properties of recoverable region and semi global stabilization in recoverable region for linear systems subject to constraints this paper investigates linear systems subject to input and state constraints it is shown that the recoverable region which is the largest domain of attraction that is theoretically achievable can be semiglobally stabilized by continuous nonlinear feedbacks while satisfying the constraints moreover when trying to compute the recoverable region a reduction technique shows that we only need to compute the recoverable region for a system of lower dimension which generally leads to a considerable simplification in the computational effort

detection of attraction domains of non linear systems using bifurcation analysis and lyapunov functions the objective of this paper is to contribute to the problem of the domain of attraction doa of locally stable non linear systems determining exactly the attraction basin is in general a difficult open problem in our proposal the main original idea is to combine tools from bifurcation theory and from lyapunov theory it is believed that this interplay is the most adequate to get a nice picture of the essential objects affecting local stability the viability of all these ideas is confirmed by the successfull application to the furuta pendulum the strategies proposed here offer the advantage of a conceptually appealing approach which obtains a conservative ellipsoidal estimation of the doa this ellipsoid is reasonably good and the computational burden is drastically reduced compared to the pure lyapunov blind search the search is improved when bifurcation information is taken into account the synthesis problem is also considered and synthesis conditions are given in the form of feasible pa

on constructing constrained control lyapunov functions for linear systems this works considers the problem of control of linear systems with input constraints the notion of control lyapunov functions clfs is first generalized and relaxed to define a constrained control lyapunov function cclf and a constructive procedure for cclf is subsequently presented

a differential complex for poisson manifolds construction du complexe canonique d une variete de poisson homologie canonique des varietes symplectiques application a l homologie de hochschild des algebres non commutatives

a lagrange approach to set theoretic control synthesis a lagrange approach to solving the nonlinear constrained optimization problem arising in the set theoretic control problem is described by introducing matrix lagrange multipliers the problem is reduced to that of solving a set of nonlinear simultaneous matrix equations one of which is the familiar matrix riccati equation frequently encountered in linear quadratic control theory the structural similarities and differences between set theoretic and linear quadratic control methods are identified the results obtained from the set theoretic control approach are compared with those obtained from the linear quadratic control approach

an 8 dimensional nonformal simply connected symplectic manifold we answer in the affirmative the question posed by babenko and taimanov 3 on the existence of nonformal simply connected compact symplectic manifolds of dimension 8

geometric quantization and multiplicities of group representations the heisenberg uncertainty principle says that it is impossible to determine simultaneously the position and momentum of a quantum mechanical particle this can be rephrased as follows the smallest subsets of classical phase space in which the presence of a quantum mechanical particle can be detected are its lagrangian submanifolds for this reason it makes sense to regard the lagrangian submanifolds of phase space as being its true points see weinstein 17 now let g be a compact lie group and g x x x a hamiltonian action of g on x see w for definitions it is well known that the fixed points of this action form a symplectic submanifold of x see for instance guillemin and sternberg 5 however what can one say about the fixed points of g we will show that they are also the points of a symplectic manifold xc this manifold is the marsden weinstein reduction of x with respect to the zero orbit in g and will be described in sect 2 it was introduced in a completely different context fl om ours by marsden and weinstein 12 problems in classical mechanics can often be reduced to the study of hamiltonian systems on symplectic manifolds and problems in quantum mechanics to the study of linear operators on hilbert space this fact has inspired a number of efforts to quantize symplectic geometry by devising schemes for associating hilbert space to symplectic manifolds the no go theorems of groenwald and van hove impose some embarrassing limitations on all such schemes however it seems to be a useful idea heuristically to think of every symplectic manifold x a ic 1 as being symbiotically associated with a hilbert space xq a t in such a way that the classical observables on the first space correspond to quantum observables on the second space the heuristics further suggests that if g is a group of symmetries of x la i j it should also be a group of symmetries of xq a tum in this heuristic spirit we will state the main conjecture of this paper

