two dimensional quantum spaces corresponding to solutions of the yang baxter equations the quantum spaces given by relation f r x x x 0 where r are eight and less vertex constant solutions of the yang baxter equations and their invariance algebras as investigated

complex earthquakes and teichmuller theory it is known that any two points in teichmuller space are joined by an earthquake path in this paper we show any earthquake path r t s extends to a proper holomorphic mapping of a simplyconnected domain d into teichmuller space where r d c these complex earthquakes relate weil petersson geometry projective structures pleated surfaces and quasifuchsian groups using complex earthquakes we prove grafting is a homeomorphism for all 1 dimensional teichmuller spaces and we construct bending coordinates on bers slices and their generalizations in the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold

robust model predictive control of constrained linear systems with bounded disturbances this paper provides a novel solution to the problem of robust model predictive control of constrained linear discrete time systems in the presence of bounded disturbances the optimal control problem that is solved online includes uniquely the initial state of the model employed in the problem as a decision variable the associated value function is zero in a disturbance invariant set that serves as the origin when bounded disturbances are present and permits a strong stability result namely robust exponential stability of the disturbance invariant set for the controlled system with bounded disturbances to be obtained the resultant online algorithm is a quadratic program of similar complexity to that required in conventional model predictive control

hyperbolic cone manifolds short geodesics and schwarzian derivatives given a geometrically finite hyperbolic cone manifold with the cone singularity sufficiently short we construct a one parameter family of cone manifolds decreasing the cone angle to zero we also control the geometry of this one parameter family via the schwarzian derivative of the projective boundary and the length of closed geodesics

positive invariant sets for fault tolerant multisensor control schemes this article deals with fault tolerant multisensor control schemes for systems with linear dynamics positive invariance is a common analysis and control design tool for systems affected by bounded constraints and disturbances this article revisits the construction of e approximations of minimal robust positive invariant sets for linear systems upon contractive set iterations the cases of switching between different sets of disturbances and the inclusion of a predefined region of the state space are treated in detail all these results are used in multisensor control schemes which have to deal with specific problems originated by the switching between different estimators and by the presence of faults in some of the sensors the construction of positive invariant sets for different operating regimes provides in this context effective fault detection information within the same framework global stability of the switching strategies can be assured if the invariant sets topology allows the exclusive se

application of shifted jacobi pseudospectral method for solving in finite horizon min max optimal control problems with uncertainty abstractthe difficulty of solving the min max optimal control problems m mocps with uncertainty using generalised euler lagrange equations is caused by the combination of split boundary conditions nonlinear differential equations and the manner in which the final time is treated in this investigation the shifted jacobi pseudospectral method sjpm as a numerical technique for solving two point boundary value problems tpbvps in m mocps for several boundary states is proposed at first a novel framework of approximate solutions which satisfied the split boundary conditions automatically for various boundary states is presented then by applying the generalised euler lagrange equations and expanding the required approximate solutions as elements of shifted jacobi polynomials finding a solution of tpbvps in nonlinear m mocps with uncertainty is reduced to the solution of a system of algebraic equations moreover the jacobi polynomials are particularly useful for boundary value problems in unbounded

controlled invariance and feedback laws we consider controlled invariance for cones translated cones polyhedra and various special polyhedral structures for any polyhedron if controlled invariance occurs then all and nothing but the admissible controls can be obtained by an inequative feedback controller for each special polyhedral structure we compare this feedback controller against piecewise affine linear and affine feedback controllers

regenerating singular hyperbolic structures from sol let m be a torus bundle over s 1 with an orientation preserving anosov monodromy the manifold m admits a geometric structure modeled on sol we prove that the sol structure can be deformed into singular hyperbolic cone structures whose singular locus m is the mapping torus of the fixed point of the monodromy the hyperbolic cone metrics are parametred by the cone angle in the interval 0 2 when 2 the cone manifolds collapse to the basis of the fibration s 1 and they can be rescaled in the direction of the fibers to converge to the sol manifold

quadratically constrained attitude control via semidefinite programming we consider the attitude control problem augmented with an arbitrary number of nonconvex quadratic constraints by utilizing the implicit magnitude constraint of the state vector i e quaternion we are able to reduce this problem to a convex quadratically constrained quadratic program or a semidefinite program this result leads to an efficient solution strategy for general quadratically constrained attitude control problems an example demonstrates the proposed methodology

