On the optimal control problem governed by the nonlinear elastic beam equation. (English) Zbl 1153.74030

Summary: We begin with using a dual variational method in proving the existence of solutions to a beam equation with Dirichlet boundary conditions and with a non-monotone nonlinear term. Later we investigate the dependence on a control parameter for the state equation. We find the optimal process for the optimal control problem in which the dynamics is governed by the nonlinear beam equation.


74M05 Control, switches and devices (“smart materials”) in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
49J20 Existence theories for optimal control problems involving partial differential equations
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[1] Amster, P.; Cárdenas Alzate, P.P., Existence of solutions for some nonlinear beam equations, Port. math. (N.S.), 63, 1, 113-125, (2006) · Zbl 1107.34010
[2] Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. math. anal. appl., 270, 2, 357-368, (2002) · Zbl 1006.34023
[3] Ekeland, I.; Temam, R., Convex analysis and variational problems, (1976), North-Holland Amsterdam
[4] Galewski, M., On the nonlinear elastic beam equation, Appl. math. comp., 202, 1, 427-434, (2008) · Zbl 1142.74021
[5] Ledzewicz, U.; Schättler, H.; Walczak, S., Optimal control systems governed by second-order ODEs with Dirichlet boundary data and variable parameters, Ill. J. math., 47, 4, 1189-1206, (2003) · Zbl 1031.49002
[6] Liu, X.-L.; Li, W.-T., Positive solutions of the nonlinear fourth-order beam equation with three parameters, J. math. anal. appl., 303, 1, 150-163, (2005) · Zbl 1077.34027
[7] Ma, T.F., Positive solutions for a beam equation on a nonlinear elastic foundation, Math. comput. modell., 39, 11-12, 1195-1201, (2004) · Zbl 1060.74035
[8] Rockafellar, R.T., Convex integral functionals and duality, (), 215-236 · Zbl 0326.49008
[9] Sanchez, L., Boundary value problems for some fourth order ordinary differential equations, Appl. anal., 38, 3, 161-177, (1990) · Zbl 0682.34020
[10] Yang, B., Positive solutions for the beam equation under certain boundary conditions, Electron. J. differen. equat., 78, (2005) · Zbl 1075.34025
[11] Yao, Q., Solvability of an elastic beam equation with caratheodory function, Math. appl. (Wuhan), 17, 3, 389-392, (2004) · Zbl 1063.34012
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