Nest Algebras And Control Theory Within The Framework Of Nest Algebras

Operator theory is a diverse area of mathematics which derives its impetus and motivation from several sources.Nest algebras of them the is important and tractable class of non-self-adjoint operator algebras on Hilbert space.In particular,a certain discrete nest algebra has important application value.At the end of twentieth Centu-ry,B.Fracisa and A.Feintuch established control theory within the framework of nest algebras.from then on,many researchers have been interested in it and have achieve many achievements in scientific research.The content of this paper is as follows:There are four parts in this paper.In Chapter 1.We give a review about the nest algebra theory,the linear system control theory and H? control theory.In Chapter 2.We give a review about the preparations with an emphasis on the basic knowledge of mathematics and control theory,such as nest algebra and complete finiteness,?strong?right representation and?strong?left representation,discrete linear time-varying systems,the tools for the stability analysis and some classical results about stability.Furthermore,we give a method of using system stabilizable to structure new system.From this readers understand control theory within the framework of nest algebras better.In Chapter 3.From the perspective of strong transitivity,a controller design method is provided to simultaneously stabilize a collection of time-vrying linear sys-tems within the framework of nest algebras.In particular,all simultaneously stabilizing controllers for a class of linear plants are characterized based on the doubly coprime factorizations.In Chapter 4.We give a bicoprime factorization approach to deal with various stability problems for time-varying linear systems within the framework of nest alge-bra.Based on the bicoprime factorizations,necessary and sufficient conditions for the close-loop stability are generalized from the time-invariant case.Moreover,the strong stabilization problem is studied for a given linear system which admits a bicoprime fac-torization.Finally,we also derive necessary and sufficient conditions for simultaneous stabilizability of two linear systems or more.Now we are going to show our main results.Theorem 1.Let L can be stabilizable,then???H ? S,L + H can be stabilizable too.If is strong right representation of L,Y,X is a causal bounded left inverse,then strong right representation of L + H can be characterized aswhich W is invertible in M₂?S?and W such thatTheorem 2.Let L₀,L₁,…,Ln ? L,and C₁,…,C_n ? L.Suppose that C_i stabilizes L_i-1 and L_i for i=1,…,n.Define ? =?YC₁L₁-XC₁??MC₁+NC₁L₁?^-1.Then the controller C?Q?for L₀,L₁,…,Ln can be characterized asuwith Q ? S is such that for j = 3,…,n.where M_Cj,N_Cj,M_Cj,N_Cj,YCj,XCj are associated with Cj and satisfy Bezout identy,respectively for j = 1,2,…,n.Theorem 3.Suppose that C_i stabilizes L_i-1 and L_i for i = 1,…,n.And C₁ ? 5.Define B=?I + L₁C₁?^-1?L₁-L₀??I + C₁L₀?^-1.Then Wwith Q ? S is such that?I + QB?^-1 ? S and?4?holds.?2?.Furthermore,all controllers C that simultaneously stabilize L₀,L₁,…,Ln are given by where ? ? S is such that I + ??I L₀C₁?^-1?L₀-Lj??I +C?Q?Lj?^-1 is invertible in S,for j = 1,…,n.Theorem 4.If P = NM-1 L K is a bicoprime factorization of P ? L,then P has a right and left coprime factorization.Theorem 5.Suppose P ?G L has a bicoprime factorization P = NM-1L + K,and C has a bicoprime factorization C = NCMC-1and such that the feedback system {P,C}is well posed.Then {P,C} is stable if and only if is invertible in M₂?S?.Theorem 6.Let P,C ? L.Suppose that P and C admit bicoprime factorizations P = NM-1 L+K,C = NCMC-1 C + KC respectively.Then {P,C} is stable if and only ifTheorem 7.Let P0,P1 ?j and N_iM_i-1 L_i + K_i be a bicoprime factorization of Pi for i= 0,1.Assume that C₀ stabilizes M0-1L₀,C₁ stabilizes M1-1L₁ and C2 stabilizes N0M0-1.Then P0,P1 can,be simultaneously stabilized if and only if there exists T ? S such that(MC₀(I+ C₀MC2?L₀-C2K0P1]?-TMC2-1?I+N0M0-1C2?^-1?P0-P1?)?I+C₁1-1L₁?^-1MC₁-1 is invertible in S,where C_i = NC_iMC_i-1 = mC_i-1NC_i is a doubly coprime factorization of C_i for i = 0,1,2,and satisfying MC2M0+NC2N0= I.Theorem 8.Let N_imM_i-L_i +K_i be a bicoprime factorization of Pi ?L for i =1.2,…,n.Assume that Co stabilizes N1M1-1,C_i stabilizes M1-1L₁ and C_i stabilizes N_iM_i-1 for i = 2,…,n.Then P1,P2,…,Pn can be simultaneously stabilized if and only if there exists Q ? S such that is invertible in S,where C_i = NC_iMC_i-1 = MC_i-1NC_i is a doubly coprime factorization of C_i for i=0,1,…,n,and satisfying M1MC₁ + L₁NC₁=I.?K)is a bicoprime factorization of P?.Ifthen {P?,C} is stable for all ? ? {A ? M₂?S?:||?||<r}.