isometric dilations for infinite sequences of noncommuting operators this paper develops a dilation theory for t n l an infinite sequence of noncommuting operators on a hilbert space when the matrix t1 t2 is a contraction a wold decomposition for an infinite sequence of isometries with orthogonal final spaces and a minimal isometric dilation for tn i are obtained some theorems on the geometric structure of the space of the minimal isometric dilation and some consequences are given this results are used to extend the sz nagy foia lifting theorem to this noncommutative setting this paper is a continuation of 5 and develops a dilation theory for an infinite sequence tn n of noncommuting operators on a hilbert space when en 1 tn tn i i is the identity on s many of the results and techniques in dilation theory for one operator 8 and also for two operators 3 4 are extended to this setting first we extend wold decomposition 8 4 to the case of an infinite sequence vn i of isometries with orthogonal final spaces in 2 we obtain a minimal isometric dilation for tn 1 by extending the schaffer construction in 6 4 using these results we give some theorems on the geometric structure of the space of the minimal isometric dilation finally we give some sufficient conditions on a sequence t7 n to be simultaneously quasi similar to a sequence rn n i of isometries on a hilbert space x with en rnrn n ix in 3 we use the above mentioned theorems to obtain the sz nagy foia5 lifting theorem 7 8 1 4 in our setting in a subsequent paper we will use the results of this paper for studying the characteristic function associated to a sequence tn lo 1 with ez 1 t t i throughout this paper a stands for the set 1 2 k k e n or the set n 1 2 received by the editors september 1 1987 1980 mathematics subject classification 1985 revision primary 47a20 secondary 47a45

tensor algebras overc correspondences representations dilations andc envelopes tensor algebras overc correspondences are noncommutative generalizations of the disk algebra they contain as special cases analytic crossed products semi crossed products and popescu s noncommutative disk algebras in this paper a dilation theorem and a commutant lifting theorem for representations of tensor algebras is proved these are used to show that for certainc correspondences thec envelopes in the sense of arveson of the tensor algebras are the cuntz pimsner algebras of the correspondences

isomorphisms of algebras associated with directed graphs given countable directed graphs g and g we show that the associated tensor algebras open image in new window g and open image in new window g are isomorphic as banach algebras if and only if the graphs g are g are isomorphic for tensor algebras associated with graphs having no sinks or no sources the graph forms an invariant for algebraic isomorphisms we also show that given countable directed graphs g g the free semigroupoid algebras open image in new window and open image in new window are isomorphic as dual algebras if and only if the graphs g are g are isomorphic in particular spatially isomorphic free semigroupoid algebras are unitarily isomorphic for free semigroupoid algebras associated with locally finite directed graphs with no sinks the graph forms an invariant for algebraic isomorphisms as well

composition operators on noncommutative hardy spaces abstract in this paper we initiate the study of composition operators on the noncommutative hardy space h ball 2 which is the hilbert space of all free holomorphic functions of the form f x 1 x n k 0 k a x f n a 2 1 where the convergence is in the operator norm topology for all x 1 x n in the noncommutative operatorial ball b h n 1 and b h is the algebra of all bounded linear operators on a hilbert space h when the symbol is a free holomorphic self map of b h n 1 we show that the composition operator c f f f h ball 2 is bounded on h ball 2 several classical results about composition operators boundedness norm estimates spectral properties compactness similarity have free analogues in our noncommutative multivariable setting the most prominent feature of this paper is the interaction between the noncommutative analytic function theory in the unit ball of b h n the operator algebras generated by the left creation operators on the full fock space with n generators and the classical complex function theory in the unit ball of c n in a more general setting we establish basic properties concerning the composition operators acting on fock spaces associated with noncommutative varieties v p 0 h b h n 1 generated by sets p 0 of noncommutative polynomials in n indeterminates such that p 0 0 p p 0 in particular when p 0 consists of the commutators x i x j x j x i for i j 1 n we show that many of our results have commutative counterparts for composition operators on the symmetric fock space and consequently on spaces of analytic functions in the unit ball of c n

invariant subspaces and hyper reflexivity for free semigroup algebras in this paper we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras this enables us to prove that they are reflexive and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces such algebras are called hyper reflexive this property is very strong but it has been established in only a very few cases moreover the prototypes of this class of algebras are the natural candidate for a non commutative analytic toeplitz algebra on n variables the case we make for this analogy is very compelling in particular in this paper the key to the invariant subspace analysis is a good analogue of the beurling theorem for invariant subspaces of the unilateral shift this leads to a notion of inner outer factorization in these algebras in a sequel to this paper 13 we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in cn a free semigroup algebra is the weak operator topology closed algebra generated by a set s1 sn of isometries with pairwise orthogonal ranges these conditions are described algebraically by s i sj iji for 1 i j n f

a linear programming oriented procedure for quadratic stabilization of uncertain systems abstract this paper gives a new necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty a constructive numerical procedure is defined to check the condition and it furthermore provides a stabilizing linear feedback gain some experiments are presented

caratheodory interpolation on the non commutative polydisk the caratheodory problem in the n variable non commutative herglotz agler class and the caratheodory fejer problem in the n variable non commutative schur agler class are posed it is shown that the caratheodory resp caratheodory fejer problem has a solution if and only if the non commutative polynomial with given operator coefficients the data of the problem indexed by an admissible set takes operator values with positive semidefinite real part resp contractive operator values on n tuples of jointly nilpotent contractive n n matrices for all n n

