Opuscula Math. 36, no. 2 (2016), 239-252
Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation
Abstract. Our goal is to study the Pareto optimal control system for a nonlinear one-dimensional extensible beam equation and its Galerkin approximation. First we consider a mathematical model of the beam equation which was obtained by S. Woinowsky-Krieger in 1950. Next we consider the Pareto optimal control problem based on this equation. Further, we describe the approximation of this system. We use the Galerkin method to approximate the solution of this control problem with respect to a spatial variable. Based on the standard finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation point of the solutions of the discrete optimal control problems exist and each of these points is the solution of the original Pareto optimal control problem.
Keywords: nonlinear beam equation, Pareto optimal control, Galerkin approximation.
Mathematics Subject Classification: 49J20, 49M25, 58E17.
- A.S. Ackleh, H.T. Banks, G.A. Pinter, A nonlinear beam equation, Appl. Math. Letters 15 (2002), 381-387.
- N. Arada, E. Casas, F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comp. Optim. Applic. 23 (2002), 201-229.
- J. Awrejcewicz, O.A. Soltykova, Yu.B. Chebotyrevskiy, V.A. Krysko, Nonlinear vibrations of the Euler-Bernoulli beam subjected to transversal load and impact actions, Nonlinear Stud. 18 (2011) 3, 329-364.
- J.M. Ball, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl. 42 (1973), 61-90.
- M. Barboteu, M. Sofonea, L.D. Tiba, The control variational method for beams in contact with deformable obstacles, ZAMM 92 (2012) 1, 25-40.
- Z. Denkowski, S. Migórski, N. Papageorgiou, Nonlinear Analysis. Applications, Kluwer Academic Publishers, Boston-Dordrecht-London, 2002.
- A. Dębinska-Nagórska, A. Just, Z. Stempień, Approximation of an optimal control problem governed by a differential parabolic inclusion, Optim. 59 (2010) 5, 707-715.
- A. Dębinska-Nagórska, A. Just, Z. Stempień, Galerkin Metod for Optimal Control of Second-Order Evolution Eqations, Math. Meth. Appl. Sci. 27 (2004), 221-230.
- M. Galewski, On the optimal control problem governed by the nonlinear elastic beam equation, Appl. Math. Comput. 203 (2008), 916-920.
- I. Hlaváček, J. Lovišek, Optimal control of semi-coercive variational inequalities with application to optimal design of beams and plates, ZAMM 78 (1998), 405-417.
- J. Hwang, Optimal control problems for an extensible beam equation, J. Math. Anal. Appl. 353 (2009), 436-448.
- E. Kącki, Z. Stempień, About approximation of optimal control of a certain process described by a partial differential equation of fourth order, Postępy Cybernetyki 7 (1984) 2, 83-91.
- W. Kotarski, Some problems of optimal and Pareto optimal control for distributed parameter systems, Scientific Publications of the University of Silesia, No. 1668 (1997).
- I. Lasiecka, Galerkin approximation of infinite-dimensional compensators for flexible structure with unbounded control action, Acta Appl. Math. 28 (1992) 2, 101-133.
- J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg-New York (1971), (Russian Edition, Mir, Moscow 1972).
- M.L. Oliveira, O.A. Lima, Exponential decay of the solutions of the beam system, Nonlinear Anal. 42 (2000), 1271-1291.
- D.C. Pereira, Existence, uniqueness and asymptotic behaviour for solutions of the nonlinear beam equation, Nonlinear Anal. 14 (1990), 613-623.
- I. Sadek, M. Abukhaled, T. Abdulrub, Coupled Galerkin and parametrization methods for optimal control of discretely connected parallel beams, Appl. Math. Modelling 34 (2010), 3949-3957.
- F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic bounduary control problems - strong convergence of optimal controls, Appl. Math. Optimz. 29 (1994), 309-329.
- S. Woinowsky-Krieger, The effect of axial force on the vibration of hinged bars, J. Appl. Mech. 17 (1950), 35-36.
- E. Zeidler, Nonlinear Functional Analysis and its Applications, Part II, Springer-Verlag, Berlin, 1990.
- Communicated by Marek Galewski.
- Received: 2015-07-09.
- Accepted: 2015-08-11.
- Published online: 2015-12-18.