Boundary Control Of Wave Equation Research Articles

Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary

Much of the boundary control of wave equations in one dimension is based on a single principle—passivity—under the assumption that control is applied through Neumann actuation on one boundary and the other boundary satisfies a homogeneous Dirichlet boundary condition. We have recently expanded the scope of tractable problems by allowing destabilizing anti-stiffness (a Robin type condition) on the uncontrolled boundary, where the uncontrolled system has a finite number of positive real eigenvalues. In this paper we go further and develop a methodology for the case where the uncontrolled boundary condition has anti-damping, which makes the real parts of all the eigenvalues of the uncontrolled system positive and arbitrarily high, i.e., the system is “anti-stable” (exponentially stable in negative time). Using a conceptually novel integral transformation, we obtain extremely simple, explicit formulae for the gain functions. For the case with only boundary sensing available (at the same end with actuation), we design backstepping observers which are dual to the backstepping controllers and have explicit output injection gains. We then combine the control and observer designs into an output-feedback compensator and prove the exponential stability of the closed-loop system.

Sufficient Optimality Conditions for Dirichlet Boundary Control of Wave Equations

We study optimal control problems for wave equations (focusing on the multidimensional wave equation) with control functions in the Dirichlet boundary conditions under pointwise control (and we admit state—by assuming weak hypotheses) constraints.

A note on geometric conditions for boundary control of wave equations with variable coefficients

The analytical condition given by Wyler for boundary stabilization of wave equations with variable coefficients is compared with the geometrical condition derived by Yao in terms of the Riemannian geometry method for exact controllability of wave equations with variable coefficients. It is shown that these two conditions are equivalent.

A mixed finite element formulation for the boundary controllability of the wave equation

Abstract : This paper introduces the mixed finite element method as a viable numerical procedure for the boundary controllability of the linear wave equation. Another numerical implementation using Galerkin finite elements has been investigated. However, due to approximation problems of the normal derivative on the boundary, the method becomes unstable as the mesh is refined. To correct for the ill-posedness of the approximate problem, a Tychonoff regularization method was implemented. The aforementioned paper also presents other possible remedies; among them is the mixed finite element method. The fixed finite element approximation is a plausible procedure to overcome these difficulties since the derivative at certain nodal values arises naturally from the formulation.