Necessary and sufficient conditions of stabilizability for first-order control systems with time-delay (Q2706644)

A linear control system with time-delayed feedback is considered: NEWLINE\[NEWLINE\dot x(t)=Ax(t)+ Bu(t-r),\quad u(t)=-Kx(t),\quad A,B,K=\text{constants}. \tag{1}NEWLINE\]NEWLINE The dynamic behaviour is thus described by NEWLINE\[NEWLINE\dot x(t)= ax(t)+ bx(t-r),\quad a,b,r=\text{constants}.\tag{2}NEWLINE\]NEWLINE Although the solutions of eq. (2) are known (e.g.: \textit{E. Kamke}, Differentialgleichungen. Lösungsmethoden und Lösungen, Becker \& Erler, Leipzig (1943; Zbl 0028.22702), p. 631, eq. 10.2), they are studied by three methods in the complex frequency domain: location of eigenvalues, frequency response and the Nyquist criterion.NEWLINENEWLINENEWLINEThe final result, eq. (24), is equivalent to Kamke's eq. 10.1, discussed on p. 630.