When Is a Linear System Conservative ?

We consider infinite-dimensional linear systems without a-priori wellposedness assumptions, in a framework based on the works of M. Livšic, M. S. Brodskĭı, Y. L. Smuljan, and others. We define the energy in the system as the norm of the state squared (other, possibly indefinite quadratic forms will also be considered). We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, well posed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control operator is one-to-one and whose observation operator has dense range, the equivalent conditions for being conservative become simpler, and reduce to three algebraic equations.