Time-Varying Regular Bilinear Systems

Well-posed systems—The LTI case and beyond

“…One such topic is the Lax-Phillips semigroup associated to a well-posed system (the connection between well-posed systems theory and scattering theory), for which we refer to [7,9,59,65,67,68]. Another topic that we are compelled to leave out are the timevarying well-posed linear systems, for which we refer to [12,33,58,59] (an incomplete list).…”

“…Well-posed linear systems have various generalizations within the LTI context. One obvious one is to replace Hilbert spaces with Banach spaces and L 2 with L p -this is one of the issues that we shall ignore in this paper, but we refer to relevant parts of [10,12,30,32,65,75,76,79] (this is an incomplete list).…”

“…12) thatρż(t) = ν∆z(t) − (z(t) · ∇)z(t) + u 1 (t) − (I − P )[ν∆z(t) − (z(t) · ∇)z(t) + u 1 (t)] (9.13) holds with each term in L 2 ([0, T ]; L 2 (Ω; R 3 )). Hence (I − P )[ν∆z − (z · ∇)z + u 1 ] ∈ L 2 ([0, T ]; G(Ω)),where G(Ω) has been defined in (9.7).…”

Well-posedness of infinite-dimensional non-autonomous passive boundary control systems

“…In the literature most attention has been devoted to autonomous control systems. However, in view of applications, the interest in non-autonomous systems has been rapidly growing in recent years, see e.g., [22,13,27,7,31,19,6,18,30] and the references therein. In particular, a class of scattering passive linear non-autonomous linear systems of the forṁ…”

“…We expect that the results in [31] can be generalized to include this general case. However, for the class of boundary control systems defined in Definition 3.2 we deal directly with (1)-(2) in combination with Fattorini's trick instead of its corresponding system (5)- (6). Our method is indeed much simpler.…”