It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition...It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.展开更多
In this paper, we consider a nonlinear hybrid dynamic （NHD） system to describe fedbatch culture where there is no analytical solutions and no equilibrium points. Our goal is to prove the strong stability with respec...In this paper, we consider a nonlinear hybrid dynamic （NHD） system to describe fedbatch culture where there is no analytical solutions and no equilibrium points. Our goal is to prove the strong stability with respect to initial state for the NHD system. To this end, we construct corresponding linear variational system （LVS） for the solution of the NHD system, also prove the boundedness of fundamental matrix solutions for the LVS. On this basis, the strong stability is proved by such boundedness.展开更多
基金supported in part by the Established Researcher Grant and the CAS Faculty Development Grant of the University of South Florida,Chunhui Plan of the Ministry of Education of China,Wang Kuancheng Foundation,the National Natural Science Foundation of China(Grant Nos.10332030,10472091 and 10502042)the Doctorate Foundation of Northwestern Polytechnical University(Grant No.CX200616).
文摘It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.
基金This work was supported by the National Science Foundation for the Youth of China （Grant Nos. 11501574, 11401073 and 11701063）, the National Natural Science Foundation of China （Grant Nos. 11771008, 61673083 and 61773086）, the National Science Foundation for the Tianyuan of China （Grant No. 11626053）, the Natural Science Foundation of Shandong Province in China （Grant No.： ZR2015FM014, ZR2015AL010 and ZR2017MA005）, the Fundamental Research Funds for the Cen- tral Universities in China （Grant No. DUT16LK07） and the Project funded by China Postdoctoral Science Foundation （Grant No. 2016M601296）.
文摘In this paper, we consider a nonlinear hybrid dynamic （NHD） system to describe fedbatch culture where there is no analytical solutions and no equilibrium points. Our goal is to prove the strong stability with respect to initial state for the NHD system. To this end, we construct corresponding linear variational system （LVS） for the solution of the NHD system, also prove the boundedness of fundamental matrix solutions for the LVS. On this basis, the strong stability is proved by such boundedness.
