Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators

Abstract

Abstract. Associated with a skew-symmetric linear operator on the spatial domain [a, b] we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated to this Dirac structure is infinite dimensional system. We parameterize the boundary port variables for which the C0-semigroup associated to this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a symmetric positive operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko Beam.

Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain

“…In [20], Section 1.5, all boundary conditions are characterized for which we have that the energy stays constant along solutions. For a formulation of this result in the language of this paper, we refer to [15]. We decompose the set of n boundary conditions into two sets, the first one contains those boundary conditions that are set to zero, whereas the other class contains the boundary conditions which are free to choose, i.e., the inputs.…”

Stabilization of infinite dimensional port-Hamiltonian systems by nonlinear dynamic boundary control

“…It follows from Assumption 1, points 1 and 2, that the system (1)-(4) is a boundary control system (see Le Gorrec et al (2005); Jacob & Zwart (2012);Jacob et al (2015)), and so for b); R n ), satisfying (2) and (3) (for t = 0), there exists a unique classical solution to (1)-(4), (Jacob & Zwart, 2012, Theorem 11.2). Thus for these dense sets of initial conditions and inputs point 3 of Assumption 1 makes sense.…”

On a comprehensive class of linear control problems

“…This yields an expression of w in terms of y, v and ζ and hence (12) becomes a boundary observation equation.…”

“…If we plug this expression for w into equality (12) we obtain a boundary control equation. Likewise we may assume that the operator matrix…”

“…A particular subclass of port-Hamiltonian systems ( [10,12,35]) can be discussed within this theory. As a by-product we give a possible generalization of boundary control systems similar to port-Hamiltonian systems to the case of more than one spatial dimension, which appeared to be, at least to the best of the authors' knowledge, an open problem.…”