Extremal Properties of Linear Dynamic Systems Controlled by Dirac’s Impulse

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