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Optimal design problems for a dynamic viscoelastic plate. I: Short memory material. (English) Zbl 0845.49001
Several optimal control problems with the same state problem – a pseudohyperbolic variational inequality with a linear operator – are considered. The control parameters appearing in the coefficients of the variational inequality as well as on the right-hand side are analyzed. The thickness of the dynamic viscoelastic plate with velocity constraints plays the role of control variable. Cost functionals correspond with adjusting the deflection (or the field moments) to a prescribed function \(z_d\) (with minimal cost). Existence of an optimal control is proven on the abstract level. Using the method of penalization an existence and uniqueness theorem for a solution of an initial-boundary value problem for a pseudohyperbolic variational inequality is proved.
49J20 Existence theories for optimal control problems involving partial differential equations
49J40 Variational inequalities
35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators
74Hxx Dynamical problems in solid mechanics
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[1] V. Barbu, T. Precupanu: Convexity and optimization. Sitjhoff-Noordhoff, Amsterdam, 1978. · Zbl 0379.49010
[2] I. Bock, J. Lovíšek: Optimal control of a viscoelastic plate bending. Mathematische Nachrichten 125 (1986), 135-151. · Zbl 0606.73104
[3] I. Bock, J. Lovíšek: An optimal control problem for a pseudoparabolic variational inequality. Applications of Mathematics 37 (1992), 62-80. · Zbl 0772.49008
[4] H. Brézis: Problémes uniltéraux. Journal de Math. Pures et Appliqué 51 (1968), 1-168.
[5] H. Brézis: Operateurs maximaux monotones et semigroupes. North Holland, Amsterdam, 1973.
[6] J. Brilla: Linear viscoelastic plate bending analysis. Proc. XI-th Congress of Applied Mechanics, München, 1964.
[7] H. Gajewski, K. Gröger, K. Zacharias: Nichlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie, Berlin, 1974. · Zbl 0289.47029
[8] J. Nečas, I. Hlaváček: Mathematical theory of elastic and elastoplastic bodies. An introduction, Elsevier, Amsterdam, 1981.
[9] O.R. Ržanicyn: Teoria polzučesti. Strojizdat, Moskva, 1968.
[10] D. Tiba: Some remarks on the control of the vibrating string with an obstacle. Revue Roumaine de Math. Pures, Appl. 29 (1984), 899-906. · Zbl 0547.49005
[11] D. Tiba: Optimal control of nonsmooth distributed parameter systems. Springer-Verlag, Berlin, 1990. · Zbl 0732.49002
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