In this paper, we will study the stability of economic systems with delay, also called delayed economic systems. We focus on the complete stability problem with respect to delay for linearized economic systems. First, we will review three types of delayed economic systems and will see that the characteristic functions are all in the form $f(\lambda ,\tau )$. Then, we will solve the complete stability problem by using the frequency-sweeping approach, which is introduced for the stability studies of economic systems (for the first time). As a consequence, the complete stability problem with delayed economic systems can be systematically solved. Finally, numerical examples will show that the whole stability set of $\tau$ for a delayed economic system may be determined accurately.

This paper addresses the stability and stabilization problems of rolling mill main drive systems with time delay. The complete stability problem with respect to the delay parameter is investigated for linearized systems with and without control. The systems are first formulated in the characteristic function form, and the corresponding complete stability problem is introduced. By employing the frequency-sweeping approach, the complete stability properties are systematically analyzed. As a result, several analytical properties related to the local asymptotic behavior of critical imaginary roots are derived. Finally, numerical examples demonstrate that the whole stability set with respect to delay can be accurately determined.

This paper studies consensus of continuous-time first-order linear multi-agent systems (MASs) with identical agent dynamics characterized by a scalar self-dynamics coefficient, under proportional derivative (PD) protocols with a common constant time delay over connected undirected graphs. Unlike existing parameter-space characterization studies, delayed derivative feedback gives rise to neutral-type dynamics, introducing an additional stability requirement associated with the neutral operator. To account for both this neutral operator constraint and the delay-induced stability variations, a unified framework combining frequency-sweeping techniques and parameter-space decomposition is developed, with the proportional gain, derivative gain, and time delay treated as free parameters. For fixed controller gains, delay-dependent stability is first characterized and the corresponding consensus delay sets are explicitly determined. The analysis is then extended to the controller gain plane, which is analytically partitioned according to the neutral operator constraint and the stability properties of the associated quasipolynomials. Algebraic conditions are established to classify the resulting regions. Based on this partition, delay-independent consensus, delay-dependent consensus, and non-consensus regions can be identified before quantitative critical-delay computation. For delay-dependent regions, the corresponding consensus sets are subsequently constructed over the joint controller-delay parameter space. Illustrative examples demonstrate the proposed characterization.