Linear passive systems and maximal monotone mappings (Q301654)

The paper studies linear systems of the form \[ x'(t)=Ax(t)+Bz(t)+u(t),\quad w(t)=Cx(t)+Dz(t)\eqno (1) \] with \[ (-z(t),w(t))\in \operatorname{graph}(M), \] where \(M: {\mathbb R}^m\to {\mathcal P}({\mathbb R}^m)\) is a given maximal monotone map. The system (1) is equivalent with a differential inclusion of the form \[ x'(t)\in -H(x(t))+u(t).\eqno (2) \] Under certain hypotheses it is proved that the set-valued map \(H(.)\) is maximal monotone. When the linear system (1) is passive, several conditions for the existence and uniqueness of solutions of the differential inclusion are obtained. Necessary and sufficient conditions for passivity by pole shifting are also provided.