  New methods are developed for the stabilization of a linear system with
general time-varying distributed delays existing at the system's states, inputs
and outputs. In contrast to most existing literature where the function of
time-varying delay is continuous and bounded, we assume it to be bounded and
measurable. Furthermore, the distributed delay kernels can be any
square-integrable function over a bounded interval, where the kernels are
handled directly by using a decomposition scenario without using
approximations. By constructing a Krasovski\u{i} functional via the application
of a novel integral inequality, sufficient conditions for the existence of a
dissipative state feedback controller are derived in terms of matrix
inequalities without utilizing the existing reciprocally convex combination
lemmas. The proposed synthesis (stability) conditions, which take dissipativity
into account, can be either solved directly by a standard numerical solver of
semidefinite programming if they are convex, or reshaped into linear matrix
inequalities, or solved via a proposed iterative algorithm. To the best of our
knowledge, no existing methods can handle the synthesis problem investigated in
this paper. Finally, numerical examples are presented to demonstrate the
effectiveness of the proposed methodologies.
