In the following, the unitary matrix Bk is designed such that it commutes with H0, k +. Theorem 4 ��Consider system (4) with the hybrid impulsive control satisfying (6) and (7). The largest invariant set is given by G = 2n?1E1E2 with E1 = , E2 = �� : , k��1,(9)where?k��2,Nk??:=��j=1k?1Bj,?=0,sl=1,2,��,ml},????sl=1,2,��,ml},Mkl???{|��?:?(?��|��j=1k?1Bj?|��f??��f|Xlsl��j=k?11Bj|��?)???????andM1l:={|��?:?(?��|��f??��f|Xlsl|��?)=0,? U0126 MAPK Xl1, Xl2,��, Xlml constitute the basis of the set (i)s[H0(s), Hl], s = 0,1, 2,��, l J. Hence, system (4) converges to G under the hybrid impulsive control.Proof ��When t = t0, from (6), we obtain l��J??(?��(t0)|��f??��f|Hl|��(t0)?)=0.

(10)The?��?[ei��?��(t0)|��f??��f|Hl|��(t0)?]=0,?thatV�B1(t0)=0?|?��(t0)|��f?| main idea of the proof is sketched as follows. The interval [tk?1, tk] is divided into nk sufficiently small intervals with duration dt. We apply the Taylor expansion on the system state and omit the high order terms of dt. By Lemma 2, the requirements V�B1(t)=0??(t��tk) and ��V1(tk) = 0 for the whole system trajectory will be transformed to the conditions on the initial state. By the Taylor expansion and commutativity between H0 and Bk, it yields ��?��f|[H0(n1),Hl]|��(t0)?)=0.(11)At???????((i)n1?��(t0)|��f?????D?V1(t1)=0??????????(i?��(t0)|��f??��f|[H0,Hl]|��(t0)?)=0,?��?��f|Hl(I?iH0dt)|��(t0)?)=0????????(?��(t0)|(I+iH0dt)|��f????(?��(t0+dt)|��f??��f|Hl|��(t0+dt)?)=0?thatV�B1(t0+dt)=0 t = tk + dt, the free evolution of |��(t) is given k=1,2,��.

(12)Similar to the?by|��(tk+dt)?=(I?iH0dt)|��(tk+)?=Bk(I?iH0dt)|��(tk)?, previous deduction, it follows from (11) and (12) s=0,1,��,n1+n2.(13)Consequently,?????????????��?��f|[H0(s),Hl]B1|��(t0)?)=0,???????((i)s?��(t0)|B1?|��f??s=0,1,��,n1+1,D?V1(t2)=0?????????????????��?��f|[H0(s),Hl]B1|��(t0)?)=0,?????????((i)s?��(t0)|B1?|��f???s=0,1,2,????????????????????????��?��f|[H0(s),Hl]B1|��(t1?dt)?)=0,???????????((i)s?��(t1?dt)|B1?|��f???s=0,1?��?��f|[H0(s),Hl]|��(t1+)?)=0,?????????((i)s?��(t1+)|��f?????(?��(t1+dt)|��f??��f|Hl|��(t1+dt)?)=0??thatV�B1(t1+dt)=0 Entinostat ??((i)s?it can be obtained thatD?V1(tk)=0��?��f|[H0(s),Hl]��j=k?11Bj|��(t0)?)=0,(14)where????????��(t0)|��j=1k?1Bj?|��f? s = 0,1,��, ��i=1kni. Noticing that the set (i)s[H0(s), Hl], s = 0,1,��, ��i=1kni, l J has finite dimension, we denote its basis to be Xl1, Xl2,��, Xlml, l J. Since the division of the interval [tk?1, tk] is random, (14) can be rewritten sl=1,��,ml.(15)For???????????????????��?��f|Xlsl��j=k?11Bj|��(t0)?)=0,?????????(?��(t0)|��j=1k?1Bj?|��f??asD?V1(tk)=0 convenience, the set of the states satisfying (15) is denoted as Mkl in (9), l J, k �� 2.