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Definition 1.1: Let us denote linear space (over complex numbers. ) .... with apriori norm )1.11 define a language Ls,~~.~~ .... Now, if we define Bi = A; C?J ... C?J A ...
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STABILITIES AND CONTROLLABILITIES OF SWITCHED SYSTEMS (WITH APPLICATIONS TO THE QUANTUM SYSTEMS)

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Leonid Gurvits Z# 174471,CCS-3

MTNS- 2002 Notre Dame University June 2002

Los Alamos NAT10NAL L A B 0RATORY

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Stabilities and Controllabilities of Switched Systems ( with applications to the Quantum Systems) Leonid Gurvits Los Alamos National Laboratory Los Alamos, NM 87545 email: [email protected]

Abstract

We study various stabilities and controllabilities of linear switched systems . 1 Main Definitions and Problems

Let us consider a set S = {A, : y E K } ,A, : C" + C". Le., S is a set of complex n x n matrices , K is an index set. Recall that linear discrete inclusion L D I ( S ) is a set of discrete dynamical systems zi+l

= A q i ) z i ;i 2 0 , R : N

+K.

(1)

I will give a very powerfull sufficient condition for hilbert apriori norms in the continuous case . A surprising feature of this result is that the proof is "infinite dimensional" though the result itself is finite dimensional . Definition 1.1: Let us denote linear space (over complex numbers ) of N x N complex matrices as M ( N ) . A positive semidefinite matrix ~ A , B: C" @ C N + C N @ C N is called bipartite unnormalized density matrix (BUDM ) , if t T ( P A , B ) = 1 then this P A , B is called bipartite density matrix . Itisconvinient torepresentbipartitep~,~ = p ( i l , i z , j1,jz) as the following block matrix :

Correspondingly , linear continuous inclusion L C I ( S ) is a set of continuous time dynamical systems . X ( t )= A R ( t ) X ( t ) ,

where R(.) is a piece-wise continuous from the right switching rule

L D I ( s ) ( L C I ( S )is) called Absolutely Asymptoticaly Stable (AAS) if all trajectories in (1) ((2)) converge to zero . It was shown in [l] , [2] that if the set S is bounded then the convergence is in fact uniform , moreover L D I ( S ) is (AAS) iff there exists a norm 11.lld on C" such that the induced norms IIA-,ll a < 1 for all y E K ([11) ; L C I ( S )is (AAS) iff there exists a norm Il.IIc on C" such that the induced norms Ilezp(A-,t) 5 ezp(at) for all y E K , t 2 0 and some a < 0. I will mostly consider in this paper the case when it is known in advance that there exists a norm 11.11* on C" such that the induced norms llA,ll 1 for all y E K in the discrete case and Ilezp(A,t) 5 111 for a l l y E K , t 2 0 in the continuous case. Let us call such norms as apriori. In the discrete case , if apriori norm is "good" , say polytope or hilbert ,then , at least , there are finite algorithms to check (AAS) property . But even for 11 norm this desision problem is NP-HARD [2] . One of the main results of this paper is that in the continuous case if apriori norm is polytope then L C I ( S ) is (AAS) iff each A,, y E K is Hurwitz . I will give also a generalization of this result for so called Lindblad's operators introduced first in the Quantum Mechanics context.


0 such that ( ( I+ r A 5 1 : A E S . But even for hilbert norms and S consisting of two elements this

Example 4.8. Consider the folowing compact set of real 2 x 2 matrices S = { U ( a ) A U ( - a ) : 0 5 a 5 2 ~ } Similarly . to Example 3.6, LCI(S)is(AAS) iff 0 + A AT. Take any Hurwitz matrix A with 0 A AT but not 0 + A AT. . It follows from Theorem(4.7) that L C I ( D ) is(AAS) for all finite subsets D of S . (Here we deal with the following C-substochastic operators : A ( X ) = AX X A T . ) Thus we consrtrusted a compact set S such that L C I ( D )is(AAS) for all finite subsets D of S ,but L C I ( S )is not(AAS) .

+

+

+

+

Example 4.9 Another counter example , based on Theorem() , is LCI(A1, A z )which is(AAS), but L D I ( C ( A 1 ) ,C ( A 2 ) )is not(AAS) . Here the Caley transform C ( A ) = ( I + A ) ( I - A ) - ' . Le. ,Caley transform can transform absolutely asymptoticaly stable continuous time systems into not absolutely asymptoticaly stable discrete systems . Of course , this also give an example of LCI(A1,A2) which is(AAS) but does not have quadratic Lyupunov function :

It is easy to see that LCI(A1,A z ) satisfies conditions of Theorem() , but p ( C ( A l ) C ( A z ) )= 1 therefore L D I ( C ( A 1 ) ,C ( A 2 ) ) is not(AAS) . I

5

Controllabilities of Switching Systems

Let us consider controlled switched systems : z(n+l) = At(,jz(n)+ Bt(n)u(n), where all the matrices involved act on some linear finite-dimensional space X . Here t : N + 1,2, ..,IC is a switching rule. Let us associate with pairs ( A ,B ) the following map which maps linear subspaces of X to linear subspaces of X :

D ( A , B ) ( Y=: ) A(Y)

+ Im(B).(discrete case),

For a given finite set of pairs { ( A I B , l ) , ..., (Ak,B k ) } we will use the following notations :

D; =: D ( A ~ , B