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

e0 dilation of strongly commuting cp0 semigroups abstract we prove that every strongly commuting pair of cp 0 semigroups has a minimal e 0 dilation this is achieved in two major steps interesting in themselves 1 we show that a strongly commuting pair of cp 0 semigroups can be represented via a two parameter product system representation 2 we prove that every fully coisometric product system representation has a fully coisometric isometric dilation in particular we obtain that every commuting pair of cp 0 semigroups on b h h finite dimensional has an e 0 dilation

properties of generalized predictive control control design using long range prediction based on a dynamic model of the plant has become an important contender for high performance applications the model can be derived analytically but more typically it is provided by simple experiments or in the adaptive context by recursive estimation many methods have been proposed in the literature depending on the assumed model structure and the choice of cost function here the approach called generalized predictive control or gpc is reviewed of special interest is the derivation of prediction equations using an observer polynomial and a demonstration that the selection of particular horizons the costing horizons and the control horizon leads to well understood basic techniques such as dead beat pole placement and lq simulated examples show that gpc is suitable for controlling a complex plant such as unstable inverse unstable systems and how the observer polynomial independently tailors the response to the disturbances preprogrammed set points as with robot trajectory control and actuator nonlinearities can also be catered for a discussion of adaptive applications of gpc to several industrial process concludes that the method is easy to use and effective

the algebraic structure of non commutative analytic toeplitz algebras the non commutative analytic toeplitz algebra is the wot closed algebra generated by the left regular representation of the free semigroup on n generators we develop a detailed picture of the algebraic structure of this algebra in particular we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n ball the k dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of h over the unit disk in 6 17 18 20 a good case is made that the appropriate analogue for the analytic toeplitz algebra in n non commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators the papers cited obtain a compelling analogue of beurling s theorem and inner outer factorization in this paper we add further evidence the main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of cn the kernel is the subgroup of quasi inner automorphisms which are trivial modulo the wot closed commutator ideal additional evidence of analytic properties comes from the structure of k dimensional completely contractive representations which have a structure very similar to the fibration of the maximal ideal space of h over the unit disk an important tool in our analysis is a detailed structure theory for wot closed right ideals curiously left ideals remain more obscure the non commutative analytic toeplitz algebra ln is determined by the left regular representation of the free semigroup fn on n generators z1 zn which acts on 2 fn by w v wv for v w in fn in particular the algebra ln is the unital wot closed algebra generated by the isometries li zi for 1 i n this algebra and its norm closed version the noncommutative disk algebra were introduced by popescu 19 in an abstract sense in connection with a non commutative von neumann inequality and 1991 mathematics subject classification 47d25 march 9 1997 october 9 1997 final draft first author partially supported by an nserc grant and a killam research fellowship second author partially supported by an nsf grant

pole assignment for uncertain systems in a specified disk by state feedback this paper presents a method for assigning the poles in a specified disk by state feedback for a linear discrete or continuous time uncertain system the uncertainty being norm bounded for this the quadratic d stabilizability concept which is the counterpart of quadratic stabilizability in the context of pole placement is defined and a necessary and sufficient condition for quadratic d stabilizability derived this condition expressed as a parameter dependent discrete riccati equation enables one to design the control gain matrix by solving iteratively a discrete riccati equation

free holomorphic functions and interpolation in this paper we obtain a noncommutative multivariable analogue of the classical nevanlinna pick interpolation problem for analytic functions with positive real parts on the open unit disc given a function f lambda to mathbb c where lambda is an arbitrary subset of the open unit ball mathbb b n z in mathbb c n z 1 we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball b mathcal h n 1 such that rm re g geq 0 quad rm and quad g z f z quad z in lambda where b mathcal h is the algebra of all bounded linear operators on a hilbert space mathcal h the proof employs several results from noncommutative multivariable operator theory and a noncommutative cayley transform introduced and studied in the present paper acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions all the results of this paper are obtained in the more general setting of free holomorphic functions with operator valued coefficients as consequences we deduce some results concerning operator valued analytic interpolation on the unit ball mathbb b n

