Regularized optimal control problem for a beam vibrating against an elastic foundation

Abstract

We deal with an optimal control problem governed by a nonlinear hyperbolic initial-boundary value problem describing the perpendicular vibrations of a clamped beam against a u elastic foundation. A variable thickness of a beam plays the role of a control variable. The original equation for the deflection is regularized in order to derive necessary optimality conditions

Regularized Optimal Design Problem for a Viscoelastic Plate Vibrating Against a Rigid Obstacle

“…In a similar way as in [5] or [6] the following theorem about Fréchet differentiability of the mapping e → u(e) can be verified. In order to derive necessary optimality conditions we assume that the cost functional…”

“…in an analogous way as in [5], where the control problem for an elastic beam vibrating against an elastic foundation of Winkler's type was considered. We remark that instead of the function g δ we can use any not negative nondecreasing function g ∈ C 2 (R) of the variable ω vanishing for ω ≤ 0 and equaled to 1 δ ω for ω ≥ δ.…”

“…The compactness method will be used in solving the minimum problem for a cost functional. We apply the approach from [5] in deriving the optimality conditions.…”

Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide

“…To derive necessary conditions for an optimal control, we would like to differentiate the map (ϕ, ψ) → T (ϕ, ψ). Since the map (ϕ, ψ) → T (ϕ, ψ) is not directly differentiable (see [28]), the idea here consists in approximating the map T (ϕ, ψ) by a family of maps T δ (ϕ, ψ) and replacing the obstacle problem given by (10) and (11) by the following smooth semilinear equation (see [12] and [27]):…”

“…Besides, if we want to design a membrane having a desired shape, we need to choose a suitable obstacle. In this case, the obstacle can be considered as a control, and the membrane as the state (see for example [11], [20], and [33]).…”