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

time optimal model predictive control for linear systems based on ellipsoidal and polytopic sets time optimal model predictive control mpc strategies are investigated in this paper we first derive a series of controllable sets described by ellipsoids or convex polytopes offline then obtain

constrained infinite horizon linear quadratic regulation of discrete time systems we are concerned with the output norm constrained infinite horizon linear quadratic regulation problem where the underlying state control constraints are specified by curved rather than polyhedral surfaces each suboptimal problem admits an exact convex synthesis condition expressed by an increasing union of linear matrix inequalities and results in an eventually time invariant controller satisfying the desired performance level subject to given constraints and closed loop stability the output norm constrained minimum time control problem is also dealt with as a corollary

nonlinear control of constrained linear systems this paper addresses the problem of obtaining nonlinear state feedback controllers for controlling constrained dynamic systems a problem which is perhaps the most important outstanding problem in control design a novel off line optimization based design procedure is employed the control law is constructed from the solution to a finite number of optimal open loop control problems the design procedure permits relatively simple modification and improvement to the control law to be made in those regions of the state space in which simulation reveals deficiencies this potential for incremental improvement constitutes an important advantage over alternative methods the results are relevant when high performance is required of systems subject lo constraints on controls and states

laminations mesurees de plissage des varietes hyperboliques de dimension 3 for a hyperbolic metric on a 3 dimensional manifold the boundary of its convex core is a surface which is almost everywhere totally geodesic but which is bent along a family of disjoint geodesics the locus and intensity of this bending is described by a measured geodesic lamination which is a topological object we consider two problems the topological characterization of those measured geodesic laminations which can occur as bending measured laminations of hyperbolic metrics and the uniqueness problem which asks whether a hyperbolic metric is uniquely determined by its bending measured lamination

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

hyperbolic manifolds with convex boundary let m m be a 3 manifold which carries a hyperbolic metric with convex boundary we consider the hyperbolic metrics on m such that the boundary is smooth and strictly convex we show that the induced metrics on the boundary are exactly the metrics with curvature k 1 and that the third fundamental forms of m are exactly the metrics with curvature k 2 each is obtained exactly once

a variable structure convex programming based control approach for a class of uncertain linear systems a variable structure convex programming based control for a class of linear uncertain systems with accessible state is presented in this note a convex programming problem is solved on line by reformulating the problem in terms of a piecewise smooth penalty function and relying on a suitable analog variable structure system implementing the gradient procedure in this way the controlled system reference movement optimal with respect to a pre specified scalar convex cost function and a set of suitable equality and inequality constraints is generated an inner control loop aimed at the finite time exact tracking of the reference movement is also designed as a result the controlled system trajectory starting in the feasible region there remains and the optimal movement in the feasible region is proved to be an asymptotically stable equilibrium point of the controlled system

collisions of particles in locally ads spacetimes ii moduli of globally hyperbolic spaces we investigate globally hyperbolic 3 dimensional ads manifolds containing particles i e cone singularities of angles less than 2 along a time like graph to each such space equipped with a time like vector field satisfying some additional properties we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities we show that this data is sufficient to recover the space locally i e in the neighborhood of a fixed metric this is a partial extension of a result of mess for non singular globally hyperbolic ads manifolds

fixed points of compositions of earthquakes let s be a closed surface of genus at least 2 and consider two measured geodesic laminations that fill s right earthquakes along these laminations are diffeomorphisms of the teichmuller space of s we prove that the composition of these earthquakes has a fixed point in the teichmuller space another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic ads manifold the proof uses some estimates from the geometry of those ads manifolds

computation of minimum time feedback control laws for discrete time systems with state control constraints the problem of finding a feedback law that drives the state of a linear discrete time system to the origin in minimum time subject to state control constraints is considered algorithms are given to obtain facial descriptions of the m step admissible sets these descriptions are then used to characterize the complete class of minimum time feedback laws moreover the characterization leads to a conceptually simple on line implementation the main ideas are illustrated with two simple examples

