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

TL;DR

This paper investigates the optimal initial velocity control of a vibrating beam governed by a hyperbolic PDE, establishing existence, uniqueness, and deriving conditions for optimality, along with a gradient-based numerical method.

Contribution

It introduces a novel variational approach to determine the optimal initial velocity for beam control, including explicit formulas and a gradient algorithm.

Findings

Proved existence and uniqueness of the optimal control.

Derived the Fréchet derivative and optimality conditions.

Outlined a gradient-based numerical solution method.

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 Fr\'echet 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.