MDV310

full article ..N1p)cfsCs00…00*afs?T(N11+…N1p)afs+afs?T��i=1p��i��v��luks��?svMvafs?��i=1p��i��v��luks��?svMs��1Mvbfs��1…��pMvbfs��p0**2��1��1D1s?Tcfs?TN21��1cfsD1s000***………****2��p��pD1s?Tcfs?TN2p��pcfsD1s0*****0) (42) ��331=(0-��i=1p��iCsTcfsT��v��lks��svPv0000*-��i=1p��i��v��lks��svPvafs-��1��v��lks��svPv[bfs��1+cfsD1s��1]…-��p��v��lks��svPv[bfs��p+cfsD1s��p]��i=1p��i��v��lks��svPvcfsD2s**0000***…00****00*****0) (43) ��332=(0-��i=1p��iCsTcfsT��v��luksPv0000*-��i=1p��i��v��luksPvafs-��1��v��luksPv[bfs��1+cfsD1s��1]…-��p��v��luksPv[bfs��p+cfsD1s��p]��i=1p��i��v��luksPvcfsD2s**0000***…00****00*****0) (44) Proof. Define Lyapunov function as V(k)=��T(k)Ps��(k) (45) V3=��j=1q��i=-��j-1��m=k+ik-1��T(m)(��s��l��svMv)��(m)��(m)=��(m+1)-��(m)?V2=��i=1p��i=k-��ik-1��T(i)(��s��l��svSv)��(i),?V1=��T(k)Ps��(k),?V(x(k),k)=V1+V2+V3, (46) Pv are the different symmetric matrices determined by transition matrices.

Define ��T(k)=(xT(k)xT��(k)) (47) The forward difference of Lyapunov function can be written as E��V11(k)=ExT(k+1)��v��lks��svPvx(k+1)-xT(k)��v��lks��svPsx(k)E��V12(k)=ExT��(k+1)��v��lks��svPvx��(k+1)-xT(k)�ġ�v��lks��svPsx(k)��?E��V1(k)=E��V11(k)+��V12(k), (48) ��V11 can be expressed as E[��V11(x(k))]=xT(k)AsT(��v��lks��svPv)Asx(k)+2xT(k)AsT(��v��lks��svPv)��i=1p��iB1sx(k-��i)+2xT(k)AsT(��v��lks��svPv)B2sw(k)+2��i=1p��ixT(k-��i)B1sT(��v��lks��svPv)B2sw(k)��i=1p��ixT(k-��i)��B1sT(��v��lks��svPv)��iB1sx(k-��i)+wT(k)B2sT(��v��lks��svPv)B2sw(k)-xT(k)(��v��lks��svPs)x(k)?��V11= (49) Define ��wT=(xT(k)xT��(k)xT(k-��1)…

xT(k-��p)��T(k)) (50) ��V11 can be written as ��V11=��wT(��111+��112+��113+��114)��w (51) Where ��311=(AsT(��v��lks��svPv)As0AsT(��v��lks��svPv)��1B1s…AsT(��v��lks��svPv)��pB1sAsT(��v��lks��svPv)B2s*00…00**��1B1sT(��v��lks��svPv)B1s��100��1B1sT(��v��lks��svPv)B2s***…0…****��pB1sT(��v��lks��svPv)B1s��p��pB1sT(��v��lks��svPv)B2s*****B2sT(��v��lks��svPv)B2s) (52) ��113=(AsT(��v��luks��svPv)As0AsT(��v��luks��svPv)��1B1s…AsT(��v��luks��svPv)��pB1sAsT(��v��luks��svPv)B2s*00…00**��1B1sT(��v��luks��svPv)B1s��100��1B1sT(��v��luks��svPv)B2s***…0…****��pB1sT(��v��luks��svPv)B1s��p��pB1sT(��v��luks��svPv)B2s*****B2sT(��v��luks��svPv)B2s) (53) ��112,��114 are the same as the terms in Theorem 1. ��111 can be written as ��111=��1T��1��1 (54) Where ��1T,��1 are the same as the terms in Theorem 1.

Define (N11-��v��luks��svPv*N21)��0 (55) we can get (xT(k)AsTxT(k-��1)��1B1sT)(N11-��v��luks��svPv*N21)(Asx(k)��1B1sx(k-��1))��0 Carfilzomib (56) It can be expressed as 2xT(k)AsT(��v��luks��svPv)��1B1sx(k-��1)��xT(k)AsTN11Asx(k)+xT(k-��1)��1B1sTN21��1B1sx(k-��1) (57) Define (N1p-��v��luks��svPv*N2p)��0 (58) We can obtain N1pAsx(k)+xT(k-��p)��pB1sTN2p��pB1sx(k-��p)?2xT(k)AsT(��v��luks��svPv)��pB1sx(k-��p)��xT(k)AsT, (59) According to Eqs.

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