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Extremal values of differential equations with application to control systems

Górecki Henryk, Zaczyk Mieczysław

Bulletin of the Polish Academy of Sciences. Technical Sciences

2021

Vol. 69, nr 1

art. no. e136041

In the paper, maximal values xe(τ) of the solutions x(t) of the linear differential equations excited by the Dirac delta function are determined. The analytical solutions of the equations and also the maximal positive values of these solutions are obtained. The analytical formulae enable the design of the system with prescribed properties. The complementary case to the earlier paper is presented. In an earlier paper it was assumed that the roots si are different, and now we consider the case when s1=s2=...=sn.

Extremal properties of linear dynamic systems controlled by Dirac’s impulse

Białas Stanisław, Górecki Henryk, Zaczyk Mieczysław

International Journal of Applied Mathematics and Computer Science

2020

Vol. 30, no. 1

75--81

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) + ∙∙∙ a₁ x’(t) + a₀ x(t) = b_m u (t) + ∙∙∙ + b₁ u’(t) + b₀ u(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).