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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 *a _{n}x^{(n)}(t) + ⋯ a_{1}x’(t) + a_{0}x(t) = b_{m}u^{(m)}(t) + ⋯ + b_{1}u’(t) + b_{0}u(t) * is considered with

*a*. In the paper we assume that the polynomials

_{i}, b_{j}> 0*M*and

_{n}(s) = a_{n}s^{n}+ ⋯ + a_{1}s + a_{0}*L*partly interlace. The solution of the above equation is denoted by

_{m}(s) = b_{m}s^{m}+ ⋯ + b_{1}s + b_{0}*x(t, L*. It is proved that the function

_{m}, M_{n})*x(t, L*is nonnegative for

_{m}, M_{n})*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*, depending on the degree of the polynomial

_{m}, M_{n})*M*or

_{n}(s)*L*(Theorems 5 and 6).

_{m}(s)**Keywords**

extremal properties, Dirac’s impulse, linear systems, transfer function