spectral determinant of schrodinger operators on graphs we study the spectral properties of the operator v x on a graph is the laplacian and v x is some potential defined on the graph in particular we derive an expression for the spectral determinant that generalizes one previously obtained for the laplacian operator

a diagram calculus for certain canonical bases we introduce a certain cellular algebra which is a quotient of the q schur algebra this is naturally equipped with a canonical basis which is compatible with lusztig s canonical bases for certain modules for the quantized enveloping algebra we describe a diagram calculus for which makes calculations involving the corresponding canonical bases easy to understand

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

controllability and observability for a class of controlled switching impulsive systems in this note we study the controllability and observability problem for a class of controlled switching impulsive systems by proposing several formula of variation of parameters for this type of time varying systems and employing the characteristic polynomial theory of matrix we establish necessary and sufficient conditions for controllability and controlled observability with respect to a given switching time sequence specializing the obtained results to the case of time invariant linear switching impulsive systems we derive some simple algebraic criteria which include the results reported in the literature for time invariant linear switching systems linear impulsive systems and classical linear systems one example is worked out for illustration

two point spectral correlations for star graphs the eigenvalues of the schrodinger operator on a graph g are related via an exact trace formula to periodic orbits on g this connection is used to calculate two point spectral statistics for a particular family of graphs called star graphs in the limit as the number of edges tends to infinity combinatorial techniques are used to evaluate both the diagonal same orbit and off diagonal different orbit contributions to the sum over pairs of orbits involved in this way a general formula is derived for terms in the short time expansion of the form factor k in powers of and the first few are computed explicitly the result demonstrates that k is neither poissonian nor random matrix but an intermediate between the two off diagonal pairs of orbits are shown to make a significant contribution to all but the first few coefficients

monomials and temperley lieb algebras we classify the fully tight simply laced coxeter groups that is the ones whoseiji avoiding kazhdan lusztig basis elements are monomials in the generatorsbsi we then investigate the basis of the temperley lieb algebra arising from the kazhdan lusztig basis of the associated hecke algebra and prove that the basis coincides with the usual monomial basis

spectral determinants and zeta functions of schrodinger operators on metric graphs a derivation of the spectral determinant of the schrodinger operator on a metric graph is presented where the local matching conditions at the vertices are of the general form classified according to the scheme of kostrykin and schrader to formulate the spectral determinant we first derive the spectral zeta function of the schrodinger operator using an appropriate secular equation the result obtained for the spectral determinant is along the lines of the recent conjecture texier 2010 j phys a math theor 43 425203

optimal control of discrete time bilinear systems with applications to switched linear stochastic systems abstract this paper aims at characterizing the most destabilizing switching law for discrete time switched systems governed by a set of bounded linear operators the switched system is embedded in a special class of discrete time bilinear control systems this allows us to apply the variational approach to the bilinear control system associated with a mayer type optimal control problem and a second order necessary optimality condition is derived optimal equivalence between the bilinear system and the switched system is analyzed which shows that any optimal control law can be equivalently expressed as a switching law this specific switching law is most unstable for the switched system and thus can be used to determine stability under arbitrary switching based on the second order moment of the state the proposed approach is applied to analyze uniform mean square stability of discrete time switched linear stochastic systems numerical simulations are presented to verify the usefulness of the theoretic results

non regular feedback linearization of nonlinear systems via a normal form algorithm in this paper the problem of non regular static state feedback linearization of affine nonlinear systems is considered first of all a new canonical form for non regular feedback linear systems is proposed using this form a recursive algorithm is presented which yields a condition for single input linearization then the left semi tensor product of matrices is introduced and several new properties are developed using the recursive framework and new matrix product a formula is presented for normal form algorithm based on it a set of conditions for single input approximate linearizability is presented

controllability properties of affine systems this paper deals with transitivity controllability of affine families of vector fields on a finite dimensional vector space v in particular we focus on affine families whose corresponding families of linear fields are transitive on v 0 and which in addition have no fixed points in v we show that such families are necessarily transitive on v and we also show that they remain transitive under small perturbations in general however affine families need not remain transitive under small perturbations for example small affine perturbations of transitive linear systems are not necessarily transitive since any affine system mathcal f naturally defines a system mathcal f r of right invariant vector fields on the semi direct product of v with gl v we also investigate transitivity properties mathcal f r our result is that if mathcal f is an affine family satisfying the preceding conditions then mathcal f r generates the full lie algebra on the semi direct product

