optimal control problem for the deffection of a thin elastic plate. We consider the perpendicular load on the plate as the control variable. In contrast to the papers [1], [2], arbitrarily large loads are edmitted. As the unicity of a solution of the state equation is not guaranteed, we consider the cost functional defined on the set of admissible controls and states, and the state equation plays the role of the constraint. The existence of an optimal couple (i.e., control and state) is verified. By using Lagrange multipliers, some necessary optimality conditions are derived. A control problem with the cost functional involving all possible solutions of the state equation for arbitrary perpendicular load-control is investigated in the last part. The optimal control problem is solved via a sequence of penalized optimal control problems.
DOI : 10.21136/AM.1987.104262
Classification : 49A22, 49B22, 49J20, 49K20, 73H05, 73K10, 74G60, 74K20, 74P99
Keywords: Lagrange multipliers; optimal control problem; system of von Kármán equations; deflection; thin elastic plate; perpendicular load; arbitrary large loads; existence proof; conditions of optimality 
@article{10_21136_AM_1987_104262,
     author = {Bock, Igor and Hlav\'a\v{c}ek, Ivan and Lov{\'\i}\v{s}ek, J\'an},
     title = {On the optimal control problem governed by the equations of von {K\'arm\'an.} {III.} {The} case of an arbitrary large perpendicular load},
     journal = {Applications of Mathematics},
     pages = {315--331},
     year = {1987},
     volume = {32},
     number = {4},
     doi = {10.21136/AM.1987.104262},
     mrnumber = {0897835},
     zbl = {0639.73028},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1987.104262/}
}
TY  - JOUR
AU  - Bock, Igor
AU  - Hlaváček, Ivan
AU  - Lovíšek, Ján
TI  - On the optimal control problem governed by the equations of von Kármán. III. The case of an arbitrary large perpendicular load
JO  - Applications of Mathematics
PY  - 1987
SP  - 315
EP  - 331
VL  - 32
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/AM.1987.104262/
DO  - 10.21136/AM.1987.104262
LA  - en
ID  - 10_21136_AM_1987_104262
ER  - 
%0 Journal Article
%A Bock, Igor
%A Hlaváček, Ivan
%A Lovíšek, Ján
%T On the optimal control problem governed by the equations of von Kármán. III. The case of an arbitrary large perpendicular load
%J Applications of Mathematics
%D 1987
%P 315-331
%V 32
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/AM.1987.104262/
%R 10.21136/AM.1987.104262
%G en
%F 10_21136_AM_1987_104262
Bock, Igor; Hlaváček, Ivan; Lovíšek, Ján. On the optimal control problem governed by the equations of von Kármán. III. The case of an arbitrary large perpendicular load. Applications of Mathematics, Tome 32 (1987) no. 4, pp. 315-331. doi: 10.21136/AM.1987.104262

[1] I. Bock I. Hlaváček J. Lovíšek: On the optimal control problem governed by the equations of von Kárman, I. The homogeneous Dirichlet boundary conditions. Aplikace mat. 29 (1984), 303-314. | MR

[2] I. Bock I. Hlaváček J. Lovíšek: On the optimal control problem governed by the equations of von Kárman. II. Mixed boundary conditions. Aplikace mat. 30 (1985), 375-392. | MR

[3] I. Hlaváček J. Neumann: In homogeneous boundary value problems for the von Kárman equations. I. Aplikace mat. 19 (1974), 253-269. | MR

[4] A. D. Joffe V. M. Tichomirov: The theory of extremal problems. (in Russian). Moskva, Nauka 1974. | MR

[5] O. John J. Nečas: On the solvability of von Kárman equations. Aplikace mat. 20 (1975), 48-62. | MR

[6] L. A. Ljusternik V. I. Sobolev: The elements of functional analysis. (in Russian). Moskva, Nauka 1965. | MR

[7] M. M. Vajnberg: Variational methods of investigating nonlinear operators. (in Russian). Moskva, Gostechizdat 1956.

[8] M. M. Vajnberg: Variational method and a method of monotone operators. (in Russian). Moskva, Nauka 1972.

Cité par Sources :