brief robust receding horizon predictive control for systems with uncertain dynamics and input saturation a receding horizon predictive control method which assures stability for systems with model uncertainty and input saturation is derived by extending earlier work in two important respects i ellipsoidal invariant sets are replaced by polyhedral invariant sets and ii the constraint that the state lie in an invariant set is invoked n steps ahead the new algorithm allows for a considerable enlargement of the stabilizable set and provides extra degrees of freedom with which to improve performance

local rigidity of hyperbolic 3 manifolds after dehn surgery it is well known that some lattices inso n 1 can be nontrivially deformed when included in so n 1 1 e g via bending on a totally geodesic hypersurface this contrasts with the super rigidity of higher rank lattices m kapovich recently gave the first examples of lattices in so 3 1 which are locally rigid inso 4 1 by considering closed hyperbolic3 manifolds obtained by dehn filling on hyperbolic two bridge knots we generalize this result to dehn filling on a more general class of one cusped finite volume hyperbolic 3 manifolds allowing us to produce the first examples of closed hyperbolic 3 manifolds which contain embedded quasi fuchsian surfaces but are locally rigid in so 4 1

the quantum general linear supergroup and braid statistics we generalize to braid statistics some of manin s work 1989 on the quantum deformation of the general linear supergroup the key ingredient in this construction is the introduction of a non standard transposition map psi psi 2 not 1 which is defined in terms of a generalized permutation operator pmu pmu 2 not 1 the dual space enveloping algebra with pmu statistics is defined consideration of coactions on quantum spaces clarifies the resulting structure

projective structures with degenerate holonomy and the bers density conjecture we prove the bers density conjecture for singly degenerate kleinian surface groups without parabolics

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

pinching constants for hyperbolic manifolds we show in this paper that for everyn 4 there exists a closedn dimensional manifoldv which carries a riemannian metric with negative sectional curvaturek but which admits no metric with constant curvaturek 1 we also estimate the pinching constantsh for which our manifoldsv admit metrics with 1 k h

nonlinear tracking control in the presence of state and control constraints a generalized reference governor this paper proposes a new approach to reference governor design as in prior literature the governor accepts input commands and modifies their evolution so that specified pointwise in time constraints on state and control variables are satisfied the new approach applies to general discrete time and continuous time nonlinear systems with uncertainties it relies on safety properties provided by sublevel sets of equilibria parameterized functions these functions need not be lyapunov functions and the corresponding sublevel sets need not be positively invariant technical conditions that capture the bare essentials of what is needed are identified and the usual desirable properties of reference governors are established the new approach significantly broadens the class of methods available for constructing the nonlinear function that is required in the implementation of the reference governors this advantage is illustrated in a nonlinear control problem where off line computer based simulation is the basis for constructing the nonlinear function

optimized robust control invariance for linear discrete time systems theoretical foundations this paper introduces the concept of optimized robust control invariance for discrete time linear time invariant systems subject to additive and bounded state disturbances a novel characterization of two families of robust control invariant sets is given the existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear convex case and a standard linear quadratic program when the constraints are polyhedral or polytopic the solution of the same optimization problem yields the corresponding feedback control law that is in general set valued a procedure for selection of a point valued nonlinear control law is provided

differential calculuses on the quantized braided groups braided differential calculuses on the quantized braided groups are constructed and their braided bialgebra hopf algebra structures are demonstrated these are a kind of generalization and unification of the differential calculuses on quantum groups braided groups quantum supergroups etc and contain the latter ones as special cases moreover it is shown that some quantum differential co vector algebras are covariant under the braided local coactions of the obtained braided differential bialgebras hopf algebras some examples are also given

time optimal control systhesis for discrete time hybrid automata in this paper the problem of time optimal control for hybrid systems with discrete time dynamics is considered the hybrid controller steers all trajectories starting from a maximal set to a given target set in minimum time we derive an algorithm that computes this maximal winning set also algorithms for the computation of level sets associated with the value function rather than the value function itself are presented we show that by solving the reachability problem for the discrete time hybrid automata we obtain the time optimal solution as well the control synthesis is subject to hard constraints on both control inputs and states for linear discrete time dynamics linear programming and quantifier elimination techniques are employed for the backward reachability analysis emphasis is given on the computation of operators for non convex sets using an extended convex hull approach a two tank example is considered in order to demonstrate the techniques of the paper