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

symplectic resolutions lefschetz property and formality we introduce a method to resolve a symplectic orbifold m omega into a smooth symplectic manifold m over tilde omega over tilde then we study how the formality and the lefschetz property of m over tilde omega over tilde are compared with that of m omega we also study the formality of the symplectic blow up of m omega along symplectic submanifolds disjoint from the orbifold singularities this allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the lefschetz property but is not formal therefore giving a counter example to a conjecture of babenko and taimanov

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

on stabilizing linear systems with input saturation via soft variable structure control laws abstract a large number of control methods deal with stabilizing single input linear systems with input saturation the linear full state feedback is one of the most used yielding either nonsaturating or high gain controls whereby the saturating effects of the latter are reduced by means of anti windup structures among the nonlinear controls the time optimal control in this case a bang bang type control yields the fastest time response but its switching surface is generally not characterizable related to the speed of the time response is the convergence rate which can be determined using invariant sets the most used ones are the ellipsoidal sets since they can be analyzed using powerful tools such as the lyapunov equation and designed via convex optimization for this reason they are also used for designing soft variable structure controls the paper presents a nonconservative design method for a stabilizing control of this type employing implicit lyapunov functions ilf a nonsaturating control law is given including some infinitely densely nested and contractive invariant sets of the equilibrium state the control law is then optimized by maximizing the ilf based lower bound of the convergence rate the maximal convergence control is shown to be of bang bang type with a parameter dependent switching scheme to overcome possible difficulties of a switching controller a saturating high gain control is also presented

a new class of lyapunov functions for the constrained stabilization of linear systems the constrained stabilization of linear uncertain systems is investigated via the set theoretic framework of control lyapunov r functions a novel composition rule allows the design of a composite control lyapunov function with external level set that exactly shapes the maximal controlled invariant set and inner sublevel sets arbitrarily close to any choice of smooth ones generalizing both polyhedral and truncated ellipsoidal control lyapunov functions the feasibility test of the proposed smooth control lyapunov functions can be cast into matrix inequalities conditions the constrained linear quadratic control is addressed as an application

reachable controllable sets and stabilizing control of constrained linear systems we consider discrete time linear systems with constrained controls we present an algorithm for computing an external representation of the n step reachable set and controllable set it is an improvement of previous methods in particular it is shown that for constructing the set of initial states that can be driven to the origin asymptotically one can restrict the search to a polyhedral set in the unstable subspace of the autonomous system which yields a drastic reduction in the computational burden application to the synthesis of stabilizing feedback controllers is discussed

feedback control of linear discrete time systems under state and control constraints in this paper the problem of stabilizing linear discrete time systems under state and control linear constraints is studied based on the concept of positive invariance existence conditions of linear state feedback control laws respecting both the constraints are established these conditions are then translated into an algorithm of linear programming

ultimate boundedness control for uncertain discrete time systems via set induced lyapunov functions in this note linear discrete time systems affected by both parameter and input uncertainties are considered the problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned rate of convergence is investigated it is shown that the problem has a solution if and only if there exists a certain lyapunov function which does not belong to a preassigned class of functions e g the quadratic ones but it is determined by the target set in which ultimate boundedness is desired one of the advantages of this approach is that we may handle systems with control constraints no matching assumptions are made for systems with linearly constrained uncertainties it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets the resulting compensator may be implemented as a linear variable structure control

control sharing and merging control lyapunov functions given two control lyapunov functions clfs a merging is a new clf whose gradient is a positive combination of the gradients of the two parents clfs the merging function is an important trade off since this new function may for instance approximate one of the two parents functions close to the origin while being close to the other far away for nonlinear control affine systems some equivalence properties are shown between the control sharing property i e the existence of a single control law which makes simultaneously negative the lyapunov derivatives of the two given clfs and the existence of merging clfs it is shown that even for linear time invariant systems the control sharing property does not always hold with the remarkable exception of planar systems the class of linear differential inclusions is also discussed and similar equivalence results are presented for this class of systems linear matrix inequalities conditions are provided to guarantee the control sharing property finally a constructive procedure based on the recently considered r functions is defined to merge two smooth positively homogeneous clfs