free pluriharmonic majorants and commutant lifting in this paper we initiate the study of sub pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball b h n 1 x1 xn b h n x1x1 xnxn 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h several classical results from complex analysis have analogues in this noncommutative multivariable setting we present basic properties for sub pluriharmonic curves characterize the class of sub pluriharmonic curves that admit free pluriharmonic majorants and find in this case the least free pluriharmonic majorants we show that for any free holomorphic function on b h n 1 the map 0 1 c r1 rn r rr1 rrn rr1 rrn is a sub pluriharmonic curve in the cuntz toeplitz algebra generated by the right creation operators r1 rn on the full fock space with n generators we prove that is in the noncommutative hardy space hball2 if and only if has a free pluriharmonic majorant in this case we find herglotz riesz and poisson type representations for the least pluriharmonic majorant of moreover we obtain a characterization of the unit ball of hball2 and provide a parametrization and concrete representations for all free pluriharmonic majorants of when is in the unit ball of hball2 in the second part of this paper we introduce a generalized noncommutative commutant lifting gncl problem which extends to our noncommutative multivariable setting several lifting problems including the classical sz nagy foias commutant lifting problem and the extensions obtained by treil volberg foias frazho kaashoek and biswas foias frazho as well as their multivariable noncommutative versions we solve the gncl problem and using the results regarding sub pluriharmonic curves and free pluriharmonic majorants on noncommutative balls we provide a complete description of all solutions in particular we obtain a concrete schur type description of all solutions in the noncommutative commutant lifting theorem

hyperbolic geometry on noncommutative polyballs in this paper we study the noncommutative balls c x1 xn b h n x1 xn 1 0 where is the joint operator radius for n tuples of bounded linear operators on a hilbert space in particular 1 is the operator norm 2 is the joint numerical radius and is the joint spectral radius we introduce a harnack type equivalence relation on c 0 and use it to define a hyperbolic distance on the harnack parts equiv alence classes of c we prove that the open ball c 0 is the harnack part containing 0 and obtain a concrete formula for the hyperbolic distance in terms of the reconstruction operator associated with the right creation operators on the full fock space with n gener ators moreover we show that the topology and the usual operator norm topology coincide on c 1 while the open ball c 1 is not a complete metric space with respect to the operator norm topology we prove that it is a complete metric space with respect to the hyper bolic metric in the particular case when 1 and h c the hyperbolic metric coincides with the poincare bergman distance on the open unit ball of c n

free holomorphic functions on the unit ball of b h n abstract in this paper we continue the study of free holomorphic functions on the noncommutative ball b h n 1 x 1 x n b h n x 1 x 1 x n x n 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h and n 1 2 or n several classical results from complex analysis have free analogues in our noncommutative setting we prove a maximum principle a naimark type representation theorem and a vitali convergence theorem for free holomorphic functions with operator valued coefficients we introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by schwarz and harnack the borel caratheodory theorem is extended to our noncommutative setting using a noncommutative generalization of schwarz s lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball b h n 1 we obtain an analogue of julia s lemma for free holomorphic functions f b h n 1 b h m 1 we also obtain pick julia theorems for free holomorphic functions with operator valued coefficients we provide a noncommutative generalization of a classical inequality due to lindelof which turns out to be sharper then the noncommutative von neumann inequality finally we introduce a pseudohyperbolic metric on b h n 1 which is invariant under the action of the free holomorphic automorphism group of b h n 1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on b n the open unit ball of c n in this setting we obtain a schwarz pick type lemma we also provide commutative versions of these results for operator valued multipliers of the drury arveson space

remarks on the interpolation and the similarity problem for nest subalgebras of von neumann algebras abstract given a nest subalgebra of a von neumann algebra a and vectors f g we give necessary and sufficient conditions for the existence of an operator a in a so that af g this generalizes a well known theorem of e c lance and the main result in anoussis 1992 katsoulis et al 1993 we also use the generalized ringrose criterion to show that the similarity problem for continuous nests in certain crossed product factors of type iii has a negative answer

controller design with regional pole constraints a design procedure is developed that combines linear quadratic optimal control with regional pole placement specifically a static and dynamic output feedback control problem is addressed in which the poles of the closed loop system are constrained to lie in specified regions of the complex plane these regional pole constraints are embedded within the optimization process by replacing the covariance lyapunov equation by a modified lyapunov equation whose solution in certain cases leads to an upper bound on the quadratic cost functional the results include necessary and sufficient conditions for characterizing static output feedback controllers with bounded performance and regional pole constraints sufficient conditions are also presented for the fixed order i e full and reduced order dynamic output feedback problem with regional pole constraints circular elliptical vertical strip parabolic and section regions are considered

on fixed points of self maps of the free ball abstract in this paper we study the structure of the fixed point sets of noncommutative self maps of the free ball we show that for such a map that fixes the origin the fixed point set on every level is the intersection of the ball with a linear subspace we provide an application for the completely isometric isomorphism problem of multiplier algebras of noncommutative complete pick spaces