the grafting map of teichmuller space one of the underlying principles in the study of kleinian groups is that aspects of the complex projective geometry of quotients of by these groups reflect properties of the three dimensional hyperbolic geometry of the quotients ofh by these groups yet even though it has been over thirty five years since lipman bers exhibited a holomorphic embedding of the teichmuller space of riemann surfaces in terms of the projective geometry of a teichmuller space of quasi fuchsian manifolds no corresponding embedding in terms of the three dimensional hyperbolic geometry has been presented one of the goals of this paper is to give such an embedding this embedding is straightforward and has been expected for some time ta97 mc98 to each member of a bers slice of the space qf of quasi fuchsian 3manifolds we associate the bending measured lamination of the convex hull facing the fixed conformal end the geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting an operation on projective structures on surfaces that traces its roots back at least to klein kl33 p 230 50 with a modern history developed by many authors ma69 he75 fa83 st83 go87 gkm00 ta97 mc98 the main technical tool in our proof that bending measures give coordinates for bers slices and the second major goal of this paper is the completion of the proof of the grafting conjecture this conjecture states that for a fixed measured lamination the self map of teichmuller space induced by grafting a surface along is a homeomorphism of teichmuller space our contribution to this argument is a proof of the injectivity of the grafting map while the principal application of this result that we give is to geometric coordinates on the bers slice of qf one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of kleinian groups see 5 2 thus we feel that this result is of interest in its own right we now state our results and methods more precisely throughout s will denote a fixed differentiable surface which is closed orientable and of genus g 2 let tg be the corresponding teichmuller space of marked conformal structures on s and let pg denote the deformation space of complex projective structures on s see 2 for definitions

brief paper the minimal disturbance invariant set outer approximations via its partial sums this paper is concerned with outer approximations of the minimal disturbance invariant set mdis of a discrete time linear system with an additive set bounded disturbance the k step disturbance reachable sets minkowski partial sums are inner approximations of mdis that converge to mdis enlarged by a suitable scaling they can lead to outer approximations of mdis three families of approximations each based on partial sums are considered theoretical properties of the families are proved and interrelated algorithmic questions including error bounds are addressed the results are illustrated by computational data from several examples

optimization over state feedback policies for robust control with constraints this paper is concerned with the optimal control of linear discrete time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input it is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence this implies that a broad class of constrained finite horizon robust and optimal control problems where the optimization is over affine state feedback policies can be solved in a computationally efficient fashion using convex optimization methods this equivalence result is used to design a robust receding horizon control rhc state feedback policy such that the closed loop system is input to state stable iss and the constraints are satisfied for all time and all allowable disturbance sequences the cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance free state and input sequences the value of the receding horizon control law can be calculated at each sample instant using a single tractable and convex quadratic program qp if the disturbance set is polytopic or a tractable second order cone program socp if the disturbance set is given by a 2 norm bound

universal bounds for hyperbolic dehn surgery this paper gives a quantitative version of thurston s hyperbolic dehn surgery theorem applications include the first universal bounds on the number of nonhyperbolic dehn fillings on a cusped hyperbolic 3 manifold and estimates on the changes in volume and core geodesic length during hyperbolic dehn filling the proofs involve the construction of a family of hyperbolic conemanifold structures using infinitesimal harmonic deformations and analysis of geometric limits

on the density of geometrically finite kleinian groups the density conjecture of bers sullivan and thurston predicts that each complete hyperbolic 3 manifold m with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3 manifolds we prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends

robust explicit model predictive control via regular piecewise affine approximation this paper proposes an explicit model predictive control design approach for regulation of linear time invariant systems subject to both state and control constraints in the presence of additive disturbances the proposed control law is implemented as a piecewise affine function defined on a regular simplicial partition and has two main positive features first the regularity of the simplicial partition allows one to efficiently implement the control law on digital circuits thus achieving extremely fast computation times moreover the asymptotic stability or the convergence to a set including the origin of the closed loop system can be enforced a priori rather than checked a posteriori via lyapunov analysis

