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(5)From the case of deciding on the Lyapunov perform based mostly over the state error defined by (five), to be able to facilitate to layout the handle laws based mostly on the Lyapunov stability theorem, the drift item appeared during the 1st buy time derivative of Lyapunov function, which can be brought about through the inner Hamiltonian, is needed to become eliminated. The current remedy would be to add a global CH5138303 phase management item �� into the control method governed by (1). This approach will not adjust the population distribution of the management technique. Therefore the dynamical equation (1) becomesi|�רB(t)?=(H0+��k=1rHkuk(t)+��I)|��(t)?.(six)Following some deduction, 1 can get the time derivative with the Lyapunov function (5) asV�B2=?(��f+��)?(?��f?�O?��?)?��k=1r?(?��f|Hk|��?)uk(t).

(seven)The manage laws which could make V�B2��0 hold is usually created as��=?��f+cf0(?(?��f?�O?��?)),(8)uk(t)=Kkfk(?(?��f|Hk|��?)),?(k=1,��,r),(9)where Kk > 0, and yk = fk(xk), (k = 0,��, r) are the monotonic raising functions via the coordinate origin in the plane xk ? yk. Based to the LaSalle invariance principle, the selleck inhibitorconvergence on the control program governed by (six) is often depicted by Theorem two.Theorem two (see [19, 23]) ��Consider the manage technique governed by (six) with handle fields united kingdom(t) created in (9) and �� developed in (eight). If (i) ��i��j�� �� ��lm, (i��, j��)��(l, m), i��, j��, l, m 1,2,��, N, ��lm = ��l ? ��m, in which ��l is definitely the lth eigenvalue of H0 corresponding to your eigenstate |?l; (ii) for any |?i �� |��f, i 1,��, N, there exists at the least a k 1,��, r this kind of that ?i|Hk|��f �� 0. Then any state trajectory will converge towards E2 = ��f, �� R.

From Theorem 2, 1 can see that to the case that the target state is definitely an eigenstate, in case the handle process governed by (six) satisfies the conditions (i)-(ii), the manage method could also converge for the equivalent state on the target eigenstate ei��|��f from any initial pure state. Lyapunov Management Primarily based on Regular Value of an Imaginary Mechanical Quantity Look at the next Lyapunov function based mostly on the regular value of an imaginary mechanical amount:V3(|��?)=?��|P|��?,(10)in which the imaginary mechanical amount P can be a favourable definite Hermitian operator. The primary order time derivative with the Lyapunov function (10) can be obtained asV�B3=i?��|[H0,P]|��?+i��k=1r?��|[Hk,P]|��?uk.(eleven)Set [H0, P] = 0 this kind of that the drift term within the suitable side of (eleven) can be eliminated.

So as to be certain V�B3��0, 1 can design and style uk(t) asuk(t)=?Kkfk(i?��|[Hk,P]|��?),?(k=1,��,r),(twelve)the place Kk > 0, and yk = fk(xk), (k = one,��, r) are monotonic raising functions as a result of the coordinate origin on the plane xk ? yk. Then based mostly over the LaSalle invariance principle, all the state trajectories with the technique will converge to your invariant set contained from the set E through which V�B=0 holds.