TY - JOUR
T1 - Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
AU - Domoshnitsky, Alexander
AU - Volinsky, Irina
AU - Polonsky, Anatoly
N1 - Publisher Copyright:
© 2019 Mathematical Institute Slovak Academy of Sciences.
PY - 2019/10/1
Y1 - 2019/10/1
N2 - There are almost no results on the exponential stability of differential equations with unbounded memory in mathematical literature. This article aimes to partially fill this gap. We propose a new approach to the study of stability of integro-differential equations with unbounded memory of the following forms x(t)+A=1m-τi(t)tbi(t)e-αi(t-s)x(s)ds=0,x(t)+A=1m0t-τi(t)bi(t)e-αi(t-s)x(s)ds=0, $$\begin{&array;} begin{split} displaystyle x"'(t)+\sum_{i=1}{m} &limitst;-τ_{i}(t)}{t}b_{i}(t)\text{e}{-α{i}(t-s) x(s)text{d} s &=0, x"'(t)+\sum_{i=1}{m}\∫&limits;0 τi(t)b_i(t)text{e}{-α i(t-s)}x(s)\text{d} s &= 0, end{split} end{&array;}$$ with measurable essentially bounded bi(t) and τi(t), i = 1, ..., m. We demonstrate that, under certain conditions on the coefficients, integro-differential equations of these forms are exponentially stable if the delays τi(t), i = 1, ..., m, are small enough. This opens new possibilities for stabilization by distributed input control. According to common belief this sort of stabilization requires first and second derivatives of x. Results obtained in this paper prove that this is not the case.
AB - There are almost no results on the exponential stability of differential equations with unbounded memory in mathematical literature. This article aimes to partially fill this gap. We propose a new approach to the study of stability of integro-differential equations with unbounded memory of the following forms x(t)+A=1m-τi(t)tbi(t)e-αi(t-s)x(s)ds=0,x(t)+A=1m0t-τi(t)bi(t)e-αi(t-s)x(s)ds=0, $$\begin{&array;} begin{split} displaystyle x"'(t)+\sum_{i=1}{m} &limitst;-τ_{i}(t)}{t}b_{i}(t)\text{e}{-α{i}(t-s) x(s)text{d} s &=0, x"'(t)+\sum_{i=1}{m}\∫&limits;0 τi(t)b_i(t)text{e}{-α i(t-s)}x(s)\text{d} s &= 0, end{split} end{&array;}$$ with measurable essentially bounded bi(t) and τi(t), i = 1, ..., m. We demonstrate that, under certain conditions on the coefficients, integro-differential equations of these forms are exponentially stable if the delays τi(t), i = 1, ..., m, are small enough. This opens new possibilities for stabilization by distributed input control. According to common belief this sort of stabilization requires first and second derivatives of x. Results obtained in this paper prove that this is not the case.
KW - Cauchy function
KW - Exponential stability
KW - distributed delays
KW - distributed input control
KW - integro-differential equations
KW - stabilization
UR - http://www.scopus.com/inward/record.url?scp=85073878174&partnerID=8YFLogxK
U2 - 10.1515/ms-2017-0298
DO - 10.1515/ms-2017-0298
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:85073878174
SN - 0139-9918
VL - 69
SP - 1165
EP - 1176
JO - Mathematica Slovaca
JF - Mathematica Slovaca
IS - 5
ER -