robust time optimal control of constrained linear systems a version of dynamic programming which computes level sets of the value function rather than the value function set itself is used to design robust non linear controllers for linear discrete time dynamical systems subject to hard constraints on controls and states the controller stabilizes the system and steers all trajectories emanating in a prescribed set to a control invariant set in minimum time for the robust regulator problem the control invariant terminal set is a neighborhood preferably small of the origin for the robust tracking problem the control invariant terminal set is a neighborhood of the invariant set in which the tracking error is zero two non linear controllers which utilize the level sets of the value function are described the first requires the controller to solve on line a modest linear program whose dimension is approximately the same as that of the control variable the second decomposes each level set into a set of simplices a piecewise linear control law affine in each simplex is then constructed

robust model predictive control using tubes a form of feedback model predictive control mpc that overcomes disadvantages of conventional mpc but which has manageable computational complexity is presented the optimal control problem solved on line yields a tube and an associated piecewise affine control law that maintains the controlled trajectories in the tube despite uncertainty computational complexity is linear rather than exponential in horizon length asymptotic stability of the controlled system is established

braided coaddition on the quantized braided matrices an additive braided coproduct also be called braided coaddition is introduced on the quantized braided matrices qbms it gives qbms another braided hopf algebra structure the coaddition is also shown to be compatible with the existing coproduct such that they together form a so called quantized braided ring and some quantized braided differential operator bialgebras hopf algebras relating to this braided coaddition are constructed these give a unification and generalization of the known results about braided and quantum matrices moreover the coaddition construction and the related differential calculi on the qbms are further extended to a kind of more general quantum matrix algebraic systems some examples are also given

systems with persistent disturbances predictive control with restricted constraints predictive regulation of linear discrete time systems subject to unknown but bounded disturbances and to state control constraints is addressed an algorithm based on constraint restrictions is presented and its stability properties are analysed

quantized braided groups algebras that represent a generalization of both quantum groups quantum supergroups and braided groups are defined they are given by a pair of solutions of the yang baxter equation that satisfy some additional conditions several examples are presented

collisions of particles in locally ads spacetimes i local description and global examples we investigate 3 dimensional globally hyperbolic ads manifolds or more generally constant curvature lorentz manifolds containing particles i e cone singularities along a graph we impose physically relevant conditions on the cone singularities e g positivity of mass angle less than 2 on time like singular segments we construct examples of such manifolds describe the cone singularities that can arise and the way they can interact the local geometry near the vertices of we then adapt to this setting some notions like global hyperbolicity which are natural for lorentz manifolds and construct some examples of globally hyperbolic ads manifolds with interacting particles

examples of braided groups and braided matrices matrix braided groups are developed as an analog of the coordinate functions on a group or supergroup the 1 in the super case is replaced by braid statistics there are braided group analogs of all the classical simple lie groups as well as braided matrix groups and braided matrices b r for every regular solution r of the quantum yang baxter equations a direct verification of b r is provided and some of the simplest examples are computed in detail

cannon thurston maps for surface groups we prove the existence of cannon thurston maps for simply and doubly degenerate surface kleinian groups as a consequence we prove that connected limit sets of nitely generated kleinian groups are locally connected

quantized braided linear algebras relating to quantized braided groups an r matrix pair r z solving a system of yang baxter type equations is needed to define a quantized braided matrix group l hlavaty j math phys 35 2560 1994 it is found that a series of such kind of r matrix pairs r n z n 0 1 2 can be constructed from a known pair r z and a series of realizations of the quantized braided matrix bialgebras a r n z in v r n 1 v r n can be obtained some covariant quantized braided linear algebras and their transformation properties under the braided coactions of the quantized braided group a r z are considered some examples are presented

