Assignment of arbitrary coefficients and stabilization of linear differential equations with distributed delay by linear static delayed output feedback

A linear control system defined by a scalar time-invariant linear differential equation with distributed delays in the form of the Stieltjes integral in the state is considered. In the system, the input is a linear combination of multiple variables and its derivatives, and the output is a multidimensional vector of linear combinations of the state and its derivatives. For this system, the problem of assigning arbitrary coefficients for the characteristic function by linear static output feedback with delay in the form of the Stieltjes integral is studied. A criterion for the solvability of the problem of assigning arbitrary coefficients by linear static delayed output feedback is obtained. Corollaries are obtained on the arbitrary finite spectrum assignment and on the stabilization of the system. An example is given illustrating the obtained results.