Minimization of a class of functionals that is applicable to the theory of pulse systems (Q1360787)

This paper deals with the minimization of pulse and continuous-pulse systems in which the functionals are of the following form \[ \begin{multlined} Q=\left.{1\over 2\pi i} \oint\right[\sum^2_{i,k=1} \bigl(q_{ik} (\zeta) w_k(\zeta) w_k(\zeta^{-1})\bigr)+ l( \zeta)w_1(\zeta) +\\ +l(\zeta^{-1}) w_1(\zeta^{-1}) +r(\zeta) w_2(\zeta) +r(\zeta^{-1}) w_2 (\zeta^{-1})\left]{d \zeta \over\zeta}\right. \end{multlined} \tag{1} \] where \(q_{jk} (\zeta)\), \(l(\zeta)\) and \(r(\zeta)\) are known linear functional (l.f.) functions, and \[ w_1(\zeta)= {1\over 1+w_\mu (\zeta) w_\rho (\zeta)}, \quad w_2(\zeta) =w_1(\zeta) w_\rho (\zeta) \tag{2} \] are unknown l.f. functions, in which \[ w_\mu (\zeta)= {\zeta d(\zeta) \over c(\zeta)} \tag{3} \] is an irreducible l.f. function with known polynomials \(c(\zeta)\) and \(d(\zeta)\). Moreover, \[ w_\rho (\zeta)= {\beta(\zeta) \over\alpha (\zeta)} \tag{4} \] is a physically realizable l.f. function with polynomials \(\alpha(\zeta)\) and \(\beta (\zeta)\) to be found so as to minimize the integral of (1) provided that the roots of the equation \[ \Delta (\zeta) =\alpha (\zeta)c (\zeta)+ \zeta\beta (\zeta)d (\zeta) \tag{5} \] lies outside the unit circle. The paper proposes a general method for solving this problem no matter if the poles of the functions \(q_{jk} (\zeta)\), \(l(\zeta)\) and \(r(\zeta)\) are simple or multiple. So the result obtained is more applicable to the theory of pulse systems than previous results. A detailed optimization algorithm for \(w_\rho (\zeta)\) has been presented.