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

moving horizon control of linear systems with input saturation and plant uncertainty part 1 robustness we present a moving horizon feedback system based on constrained optimal control algorithms for linear plants with input saturation the system is a non conventional sampled data system its sampling periods vary from sampling instant to sampling instant and the control during the sampling time is not constant but determined by the solution of an open loop optimal control problem this is a two part paper in this part we show that the proposed moving horizon control system is robustly stable whether the state of the plant is measurable or not in the second part we show that the proposed moving horizon control system is capable of following a class of reference inputs and suppressing a class of disturbances experimental results show that the behaviour of the moving horizon control system is superior to that resulting from some alternative control laws

moving horizon control of linear systems with input saturation and plant uncertainty part 2 disturbance rejection and tracking we present a moving horizon feedback system based on constrained optimal control algorithms for linear plants with input saturation the system is a non conventional sampled data system its sampling periods vary from sampling instant to sampling instant and the control during the sampling time is not constant but determined by the solution of an open loop optimal control problem in part 1 we showed that the proposed moving horizon control system is robustly stable in this paper we show that it is capable of asymptotically suppressing a class of l bounded disturbances that are restricted to a subspace and of asymptotically following a class of reference inputs that are continuously differentiable bounded in magnitude and slope and restricted to a subspace

optimal estimation on the order of local testability of finite automata abstract a locally testable language l is a language with the property that for some nonnegative integer k called the order of local testability whether or not a word u is in the language l depends on 1 the prefix and suffix of the word u of length k 1 and 2 the set of subwords of length k of the word u for given k the language is called k testable we improve the upper bound on the order of local testability of a locally testable deterministic finite automaton with n states to 1 2 n 2 n 1 this bound is the best possible we give an answer to the following conjecture of kim mcnaughton and mccloskey for deterministic finite locally testable automata with n states is the order of local testability no greater than o n 1 5 when the alphabet size is two our answer is negative in the case of size two the situation is the same as in general case the order of local testability is n 2 the necessary and sufficient conditions for the language of an automaton to be k testable are given in terms of the length of paths of a related graph some estimates of the bounds on the order of local testability follow from these results

a linear framework on the distributed model predictive control of positive systems abstract this paper investigates the distributed model predictive control dmpc for positive systems with interval and polytopic uncertainties respectively different from the traditional quadratic dmpc a new linear dmpc framework is established for positive systems first a linear cooperative cost function performance index function is introduced for the dmpc of positive systems accordingly some linear constraint conditions are imposed on the systems then a linear co positive lyapunov function is constructed using the linear lyapunov function the dmpc controllers are designed for interval positive systems and polytopic positive systems respectively the controller of each subsystem contains not only the state information of the subsystem but also the state information of other correlated subsystems linear programming based optimization algorithm is addressed to compute the controller parameters to guarantee the feasibility and stability of the systems a cone invariant set is proposed finally two case studies are provided to illustrate the effectiveness of the theoretical findings

tensor algebras of c correspondences and their c envelopes abstract we show that the c envelope of the tensor algebra of an arbitrary c correspondence x coincides with the cuntz pimsner algebra o x as defined by katsura t katsura on c algebras associated with c correspondences j funct anal 217 2004 366 401 this improves earlier results of muhly and solel p s muhly b solel tensor algebras over c correspondences representations dilations and c envelopes j funct anal 158 1998 389 457 and fowler muhly and raeburn n fowler p muhly i raeburn representations of cuntz pimsner algebras indiana univ math j 52 2003 569 605 who came to the same conclusion under the additional hypothesis that x is strict and faithful

robust constrained model predictive control using linear matrix inequalities abstract the primary disadvantage of current design techniques for model predictive control mpc is their inability to deal explicitly with plant model uncertainty in this paper we present a new approach for robust mpc synthesis that allows explicit incorporation of the description of plant uncertainty in the problem formulation the uncertainty is expressed in both the time and frequency domains the goal is to design at each time step a state feedback control law that minimizes a worst case infinite horizon objective function subject to constraints on the control input and plant output using standard techniques the problem of minimizing an upper bound on the worst case objective function subject to input and output constraints is reduced to a convex optimization involving linear matrix inequalities lmis it is shown that the feasible receding horizon state feedback control design robustly stabilizes the set of uncertain plants several extensions such as application to systems with time delays problems involving constant set point tracking trajectory tracking and disturbance rejection which follow naturally from our formulation are discussed the controller design is illustrated with two examples