on quadratic lyapunov functions a topological structure as a subset of 0 2 spl pi sup l spl times spl ropf sub sup n 1 is proposed for the set of quadratic lyapunov functions qlfs of a given stable linear system a necessary and sufficient condition for the existence of a common qlf of a finite set of stable matrices is obtained as the positivity of a certain integral the structure and the conditions are considerably simplified for planar systems it is also proved that a set of block upper triangular matrices share a common qlf if each set of diagonal blocks share a common qlf

stability condition for sampled data based control of linear continuous switched systems abstract most practical systems are continuous in nature but with discrete sampled feedback when digital control is utilized this paper investigates the stabilization problem of switched linear continuous time systems with sampled data based control for both known and unknown arbitrary switching processes on the basis of lyapunov stability theory a sufficient global exponential stability condition related to dwell time and sampling period is established for the latter case on line one step adaptive estimation algorithm is derived and integrated with sampled feedback for control design validation and verification of the established result are conducted through cruise control of train systems

a constructive approach to reachability realization of discrete time switched linear systems this paper studies the reachability of discrete time switched linear systems a new constructive approach to designing switching sequences is developed so that not only the reachable subspace is realized but also the required number of switchings is reduced significantly detailed comparison between the obtained result and that reported in the open literature is also made which verifies the advantage of the proposed approach

inverse eigenvalue problems on directed graphs the differential operators id and d2 p are constructed on certain finite directed weighted graphs two types of inverse spectral problems are considered first information about the graph weights and boundary conditions is extracted from the spectrum of d2 second the compactness of isospectral sets for d2 p is established by computation of the residues of the zeta function

on stability of switched homogeneous nonlinear systems in this paper the problem of stability of switched homogeneous systems is addressed first of all if there is a quadratic lyapunov function such that nonlinear homogeneous systems are asymptotically stable a matrix lyapunov like equation is obtained for a stable nonlinear homogeneous system using semi tensor product of matrices and lyapunov equation of linear system is just its particular case following the previous results a sufficient condition is obtained for stability of switched nonlinear homogeneous systems and a switching law is designed by partition of state space in particular a constructive approach is provided to avoid chattering phenomena which is caused by the switching rule then for planar switched homogeneous systems an lmi approach to stability of planar switched homogeneous systems is presented similar to the condition for linear systems the lmi type condition is easily verifiable an example is given to illustrate that candidate common lyapunov function is a key point for design of switching law

generating series for bilinear hybrid systems in this paper we introduce the novel concept of a hybrid generating series and show that continuous state and output trajectories of bilinear hybrid systems can be described in terms of these series the results represent an extension of the fliess series expansion for bilinear systems to hybrid systems

stabilization of planar switched systems this paper considers the problem of stabilization of single input planar switched systems we assume the switching law is observable a formula is presented which provides a necessary and sufficient condition for the system to be quadratically stabilizable a set of linear inequalities are given to describe the set of all quadratic lyapunov functions the solvability and the control design technique are clearly described in a straightforward computation algorithm

value distribution of the eigenfunctions and spectral determinants of quantum star graphs we compute the value distributions of the eigenfunctions and spectral determinant of the schrodinger operator on families of star graphs the values of the spectral determinant are shown to have a cauchy distribution with respect both to averages over bond lengths in the limit as the wavenumber tends to infinity and to averages over wavenumber when the bond lengths are fixed and not rationally related this is in contrast to the spectral determinants of random matrices for which the logarithm is known to satisfy a gaussian limit distribution the value distribution of the eigenfunctions also differs from the corresponding random matrix result we argue that the value distributions of the spectral determinant and of the eigenfunctions should coincide with those of seba type billiards