a set induced norm approach to the robust control of constrained systems most realistic control problems involve both some type of time domain constraints and model uncertainty however the majority of controller design procedures currently available focus only on one aspect of the problem with only a handful of methods capable of simultaneously addressing both issues recently it has been proposed to address this class of problems by using a constrained robustness measure generated by a constraint set induced operator norm to assess the stability properties of a family of systems in this paper the properties of this constrained robustness measure are explored and the theoretical framework is extended to include control as well as state constraints these results are applied to the problem of designing fixed order stabilizing feedback controllers for systems subject to structured parametric model uncertainty and time domain constraints

constrained stabilization of continuous time linear systems in this paper we consider the stabilization problem for linear continuous time systems under state and control constraints we show that the largest domain of attraction to the origin can be arbitrarily closely approximated by a polyhedral domain of attraction associated to a certain continuous feedback stabilizing control and we show how to use existing numeric procedures for discrete time systems to solve the continuous time problem we propose a new discontinuous stabilizing control law for scalar input systems which has the advantage of being successfully applicable to systems with quantized control

positively invariant sets for constrained continuous time systems with cone properties this note deals with some properties of particular bounded sets w r t linear continuous time systems described by x spl dot t a 0 x t c t where c t spl isin spl omega spl sub r sup n spl omega a compact set and matrix e sup ta 0 has the property of leaving a proper cone k positively invariant that is e sup ta 0 k spl sub k the considered bounded sets spl dscr k a b are described as the intersection of shifted cones necessary and sufficient conditions are given they guarantee that such sets are positively invariant w r t the considered system the trajectories starting from x sub 0 spl isin r sup n spl dscr k a b respectively to x sub 0 spl isin r sup n are studied in terms of attractivity and contractivity of the set spl dscr k a b the results are applied to the study of the constrained state feedback regulator problem

a novel unified approach to invariance conditions for a linear dynamical system in this paper we propose a novel simple and unified approach to explore sufficient and necessary conditions i e invariance conditions under which four classic families of convex sets namely polyhedra polyhedral cones ellipsoids and lorenz cones are invariant sets for a linear discrete or continuous dynamical system for discrete dynamical systems we use the theorems of alternatives i e farkas lemma and s lemma to obtain simple and general proofs to derive invariance conditions this novel method establishes a solid connection between optimization theory and dynamical system also using the s lemma allows us to extend invariance conditions to any set represented by a quadratic inequality such sets include nonconvex and unbounded sets for continuous dynamical systems we use the forward or backward euler method to obtain the corresponding discrete dynamical systems while preserves invariance this enables us to develop a novel and elementary method to derive invariance conditions for continuous dynamical systems by using the ones for the corresponding discrete systems finally some numerical examples are presented to illustrate these invariance conditions

bounded symbols and reproducing kernel thesis for truncated toeplitz operators compressions of toeplitz operators to coinvariant subspaces of h2 are called truncated toeplitz operators we study two questions related to these operators the first raised by sarason is whether boundedness of the operator implies the existence of a bounded symbol the second is the reproducing kernel thesis we show that in general the answer to the first question is negative and we exhibit some classes of spaces for which the answers to both questions are positive

simultaneous quadratic stabilization for a class of non linear systems with input saturation using dynamic surface control in this paper a new method to estimate the initial condition set which guarantees the quadratic stability for a class of non linear systems via dynamic surface control dsc in the presence of input saturation has been proposed the estimated set is enlarged to allow some degree of input saturation while achieving the simultaneous quadratic stability of the system in the domain a convex optimization problem to obtain ellipsoidal initial condition sets via a linear matrix inequality lmi approach as well as an illustrative example will be presented

