In recent years, a novel pole placement paradigm has emerged for linear time-invariant systems of functional differential equations, addressing the complexities posed by their infinite spectrum and intricate dynamics. This paradigm is built upon the multiplicity-induced-dominancy (MID) property, which asserts that a characteristic root with sufficiently high multiplicity can dominate the spectral behavior of the system, thereby influencing its dynamic response. While the MID property has proven to be a powerful tool in control design, its applicability often depends on specific system configurations and parameter constraints. In this study, we propose new sufficient conditions ensuring the validity of the MID property in the context of the lowest intermediate over-order multiplicity. Our approach establishes a connection between the MID property and Laguerre polynomials by exploiting their inherent properties. Thanks to their structural properties, these conditions are tailored to the prescribed stabilization of conservative mechanical systems, which are characterized in the Laplace domain by even polynomials. By leveraging these conditions, we present a systematic framework for the precise assignment of dominant roots in the spectrum of conservative mechanical systems stabilized via delayed state feedback, enabling accurate control over both the system's solution's long-time behavior as well as its exact exponential decay.
Balogh et al. (Fri,) studied this question.