On the optimal control of initial velocity in a hyperbolic beam equation by the variational method

TL;DR

The paper addresses controlling the initial velocity in a hyperbolic Euler–Bernoulli beam equation to drive the beam displacement $u$ toward a target $y$ in $L^2(\Omega)$. It builds a variational framework, proves existence and uniqueness of the forward weak solution and of the optimal control, and derives the Fréchet gradient of the cost functional via an adjoint problem, with $J'_{\alpha}(v) = -\psi(\cdot,0) + 2\alpha v$ where $\psi$ solves $\psi_{tt}+(k(x)\psi_{xx})_{xx}=-2[u-y]$. A gradient-type numerical method is proposed, using Galerkin discretization to compute forward and adjoint states and a Lipschitz constant $\mathcal{L}$ to ensure stable step sizes. These results yield a rigorous and implementable framework for initial-condition control of beam vibrations, with potential extensions to higher-regularity control spaces.

Abstract

We study the problem of controlling the initial condition of a vibrating beam. The optimal control problem seeks to determine solutions of initial velocity that assure the approach of the state of the beam to a given target function in the $L^2-$norm. We prove both the existence and uniqueness of the optimal solution. Employing identities based on the adjoint and difference problems, we determine the Fréchet derivative of the cost functional. We further derive the necessary optimality conditions of this control problem. Finally, we provide a sketch of a gradient-based algorithm, that rests on the explicit formula of the gradient of the cost functional, to obtain numerical solutions.

Theorems & Definitions (13)

• Definition 1
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• Definition 6
• Lemma 1
• Theorem 1
• Proposition 1
• Lemma 2