examples of non kahler hamiltonian circle manifolds with the strong lefschetz property abstract in this paper we construct six dimensional compact non kahler hamiltonian circle manifolds which satisfy the strong lefschetz property themselves but nevertheless have a non lefschetz symplectic quotient this provides the first known counterexamples to the question whether the strong lefschetz property descends to the symplectic quotient we also give examples of hamiltonian strong lefschetz circle manifolds which have a non lefschetz fixed point submanifold in addition we establish a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a strong lefschetz manifold we then use it to show the existence of lefschetz four manifolds with non lefschetz finite covering spaces

a new class of universal lyapunov functions for the control of uncertain linear systems the authors analyze the problem of synthesizing a state feedback control for the class of uncertain continuous time linear systems affected by time varying memoryless parametric uncertainties they consider as candidate lyapunov functions the elements of the class spl sigma sub p sup z which is formed by special homogeneous positive definite functions they show that this class is universal in the sense that a lyapunov function exists if and only if there exists a lyapunov function in spl sigma sub p sup z they prove this result in a constructive way showing that such a lyapunov function can always be obtained by smoothing a polyhedral function for which construction algorithms are available the authors show that unlike the polyhedral lyapunov functions these functions allow us to derive explicit formulas for the stabilizing controller

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

nonquadratic lyapunov functions for robust control abstract this paper deals with the problem of controlling a linear continuous time system with structured time varying parameter uncertainties and input disturbances with a lyapunov function approach in contrast with most of the previous results in the literature we do not confine our attention to the class of quadratic lyapunov functions conversely the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate lyapunov functions this question has a positive answer the proposed class is that of polyhedral norms or more generally of polyhedral minkowski functionals we show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a lyapunov function and a continuous state feedback compensator then it can be solved via a polyhedral lyapunov function and a possibly different continuous control moreover we show that the control can be piecewise linear a numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral

eigenstructure assignment for state constrained linear continuous time systems abstract linear state constraints define a polyhedron in the state space of a dynamical system if this polyhedron can be made positively invariant by state feedback the constraints are satisfied all along the trajectory for any initial point in this polyhedron in the context of continuous time linear systems some algebraic positive invariance conditions are established for unbounded symmetrical polyhedra the existence of a solution to the corresponding set of equations and inequalities depends on the structural properties of an associated input output system in particular the number of zeros of this system should be appropriate and they should be stable if these conditions are satisfied the problem can easily be solved by eigenvalue and eigenvector assignment

constrained stabilization via smooth lyapunov functions abstract we consider the problem of stabilizing a dynamic system by means of bounded controls we show that the largest domain of attraction can be arbitrarily closely approximated by a smooth domain of attraction for which we provide an analytic expression such an expression allows for the determination of a non linear control law in explicit form

the regulator problem for class of linear systems with constrained control abstract this paper deals with the regulator problem for a class of linear discrete time systems with nonsymmetrical constrained controls for such a class a nonsymmetrical set of positive invariance and global attractivity exists this allows a globally asymptotically stable regulator to be obtained despite the existence of the constrained controls

the lefschetz property formality and blowing up in symplectic geometry in this paper we study the behaviour of the lefschetz property under the blow up construction we show that it is possible to reduce the dimension of the kernel of the lefschetz map if we blow up along a suitable submanifold satisfying the lefschetz property we use this together with results about massey products to construct compact nonformal symplectic manifolds satisfying the lefschetz property

constrained regulation of linear continuous time dynamical systems abstract in this paper the linear constrained regulation problem lcrp for continuous time dynamical systems is studied the first part of the paper deals with the existence of linear state feedback control laws that transfer asymptotically to the origin all initial states belonging to a given polyhedral subset of the state space while respecting linear constraints on both state and control vectors then the lcrp is formulated as an optimization problem which can be solved by applying linear programming techniques

semi global exponential stabilization of linear discrete time systems subject to input saturation via linear feedbacks abstract in this paper we show that a linear discrete time system subject to input saturation is semi globally exponentially stabilizable via linear state and or output feedback laws as long as the system in the absence of input saturation is stabilizable and detectable and has all its poles located inside or on the unit circle furthermore the semi globally stabilizing feedback laws are explicitly constructed the results presented here are parallel to our earlier results on the continuous time counterpart lin and saberi 1993