finite time stability analysis of impulsive discrete time switched systems with nonlinear perturbation the problem of finite time stability for a class of discrete time switched systems in the presence of both non lipschitz perturbation and impulse effects is studied in this paper based on the average dwell time approach a criterion is proposed which ensures that the system s state trajectory remains in a bounded region of the state space over a pre specified finite time interval if we impose a bound on the initial condition it is shown that the finite time stability degree could be greater than one which is quite different from the existing results for asymptotic stability moreover the total activation time of the schur stable subsystems does not need to be greater than that of the unstable subsystems a numerical example is presented to illustrate the effectiveness of the proposed design method

design of switched linear systems in the presence of actuator saturation for a group of linear systems each under a saturated linear not necessarily stabilizing feedback law we design a switching scheme such that the resulting switched system is locally asymptotically stable at the origin with a large domain of attraction by expressing each saturated linear feedback in a convex hull of a group of auxiliary linear feedbacks we formulate and solve the problem of designing such a switching scheme as a constrained optimization problem with the objective of maximizing an estimate of the domain of attraction simulation results indicate that the resulting domain of attraction extends well beyond the linear regions of the actuators

null controllable region of delta operator systems subject to actuator saturation abstractin this paper we give exact description of null controllable regions for delta operator systems subject to actuator saturation the null controllable region is in terms of a set of extremal trajectories of anti stable subsystems for the delta operator system with real eigenvalues or complex eigenvalues the description is simplified to an explicit formula which is used to characterise the boundary of a null controllable region the relations of null controllable regions are shown separately for continuous time systems discrete time systems and delta operator systems two numerical examples are given to illustrate the effectiveness of the proposed techniques on null controllable regions

controllability of switched bilinear systems the controllability of switched bilinear systems sblss is considered three kinds of controllabilities including weak controllability approximate controllability and global controllability are investigated one by one sets of easily verifiable sufficient conditions are obtained for each case which are applicable to a large class of switched bilinear systems

brief paper l2 gain analysis for a class of switched systems this paper considers the problem of disturbance tolerance rejection for a family of linear systems subject to actuator saturation and l 2 disturbances for a given set of linear feedback gains a given switching scheme and a given bound on the l 2 norm of the disturbances conditions are established in terms of linear or bilinear matrix inequalities under which the resulting switched system is bounded state stable that is trajectories starting from a bounded set will remain inside the set or a larger bounded set with these conditions both the problem of assessing the disturbance tolerance rejection capability of the closed loop system and the design of feedback gain and switching scheme can be formulated and solved as constrained optimization problems disturbance tolerance is measured by the largest bound on the disturbances for which the trajectories from a given set remain bounded disturbance rejection is measured by the restricted l 2 gain over the set of tolerable disturbances in the event that all systems in the family are identical the switched system reduces to a single system under a switching feedback law it will be shown that such a single system under a switching feedback law has stronger disturbance tolerance rejection capability than a single linear feedback law can achieve

span reachability and observability of bilinear hybrid systems the paper presents an algebraic characterization of observability and span reachability of bilinear hybrid systems without guards i e hybrid systems whose continuous dynamics are determined by bilinear control systems and whose discrete transitions are triggered externally the proposed characterization provides necessary and sufficient conditions for observability and span reachability in terms of rank conditions for generalized observability and controllability matrices this characterization is useful for the system theory of bilinear hybrid systems in particular it is used in the realization theory of bilinear hybrid systems petreczky 2006

brief paper periodic stabilizability of switched linear control systems stabilizability via direct observer based state feedback control for discrete time switched linear control systems slcss is investigated in this paper for an slcs the control factors are not only the control input but also the switching signal and they need to be designed in order to stabilize the system as a result stabilization design for slcss is more complicated than that for non switched ones differently from the existing approaches a periodic switching signal and piecewise constant linear state feedback control are adopted to achieve periodic stabilizability for such systems it is highlighted that multiple feedback controllers need to be designed for one subsystem for discrete time slcss it is proved that reachability implies periodic stabilizability via state feedback a necessary and sufficient criterion for periodic stabilizability is also established two stabilization design algorithms are presented for real application moreover it is proved that reachability and observability imply periodic stabilizability via observer based feedback for discrete time slcss periodic detectability as the dual concept of periodic stabilizability is discussed and the stabilization design algorithms via observer based feedback are presented as well