multivariable nehari problem and interpolation abstract in this paper we obtain a multivariable nehari type theorem for hankel operators extending the classical result as well as treil volberg generalization the result is based on a new generalization of the noncommutative commutant lifting theorem which extends to several variables the commutant lifting result obtained by treil and volberg and recently generalized by biswas foias and frazho an extension of i s iokhvidov ky fan theorem is obtained and used to prove a multivariable adamian arov krein type theorem for generalized hankel operators as consequences we obtain caratheodory and nevanlinna pick type interpolation results for meromorphic operators on fock spaces resp operator valued meromorphic functions on the unit ball of c n the nehari type results are used to obtain descriptions of hankel operators acting on fock spaces or weighted fock spaces multivariable dirichlet type spaces and to solve new weighted interpolation problems e g sarason for noncommutative analytic toeplitz algebras which have consequences to the operator valued analytic interpolation in the unit ball of c n moreover using central intertwining liftings we obtain explicit formulas for certain solutions of the weighted interpolation problems as another application of the generalized noncommutative commutant lifting theorem we derive an abstract interpolation problem for multipliers this is used to solve a left tangential nevanlinna pick type interpolation problem with operatorial argument for noncommutative analytic toeplitz algebras

free biholomorphic functions and operator model theory ii let f f1 fn be an n tuple of formal power series in noncommutative indeterminates z1 zn such that f 0 0 and the jacobian detjf 0 0 and let g g1 gn be its inverse with respect to composition we assume that f and g have nonzero radius of convergence and g is a bounded free holomorphic function on the open unit ball b h n 1 where b h is the algebra of bounded linear operators an a hilbert space h in this paper several results concerning the noncommutative multivariable operator theory on the unit ball b h n 1 are extended to the noncommutative domain bf h x b h n g f x x and f x 1 for an appropriate evaluation x f x we develop an operator model theory and dilation theory for bf h where the associated universal model is an n tuple mz1 mzn of left multiplication operators acting on a hilbert space h2 f of formal power series all the results of this paper have commutative versions

orthonormal basis functions applied to optimal control design with closed loop pole location constraints this article proposes an optimal control design technique to ensure that the closed loop poles are assigned to prescribed locations in the complex plane a key element in this approach is the usage of orthonormal basis functions to parametrise the main design parameter this goal is achieved by using a quadratic cost based optimisation approach considering control effort penalisation as well as cheap control youla parametrisation is used subject to a set of constraints which guarantee stability zero steady error to constant references and disturbances and yielding the prescribed set of closed loop poles

representations of product systems over semigroups and dilations of commuting cp maps we prove that every pair of commuting cp maps on a von neumann algebra m can be dilated to a commuting pair of endomorphisms on a larger von neumann algebra to achieve this we first prove that every completely contractive representation of a product system of c correspondences over the semigroup n2 can be dilated to an isometric or toeplitz representation

a primal dual potential reduction method for problems involving matrix inequalities we describe a potential reduction method for convex optimization problems involving matrix inequalities the method is based on the theory developed by nesterov and nemirovsky and generalizes gonzaga and todd s method for linear programming a worst case analysis shows that the number of iterations grows as the square root of the problem size but in practice it appears to grow more slowly as in other interior point methods the overall computational effort is therefore dominated by the least squares system that taust be solved in each iteration a type of conjugate gradient algorithm can be used for this purpose which results in important savings for two reasons first it allows us to take advantage of the special structure the problems often have e g lyapunov or algebraic riccati inequalities second we show that the polynomial bound on the number ofiterations remains valid even if the conjugate gradient algorithm is not run until completion which in practice can greatly reduce the computational effort per iteration we describe in detail how the algorithm works for optimization problems with l lyapunov inequaliues each of size m we prove an overall worst case operation count of o mssll5 the average case complexity appears to be closer to o m4ll5 this estimate is justified by extensive numerical experimentation and is consistent with other researchers experience with the practical performance of interior point algorithms for linear programming this result means that the computational cost of extending current control theory based on the solution of lyapunov or riccati equations to a theory that is based on the solution of multiple coupled lyapunov or riccati inequalities is modest

computing the order of a locally testable automaton a locally testable language is a language with the property that for some positive integer j whether or not a string x is in the language depends on 1 the prefix and suffix of x of length j 1 and 2 the set of substrings of x of length j without regard to the order in which these substrings occur or the number of times each substring occurs for any j for which this is true it is said that the language is j testable for a given locally testable language the smallest such number j is called the order of the language locally testable languages are regular and therefore these concepts apply to the finite automata that recognize the languages the authors show that computing the order of a given locally testable deterministic automaton is np hard and present a polynomial time epsilon approximation algorithm for computing it in addition an upper bound of 2n 2 1 on the order of a locally testable automaton of n states is obtained and the co np completeness of the problem of whether for a given j a given deterministic automaton is j testable is proven

