<abstract xmlns="http://www.w3.org/1999/xhtml">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<italic>n</italic><italic>x</italic>(<italic>n</italic>)(<italic>t</italic>)+···+a1<italic>x</italic>′ (<italic>t</italic>)+a0<italic>x</italic>(<italic>t</italic>)= <italic>b</italic><italic>m</italic><italic>u</italic>(<italic>m</italic>)(<italic>t</italic>)+···+<italic>b</italic>1<italic>u′</italic> (<italic>t</italic>)+<italic>b</italic>0<italic>u</italic>(<italic>t</italic>) is considered with <italic>a</italic><italic>i, b</italic><italic>j ></italic> 0. In the paper we assume that the polynomials <italic>M</italic><italic>n</italic>(<italic>s</italic>)= <italic>a</italic><italic>n</italic><italic>s</italic><italic>n</italic> + ··· + <italic>a</italic>1<italic>s</italic> + <italic>a</italic>0 and <italic>L</italic><italic>m</italic>(<italic>s</italic>)= <italic>b</italic><italic>m</italic><italic>s</italic><italic>m</italic> + ··· + <italic>b</italic>1<italic>s</italic> + <italic>b</italic>0 partly interlace. The solution of the above equation is denoted by <italic>x</italic>(<italic>t, L</italic><italic>m</italic>,<italic>M</italic><italic>n</italic>). It is proved that the function <italic>x</italic>(<italic>t, L</italic><italic>m</italic>,<italic>M</italic><italic>n</italic>) is nonnegative for <italic>t</italic> ∈ (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 <italic>x</italic>(<italic>t, L</italic><italic>m</italic>,<italic>M</italic><italic>n</italic>), depending on the degree of the polynomial <italic>M</italic><italic>n</italic>(<italic>s</italic>) or <italic>L</italic><italic>m</italic>(<italic>s</italic>) (Theorems 5 and 6).</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).

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

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Extremal Properties of Linear Dynamic Systems Controlled by Dirac\u2019s Impulse"}

The paper concerns the properties of linear dynamical systems described by linear differential equations, excited by the Dirac delta function. A differential...

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

Published Online: Apr 03, 2020

Page range: 75 - 81

Received: Apr 08, 2019

Accepted: Nov 16, 2019

DOI: https://doi.org/10.34768/amcs-2020-0006

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

© 2020 Stanisław Białas et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Published Online: Apr 03, 2020

Page range: 75 - 81

Received: Apr 08, 2019

Accepted: Nov 16, 2019

DOI: https://doi.org/10.34768/amcs-2020-0006

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

© 2020 Stanisław Białas et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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