Optimal control problems for the equation of motion of membrane with strong viscosity

Abstract

Optimal control problems are studied for the equation of membrane with strong viscosity. The Gâteaux differentiability of solution mapping on control variables is proved and the various types of necessary optimality conditions corresponding to the distributive and terminal values observations are established.

Parameter identification problem for the equation of motion of membrane with strong viscosity

“…We can prove that the solution map q → y(q) of P into W (0, T ) is Gâteaux differentiable. The following theorem gives the characterization of the Gâteaux derivatives as in [5].…”

“…In the previous paper [5], we studied the quadratic optimal control problems for (1.1) and established the necessary conditions for the costs of distributive and terminal values observations based on the well-posedness of weak solutions in [4].…”

“…For this we prove the strong Gâteaux differentiability of the nonlinear mapping q → y(q). We also emphasize that in the velocity's measurements case (1.3), a first order Volterra integro-differential equation is utilized as a proper adjoint system as in [5].…”

Weak and Strong Solutions for a Strongly Damped Quasilinear Membrane Equation

“…Besides, the well-posedness of less regular solutions is proved in [1], called weak solutions in the framework of the variational method in Dautray and Lions [3]. Based on these results, we have treated the associated optimal control and identification problems in [6] and [7], respectively. Furthermore, in [8] we have extended the results in [1] to more general quasilinear nonautonomous wave equation with strong damping term.…”

Existence and Uniqueness of a Solution in the Space of BV Functions to the Equation of a Vibrating Membrane with a “Viscosity” Term

“…On the other hand, the equation with the strong viscosity term −Δ is investigated by several authors. For example, in [5], it is investigated in the context of control theory, and it is asserted that if (0) ∈ is a smaller class than the space of BV functions, this suggests that the influence of the term −Δ is too strong. In this paper, replacing the strong viscosity term −Δ with −(div(∇ / √ 1 + |∇ | 2 )) , we investigate it in the space of BV functions.…”