multi toeplitz operators and free pluriharmonic functions abstract we initiate the study of weighted multi toeplitz operators associated with noncommutative regular domains d q m h b h n m n 1 where b h is the algebra of all bounded linear operators on a hilbert space h these operators are acting on the full fock space with n generators and have as symbols free pluriharmonic functions on the interior of the domain d q m h we prove that the set of all weighted multi toeplitz operators coincides with a d q m a d q m w o t where the domain algebra a d q m is the norm closed unital non selfadjoint algebra generated by the universal model w 1 w n of the noncommutative domain d q m h these results are used to study the class of free pluriharmonic functions on d q m h several classical results from complex analysis concerning harmonic functions have analogues in our noncommutative setting in particular we show that the bounded free pluriharmonic functions are precisely those which are noncommutative berezin transforms of weighted multi toeplitz operators and solve the dirichlet extension problem in this setting using noncommutative cauchy transforms we provide a free analytic functional calculus for n tuples of operators which extends to free pluriharmonic functions our study of weighted multi toeplitz operators on fock spaces is a blend of multi variable operator theory noncommutative function theory operator spaces and harmonic analysis

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

dilating covariant representations of the non commutative disc algebras let be an isometric automorphism of the non commutative disc algebra an for n 2 we show that every contractive covariant representation of an dilates to a unitary covariant representation of on hence the c envelope of the semicrossed product an z is on z

hardy algebras w correspondences and interpolation theory given a von neumann algebra m and a w correspondence e over m we construct an algebra h e that we call the hardy algebra of e when m open image in new window e h e is the classical hardy space h open image in new window of bounded analytic functions on the unit disc when m open image in new window and e open image in new window h e is the free semigroup algebra studied by popescu davidson and pitts and many others we show that given any faithful normal representation of m on a hilbert space h there is a natural correspondence e over the commutant m called the dual of e and that h e can be realized in terms of b h valued functions on the open unit ball open image in new window e in the space of adjoints of elements in e we prove analogues of the nevanlinna pick theorem in this setting and discover other aspects of the value distribution theory for elements in h e we also analyze the boundary behavior of elements in h e and obtain generalizations of the sz nagy foias functional calculus and the functional calculus of popescu for c n c row contractions the correspondence e has a dual that is naturally isomorphic to e and the commutants of certain so called induced representations of h e can be viewed as induced representations of h e for these induced representations a double commutant theorem is proved

nonconvex optimization problem the infinite horizon linear quadratic control problem with quadratic constraints abstract we present a general method for the solution of the problem of minimization of a quadratic functional in a hilbert space over the intersection of some subspace and a set defined by special quadratic constraints many optimal control problems can be formulated in this abstract form as an example a stationary infinite horizon linear quadratic optimization problem with quadratic constraints is posed and solved

effective noncommutative nevanlinna pick interpolation in the row ball and applications abstract we provide an effective single matrix criterion in terms of what we call the elementary pick matrix for the solvability of the noncommutative nevanlinna pick interpolation problem in the row ball and provide some applications in particular we show that the so called column row property fails for the free semigroup algebras in stark contrast to the analogous commutative case additional applications of the elementary pick matrix include a local dilation theorem for matrix row contractions and interpolating sequences in the noncommutative setting finally we present some numerical results related to the failure of the column row property

quasi min max mpc algorithms for lpv systems in this paper a new model predictive controller mpc is developed for polytopic linear parameter varying lpv systems we adopt the paradigm used in gain scheduling and assume that the time varying parameters are measured on line but their future behavior is uncertain and contained in a given polytope at each sampling time optimal control action is computed by minimizing the upper bound on the quasi worst case value of an infinite horizon quadratic objective function subject to constraints on inputs and outputs the mpc algorithm is called quasi because the first stage cost can be computed without any uncertainty this allows the inclusion of the first move u k k separately from the rest of the control moves governed by a feedback law and is shown to reduce conservatism and improve feasibility characteristics with respect to input and output constraints proposed optimization problems are solved by semi definite programming involving linear matrix inequalities it is shown that closed loop stability is guaranteed by the feasibility of the linear matrix inequalities a numerical example demonstrates the unique features of the mpc design

interpolation problems in several variables abstract we obtain a noncommutative nudel man type interpolation theorem in several variables and a commutative analogue forh b n the hardy space of bounded analytic functions in the open ball of c n using this result we obtain an operatorial analogue in several variables of the rosenblum rovnyak restriction theorem as well as an extension to the noncommutative analytic toeplitz algebraf we provide new extensions of the caratheodory fejer and nevanlinna pick interpolation theorems forf

exponential stability of constrained receding horizon control with terminal ellipsoid constraints in this paper state and output feedback receding horizon controllers are proposed for linear discrete time systems with input and state constraints the proposed receding horizon controllers are obtained from the finite horizon optimization problem with the finite terminal weighting matrix and the artificial invariant ellipsoid constraint which is less restrictive than the conventional terminal equality constraint both hard constraints and mixed constraints are considered in the state feedback case and mixed constraints are considered in the output feedback case it is shown that all proposed state and output feedback receding horizon controllers guarantee the exponential stability of closed loop systems for all feasible initial sets using the lyapunov approach

