Kernel characterization of maximal monotonicity

Characterize maximal monotonicity directly in terms of kernel representations defined by matrix pairs F,G for dissipative port-Hamiltonian differential-algebraic systems.

Background

In the linear dissipative port-Hamiltonian setting, a maximally monotone structure can be represented in image form as M = im [V; W], with maximal monotonicity characterized by W^T V + V^T W \geq 0 together with a rank condition. To obtain differential-algebraic equations, however, the structure must instead be expressed in kernel form, M = ker [F\;G].

The paper states that a direct characterization in terms of the kernel matrices F and G is not yet available at that point, and then supplies Proposition \ref{prop:maxmon}, which gives a condition involving every maximal annihilator. The unresolved issue is therefore the development of a direct, intrinsic kernel-representation characterization of maximal monotonicity, rather than one obtained through image representations or quantified over maximal annihilators.