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Paper details
Number 1 - March 2020
Volume 30 - 2020
Extremal properties of linear dynamic systems controlled by Dirac’s impulse
Stanisław Białas, Henryk Górecki, Mieczysław Zaczyk
Abstract
The paper concerns the properties of linear dynamical systems described by linear differential equations, excited by the Dirac delta function. A differential equation of the form anx(n)(t) + ⋯ a1x’(t) + a0x(t) = bmu(m)(t) + ⋯ + b1u’(t) + b0u(t) is considered with ai, bj > 0. In the paper we assume that the polynomials Mn(s) = ansn + ⋯ + a1s + a0 and Lm(s) = bmsm + ⋯ + b1s + b0 partly interlace. The solution of the above equation is denoted by x(t, Lm, Mn). It is proved that the function x(t, Lm, Mn) is nonnegative for t ∊ (0, ∞) , and does not have more than one local extremum in the interval (0, ∞) (Theorems 1, 3 and 4). Besides, certain relationships are proved which occur between local extrema of the function x(t, Lm, Mn), depending on the degree of the polynomial Mn(s) or Lm(s) (Theorems 5 and 6).
Keywords
extremal properties, Dirac’s impulse, linear systems, transfer function