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

gleason s problem and schur multipliers in the multivariable quaternionic setting abstract we define and study the counterparts of gleason s problem of the drury arveson s space and of schur multipliers when the unit ball of c n is replaced by the unit ball of h n where h denotes the algebra of real quaternions schur multipliers are characterized in terms of coisometric operator matrices in quaternionic spaces finally we define the counterpart of blaschke factors in this setting and study a one point interpolation problem

noncommutative reproducing kernel hilbert spaces abstract the theory of positive kernels and associated reproducing kernel hilbert spaces especially in the setting of holomorphic functions has been an important tool for the last several decades in a number of areas of complex analysis and operator theory an interesting generalization of holomorphic functions namely free noncommutative functions e g functions of square matrix arguments of arbitrary size satisfying additional natural compatibility conditions is now an active area of research with motivation and applications from a variety of areas e g noncommutative functional calculus free probability and optimization theory in linear systems engineering the purpose of this article is to develop a theory of positive kernels and associated reproducing kernel hilbert spaces for the setting of free noncommutative function theory

a polynomial time algorithm for the local testability problem of deterministic finite automata the local testability problem of deterministic finite automata is investigated a locally testable language is a language with the property that for some nonnegative integer k whether or not a word w is in the language depends on 1 the prefix and suffix of w of length k and 2 the set of substrings of w length k 1 without regard to the order in which these substrings occur the local testability problem is given a deterministic finite automation to decide whether or not it accepts a locally testable language the authors present an o n sup 2 time algorithm for the local testability problem based on two simple properties that characterize locally testable automata

on linear programming and robust modelpredictive control using impulse responses abstract campo and morari have derived a linear programming problem with a potentially large number of constraints which is equivalent to a min max formulation for robust model predictive control of linear systems that formulation involves minimization with respect to the controls of the maximum with respect to the system s impulse response from a set of possible impulse responses of the infinity norm of the error between the predicted and required system output sequences here an alternative linear programming problem is derived which has a smaller number of constraints and is therefore potentially more convenient for on line control

on stability of constrained receding horizon control with finite terminal weighting matrix in this paper a new stabilizing receding horizon control based on a finite input and state horizon cost with a finite terminal weighting matrix is proposed for time varying discrete linear systems with constraints we propose matrix inequality conditions on the terminal weighting matrix under which closed loop stability is guaranteed for both cases of unconstrained and constrained systems with input and state constraints we show that such a terminal weighting matrix can be obtained by solving a linear matrix inequality lmi in the case of constrained time invariant systems an artificial invariant ellipsoid constraint is introduced in order to relax the conventional terminal equality constraint and to handle constraints using the invariant ellipsoid constraints a feasibility condition of the optimization problem is presented and a region of attraction is characterized for constrained systems with the proposed receding horizon control

on the morita equivalence of tensor algebras our objective is two fold first we want to develop a notion of morita equivalence for c correspondences that guarantees that if two c correspondences e and f are morita equivalent then the tensor algebras of e and f taoeu and taofu are strongly morita equivalent in the sense of 8 the toeplitz algebras toeu and tofu are strongly morita equivalent in the sense of rieffel 32 see 33 also and the cuntz pimsner algebras 28 ooeu and oofu are strongly morita equivalent in the same sense we were inspired by earlier work of curto williams and the first author in 15 and combes 14 these papers investigate conditions implying that the crossed product of two c dynamical systems are strongly morita equivalent given that the coefficient algebras are strongly morita equivalent abadie eilers and exel 1 were similarly inspired and developed a theory of morita equivalence for special c correspondences our analysis extends theirs but our arguments are different see example 2 5 and remarks 3 3 and 3 6 for a comparison of their results and ours second we want to investigate the converse implication that is we study the following question if taoeu and taofu are strongly morita equivalent in the sense of 8 when are e and f morita equivalent in the sense we define for this converse direction we focus on the tensor algebras rather than the toeplitz and cuntz pimsner algebras because very little can be said in the self adjoint setting here we were inspired by arveson s paper 2 in which it is proved that two ergodic measure preserving transformations are conjugate if and only if certain non self adjoint crossed products associated with the transformations are algebraically isomorphic see 3 also his analysis was further extended by k saito and the first author to cover non self adjoint crossed products built from automorphisms of general von neumann algebras they showed that non self adjoint crossed products built from properly outer automorphisms of von neumann algebras are isomorphic if and only if the automorphisms are conjugate see 22 and 23 our principal result in the converse direction theorem 7 2 asserts that if e and f are two aperiodic correspondences and if taoeu is strongly morita equivalent to taofu in the sense of 8 then e and f are morita equivalent as we shall see in 5 the notion of aperiodicity is an exact analogue for correspondences of the concept of free action in ergodic theory free action was a central hypothesis in arveson s