brief a polytopic game we consider a discrete time linear dynamic game in state space form where the actions of two players are confined to belong to prescribed convex polytopes at all times each player has a home set a polyhedral domain in state space and should devise a cooperative or a non cooperative strategy contingent on the known initial state to reach her own home set in a finite number of steps and stay there irrespective of the action taken by the opponent players act independently on the basis of the state at each time using feedback strategies purpose of the analysis is to classify the set of initial states into winning sets and draw sets for each player and prescribe strategies able to implement the kind of invariance home sets should enjoy in the paper necessary and sufficient conditions for the existence of invariant sets with the above properties are derived these conditions and the relative strategies are characterized in terms of linear programming problems that provide a straightforward algorithmic solution to the game

on the robustness of nominal nonlinear minimum time control and extension to non robustly controllable target sets this work deals with the analysis and the design of minimum time control laws for a class of nonlinear discrete time dynamical systems characterized by k continuous transition maps and bounded control inputs in the paper it is shown that the reachability properties of the target set even if not robust positively controllable in one state transition can be exploited to assess the existence of a robust positively controllable set including the target in its interior this result allows the formulation of a robustified minimum time control policy based on iterated online optimizations and guaranteeing the ultimate boundedness of the state trajectories in the presence of bounded uncertainties even if the target set is not robust positively controllable

quasi fuchsian manifolds with particles we consider 3 dimensional hyperbolic cone manifolds which are convex cocompact in a natural sense with cone singularities along infinite lines such singularities are sometimes used by physicists as models for massive spinless point particles we prove an infinitesimal rigidity statement when the angles around the singular lines are less than any infinitesimal deformation changes either these angles or the conformal structure at infinity with marked points corresponding to the endpoints of the singular lines moreover any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone manifold structure these results hold also when the singularities are along a graph i e for interacting particles

parameterized robust control invariant sets for linear systems theoretical advances and computational remarks we characterize a family of parametrized robust control invariant sets for linear discrete time systems subject to additive but bounded state disturbances the existence of a member of the introduced family of parametrized robust control invariant sets can be verified by solving a tractable convex optimization problem in the linear convex case which reduces to the standard linear or convex quadratic programme in the linear polytopic case the developed method can also be utilized to detect and obtain an implicit representation of local control lyapunov functions in the linear convex case from the solution of a single and tractable convex optimization problem the offered method permits for the computation of polytopic robust control invariant sets and local control lyapunov functions of indirectly controlled and limited complexity in the linear polytopic case

invariant approximations of the minimal robust positively invariant set this note provides results on approximating the minimal robust positively invariant mrpi set also known as the 0 reachable set of an asymptotically stable discrete time linear time invariant system it is assumed that the disturbance is bounded persistent and acts additively on the state and that the constraints on the disturbance are polyhedral results are given that allow for the computation of a robust positively invariant outer approximation of the mrpi set conditions are also given that allow one to a priori specify the accuracy of this approximation

min max optimal control of linear systems with uncertainty and terminal state constraints in this paper a class of min max optimal control problems with continuous dynamical systems and quadratic terminal constraints is studied the main contribution is that the original terminal state constraint in which the disturbance is involved is transformed into an equivalent linear matrix inequality without disturbance under certain conditions then the original min max optimal control problem is solved via solving a sequence of semi definite programming problems an example is presented to illustrate the proposed method

a framework for multiple robust explicit mpc controllers for linear systems this article presents a multi mode explicit controller for constrained linear systems with bounded disturbances using a switching strategy based on model predictive control mpc in the proposed approach the system switches among several mpc controllers having different performance levels the switching is done so as to achieve increasing levels of performance as time evolves reaching the desired controller in finite time steps the conditions needed for switching and robust convergence of the multi mode mpc controllers are provided compared with standard robust explicit mpc implementations the proposed approach has the flexibility of having a large domain of attraction a good asymptotic behaviour and a small number of partitions

quantum and braided linear algebra quantum matrices a r are known for every r matrix obeying the quantum yang baxter equations it is also known that these act on vectors given by the corresponding zamalodchikov algebra this interpretation is developed in detail distinguishing between two forms of this algebra v r vectors and v r covectors a r v r21 v r is an algebra homomorphism i e quantum matrices are realized by the tensor product of a quantum vector with a quantum covector while the inner product of a quantum covector with a quantum vector transforms as a scaler it is shown that if v r and v r are endowed with the necessary braid statistics then their braided tensor product v r v r is a realization of the braided matrices b r introduced previously while their inner product leads to an invariant quantum trace introducing braid statistics in this way leads to a fully covariant quantum braided linear algebra the braided groups obtained from b r act on themselves by conjugation in a way impossible