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

lie algebraic stability criteria for switched systems it was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable lie algebra possesses a quadratic common lyapunov function which implies that the corresponding switched linear system is exponentially stable for arbitrary switching in this paper we prove that the same properties hold under the weaker condition that the lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact lie group the corresponding local stability result for nonlinear switched systems is also established moreover we demonstrate that if a lie algebra fails to satisfy the above condition then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable relevant facts from the theory of lie algebras are collected at the end of the paper for easy reference

reachable sets and controllability of bilinear time invariant systems a qualitative approach this paper is concerned with a qualitative method for the study of reachable sets and controllability of bilinear time invariant systems a nonlinear decomposition technique which reduces the solution of the problem to the analysis of the qualitative behavior of a finite number of linear uncontrolled systems is introduced special consideration is given to the homogeneous bilinear systems with commutative matrices two and three dimensional examples are discussed

quantum oscillations in mesoscopic rings and anomalous diffusion we consider the weak localization correction to the conductance of a ring connected to a network we analyse the harmonics content of the al tshuler aronov spivak aas oscillations and we show that the presence of wires connected to the ring is responsible for a behaviour different from the one predicted by aas the physical origin of this behaviour is the anomalous diffusion of brownian trajectories around the ring due to the diffusion in the wires we show that this problem is related to the anomalous diffusion along the skeleton of a comb we study in detail the winding properties of brownian curves around a ring connected to an arbitrary network our analysis is based on the spectral determinant and on the introduction of an effective perimeter probing the different time scales a general expression of this length is derived for arbitrary networks more specifically we consider the case of a ring connected to wires to a square network and to a bethe lattice

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

brief paper design of switching sequences for controllability realization of switched linear systems in this paper we study the problem of designing switching sequences for controllability of switched linear systems each controllable state set of designed switching sequences coincides with the controllable subspace both aperiodic and periodic switching sequences are considered for the aperiodic case a new approach is proposed to construct switching sequences and the number of switchings involved in each designed switching sequence is shown to be upper bounded by d d d 1 1 here d is the dimension of the controllable subspace d email protected i 1 m where a i b i are subsystems for the periodic case we show that the controllable subspace can be realized within d switching periods

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

structural minimum controllability problem for switched linear continuous time systems abstract this paper addresses a structural design problem in control systems and explicitly takes into consideration the possible application to large scale systems more precisely we aim to determine and characterize the minimum number of manipulated state variables ensuring structural controllability of switched linear continuous time systems towards this goal we provide a new necessary and sufficient condition that leverages both graph theoretic and algebraic properties required to ensure feasibility of the solutions with this new condition we show that a solution can be determined by an efficient procedure i e polynomial in the number of state variables in addition we also discuss the switching signal properties that ensure structural controllability and the computational complexity of determining these sequences in particular we show that determining the minimum number of modes that a switching signal requires to ensure structural controllability is np hard

controllability and stabilizability of switched linear systems in this paper we study the conjecture made in ieee trans automat control 47 8 2002 1401 concerning the existence of basic switching sequence for switched systems and give it an affirmative answer and a construction method this paper proves that for switched linear system there exists a switching sequence such that the controllable state set of this switching sequence is equal to the controllable state set of the system hence the controllability can be realized by using only one switching sequence then the result is extended to the systems with time delay in controls furthermore we define the stabilizability of switched linear systems and based on certain transformation of state coordinate we give a necessary and sufficient condition for the stabilizability two examples are presented to illustrate the obtained results

controllability and observability of hybrid systems abstract a special class of hybrid systems is considered whose state space is a cross product space of a euclidean space and a finite state space such models may be used to represent systems subject to known abrupt parameter variations such as commu tated networks they may also be used to approximate some types of time varying systems controllability observability and stability of hybrid systems are inves tigated in particular a necessary and sufficient algebraic condition a simple algebraic criterion and a computationally simple algebraic sufficient test for controllability and observability are derived moreover a simple sufficient stability condition is provided