brief optimizing the end point state weighting matrix in model based predictive control in this paper a linear model based predictive control mpc algorithm is presented for which nominal closed loop stability is guaranteed the input is obtained by minimizing a quadratic performance index over a finite horizon plus an end point state eps penalty subject to input state and output constraints under certain conditions the weighting matrix in the eps penalty enables one to specify an invariant ellipsoid in which the input state and output constraints are satisfied in existing mpc algorithms this weighting matrix is calculated off line the main contribution of this paper is to incorporate the calculation of the eps weighting matrix into the on line optimization problem of the controller the main advantage of this approach is that a natural and automatic trade off between feasibility and optimality is obtained this is demonstrated in a simulation example

poisson transforms on some c algebras generated by isometries abstract a noncommutative poisson transform associated to a certain class of sequences of operators on hilbert spaces with property p is defined on some universal c algebras resp nonselfadjoint algebras generated by isometries its properties are described and used to study these universal algebras and their representations as consequences we obtain a functional calculus isometric resp unitary dilations and commutant lifting theorem for the class of sequences of operators with property p our geometrical approach leads also to new and elementary proofs as well as extensions of some classical results

uniting bounded control and mpc for stabilization of constrained linear systems in this work a hybrid control scheme uniting bounded control with model predictive control mpc is proposed for the stabilization of linear time invariant systems with input constraints the scheme is predicated upon the idea of switching between a model predictive controller that minimizes a given performance objective subject to constraints and a bounded controller for which the region of constrained closed loop stability is explicitly characterized switching laws implemented by a logic based supervisor that constantly monitors the plant are derived to orchestrate the transition between the two controllers in a way that safeguards against any possible instability or infeasibility under mpc reconciles the stability and optimality properties of both controllers and guarantees asymptotic closed loop stability for all initial conditions within the stability region of the bounded controller the hybrid control scheme is shown to provide irrespective of the chosen mpc formulation a safety net for the practical implementation of mpc for open loop unstable plants by providing a priori knowledge through off line computations of a large set of initial conditions for which closed loop stability is guaranteed the implementation of the proposed approach is illustrated through numerical simulations for an exponentially unstable linear system

implementation of stabilizing receding horizon controls for time varying systems in this paper a new stabilizing receding horizon control rhc scheme is proposed for linear discrete time varying systems which can be easily implemented by using linear matrix inequality lmi optimization the control scheme is based on the minimization of the finite horizon cost with a finite terminal weighting matrix the resulting stabilizing rhc scheme leads to time varying finite terminal weighting matrices even for time invariant systems which is more general than in the case of using constant matrices based on the proposed scheme another implementation method is also discussed for easy computation and numerical feasibility consideration of lmi optimization although the second method does not guarantee the closed loop stability theoretically through a simulation example the effectiveness of the proposed schemes is illustrated

an efficient model predictive controller with pole placement for model predictive control mpc of constrained systems enlarging the feasible region is usually in conflict with improving the dynamic performance to resolve the conflict we proposed an efficient model predictive controller with pole placement for a class of discrete time linear systems by specifying a group of circular regions that contain the desired closed loop poles appropriate terminal weighting matrices and local controllers are calculated to construct a time varying terminal convex set which is a significant constraint for the online optimization problem during the online optimization the size of the terminal convex set can adjust itself according to the actual state at each sampling time in this way a large initial feasible region can be achieved while maintaining the good dynamic performance an illustrative example is used to show the effectiveness of the proposed approach

reflexivity of non commutative hardy algebras abstract let h e be a non commutative hardy algebra associated with a w correspondence e these algebras were introduced in 2004 by muhly and solel and generalize the classical hardy algebra of the unit disc h d as a special case one obtains also the algebra f d of popescu which is h c d in our setting in this paper we view the algebra h e as acting on a hilbert space via an induced representation we write it h e and we study the reflexivity of h e this question was studied by arias and popescu in the context of the algebra f d and by other authors in several other special cases as it will be clear from our work the extension to the case of a general w correspondence e over a general w algebra m requires new techniques and approach we obtain some partial results in the general case and we turn to the case of a correspondence over a factor under some additional assumptions on the representation m b h we show that h e is reflexive then we apply these results to analytic crossed products h m and obtain their reflexivity for any automorphism aut m whenever m is a factor finally we show also the reflexivity of the compression of the hardy algebra to a suitable coinvariant subspace m which may be thought of as a generalized symmetric fock space