robust synergetic control design under inputs and states constraints abstractin this paper a novel robust constrained control methodology for discrete time linear parameter varying dt lpv systems is proposed based on a synergetic control theory sct approach it is shown that in dt lpv systems without uncertainty and for any unmeasured bounded additive disturbance the proposed controller accomplishes the goal of stabilizing the system by asymptotically driving the error of the controlled variable to a bounded set containing the origin and then maintaining it there moreover given an uncertain dt lpv system jointly subject to unmeasured and constrained additive disturbances and constraints in states input commands and reference signals setpoints then invariant set theory is used to find an appropriate polyhedral robust invariant region in which the proposed control framework is guaranteed to robustly stabilize the closed loop system furthermore this is achieved even for the case of varying non zero control setpoints in such uncertain dt lpv systems the contr

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

an ellipsoidal off line mpc scheme for uncertain polytopic discrete time systems an off line model predictive control mpc method based on ellipsoidal calculus and viability theory is described in order to address control problems in the presence of state and input constraints for uncertain polytopic linear plants subject to persistent disturbances in order to reduce the computational burdens and conservativeness of traditional polytopic mpc schemes the present approach carries out off line most of the computations and it makes use of closed loop predictions to improve the control performance this is done by recursively pre computing suitable ellipsoidal inner approximations of the exact controllable sets and solving on line a simple and numerically low demanding optimization problem subject to a set membership constraint comparisons with three other recent off line mpc approaches are also provided in the final example

infinitesimal rigidity of cone manifolds and the stoker problem for hyperbolic and euclidean polyhedra the deformation theory of hyperbolic and euclidean cone manifolds with all cone angles less than 2 plays an important role in many problems in low dimensional topology and in the geometrization of 3 manifolds furthermore various old conjectures dating back to stoker about the moduli space of convex hyperbolic and euclidean polyhedra can be reduced to the study of deformations of cone manifolds by doubling a polyhedron across its faces this deformation theory has been understood by hodgson and kerckhoff 7 when the singular set has no vertices and by weis 32 when the cone angles are less than we prove here an infinitesimal rigidity result valid for cone angles less than 2 stating that infinitesimal deformations which leave the dihedral angles fixed are trivial in the hyperbolic case and reduce to some simple deformations in the euclidean case the method is to treat this as a problem concerning the deformation theory of singular einstein metrics and to apply analytic methods about elliptic operators on stratified spaces this work is an important ingredient in the local deformation theory of cone manifolds by the second author 22 see also the concurrent work by weis 33

cusp shapes under cone deformation a horospherical torus about a cusp of a hyperbolic manifold inherits a euclidean similarity structure called a cusp shape we bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp the bounds are in terms of the change in structure in a neighborhood of the singular locus alone we then apply this result to provide information on the cusp shape of many hyperbolic knots given only a diagram of the knot specifically we show there is a universal constant c such that if a knot admits a prime twist reduced diagram with at least c crossings per twist region then the length of the second shortest curve on the cusp torus is bounded the bound is linear in the number of twist regions of the diagram

local rigidity of 3 dimensional cone manifolds we investigate the local deformation space of 3 dimensional conemanifold structures of constant curvature 1 0 1 and coneangles under this assumption on the cone angles the singular locus will be a trivalent graph in the hyperbolic and the spherical case our main result is a vanishing theorem for the first l 2 cohomology group of the smooth part of the cone manifold with coefficients in the flat bundle of infinitesimal isometries weconclude local rigidity from this in the euclidean case we prove that the first l 2 cohomology group of the smooth part with coefficients in the flat tangent bundle is represented by parallel forms

examples of relative deformation spaces that are not locally connected we provide an infinite family of pared manifolds with the property that the relative deformation spaces of hyperbolic structures on these manifolds are not locally connected this is a natural extension of the recent result of bromberg that shows the space of kleinian punctured torus groups is not locally connected