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

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 a_nx(ⁿ)(t)+···+a₁x′ (t)+a₀x(t)= b_mu(^m)(t)+···+b₁u′ (t)+b₀u(t) is considered with a_i, b_j > 0. In the paper we assume that the polynomials M_n(s)= a_nsⁿ + ··· + a₁s + a₀ and L_m(s)= b_ms^m + ··· + b₁s + b₀ partly interlace. The solution of the above equation is denoted by x(t, L_m,M_n). It is proved that the function x(t, L_m,M_n) 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, L_m,M_n), depending on the degree of the polynomial M_n(s) or L_m(s) (Theorems 5 and 6).