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Additional dynamics in transformed time-delay systems. (English) Zbl 0986.34066
The authors compare the stability properties of the linear system with one single delay \[ \dot x(t)= Ax(t)+ A_dx(t- r)\tag{1} \] and the transformed system with distributed delay \[ \dot x(t)= (A+ A_d)x(t)- A_d \int^0_{-r} [Ax(t+\tau)+ A_dx(t+ \tau- r)] d\tau,\tag{2} \] where \(r> 0\) is a constant and \(A\), \(A_d\) are (constant) \(n\times n\)-matrices with real entries. The characteristic values of the transformed system (2) consist of those to the original system (1) and some additional characteristic values. It is shown that if \(r\) is sufficiently small, then all additional characteristic values lie in the left half plane. Moreover, the critical value of the delay when such characteristic values cross the imaginary axis can be explicitly calculated.

34K20 Stability theory of functional-differential equations
93C23 Control/observation systems governed by functional-differential equations
34K35 Control problems for functional-differential equations
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