almost equitable partitions and new necessary conditions for network controllability abstract in this paper we consider the controllability problem for multi agent networked control systems the main results of the paper are new graph theoretic necessary conditions for controllability involving almost equitable graph vertex partitions we generalize the known results on the role of graph symmetries and uncontrollability to weighted digraphs with multiple leaders and we also consider the broadcasted control scenario our results show that the internal structure of communities in a graph can induce obstructions to controllability that cannot be characterized by symmetry arguments alone and that in some cases depend on the number theoretic properties of the communities we show via examples that our results can be used to account for a large portion of uncontrollable inducing leader selections that could not have otherwise been accounted for using symmetry results

on almost cylindrical languages and the decidability of the d0l and pwd0l primitivity problems abstract primitive words and their properties have always been of fundamental importance in the study of formal language theory head and lando in periodic d0l languages proposed the idea of deciding whether or not a given d0l language has the property that every word in it is a primitive word after reducing the general problem to the case in which h is injective it will be shown that primitivity is decidable when a h is an almost cylindrical set moreover in this case it is shown that the set of words which generate primitive sequences given a particular d0l scheme is an algorithmically constructible context sensitive language an undecidability result for the pwd0l primitivity problem and decidability results for cases of the rwd0l primitivity problem are also given

necessary and sufficient graphical conditions for formation control of unicycles the feasibility problem is studied of achieving a specified formation among a group of autonomous unicycles by local distributed control the directed graph defined by the information flow plays a key role it is proved that formation stabilization to a point is feasible if and only if the sensor digraph has a globally reachable node a similar result is given for formation stabilization to a line and to more general geometric arrangements

on the controllability properties of circulant networks this paper examines the controllability of a group of first order agents adopting a weighted consensus type coordination protocol over a circulant network specifically it is shown that a circulant network with laplacian eigenvalues of maximum algebraic multiplicity q is controllable from q nodes our approach leverages on the cauchy binet formula which in conjunction with the popov belevitch hautus test leads to new insights on structural aspects of network controllability

controllability of multi agent systems with periodically switching topologies and switching leaders abstractthis paper considers controllability of multi agent systems with periodically switching topologies and switching leaders the concept of m periodic controllability is proposed and a criterion for m periodic controllability is established the effect of the duration of subsystems on controllability is analysed by utilising a property of analytic functions in addition the influence of switching periods on controllability is investigated and an algorithm is proposed to search for the fewest periods to ensure controllability a necessary condition for m periodic controllability is obtained from the perspective of eigenvectors of the subsystems laplacian matrices for a system with switching leaders it is proved that switching leader controllability is equivalent to multiple leader controllability furthermore both the switching order and the tenure of agents being leaders have no effect on the controllability some examples are provided to illustrate the theoretical results

controllability of heterogeneous multi agent systems under directed and weighted topology abstractthis paper considers the controllability problem for both continuous and discrete time linear heterogeneous multi agent systems with directed and weighted communication topology first two kinds of neighbour based control protocols based on the distributed protocol of first order and second order multi agent systems are proposed under which it is shown that a heterogeneous multi agent system is controllable if the underlying communication topology is controllable then under special leader selection the result shows that the controllability of a heterogeneous multi agent system is solely decided by its communication topology graph furthermore some necessary and or sufficient conditions are derived for controllability of communication topology from algebraic and graphical perspectives finally simulation examples are presented to demonstrate the effectiveness of the theoretical results

quadratic stabilisability of multi agent systems under switching topologies this paper addresses the stabilisability of multi agent systems mass under switching topologies necessary and or sufficient conditions are presented in terms of graph topology these conditions explicitly reveal how the intrinsic dynamics of the agents the communication topology and the external control input affect stabilisability jointly with the appropriate selection of some agents to which the external inputs are applied and the suitable design of neighbour interaction rules via a switching topology an mas is proved to be stabilisable even if so is not for each of uncertain subsystem in addition a method is proposed to constructively design a switching rule for mass with norm bounded time varying uncertainties the switching rules designed via this method do not rely on uncertainties and the switched mas is quadratically stabilisable via decentralised external self feedback for all uncertainties with respect to applications of the stabilisability results the formation control and the cooper

performance and design of cycles in consensus networks abstract this work explores the role of cycles in consensus seeking networks for analysis and synthesis purposes cycles are critical for many reasons including improving the convergence rate of the system resilience to link failures and the overall performance of the system the focus of this work examines how cycles impact the h 2 performance of consensus networks a first contribution shows that the addition of cycles always improves the performance of the system we provide an analytic characterization of how the addition of edges improves the performance and show that it is related to the inverse of the cycle lengths and the number of shared edges between independent cycles these results are then used to consider the design of consensus networks in this direction we present an l 1 relaxation method that leads to a convex program for adding a fixed number of edges to a consensus networks we also demonstrate how this relaxation can be used to embed additional performance criteria such as maximization of the algebraic connectivity of the graph

zero forcing sets and controllability of dynamical systems defined on graphs in this technical note controllability of systems defined on graphs is discussed we consider the problem of controllability of the network for a family of matrices carrying the structure of an underlying directed graph a one to one correspondence between the set of leaders rendering the network controllable and zero forcing sets is established to illustrate the proposed results special cases including path cycle and complete graphs are discussed moreover as shown for graphs with a tree structure the proposed results of the present technical note together with the existing results on the zero forcing sets lead to a minimal leader selection scheme in particular cases

controllability of the multi agent system modeled by the threshold graph with one repeated degree abstract the controllability issue of multi agent systems modeled by a special class of graphs is studied suppose the modeling graph is simple connected and driven by only one controller the author shows that the system is controllable if the graph has exactly two vertices with the same degree and either one of these two vertices is selected to receive the control input this result is extended to the case of the threshold graph that has only one repeated degree specifically it is shown that if the threshold graph has only one repeated degree and its multiplicity is m then m 1 is the minimum number of the controllers required to make the system controllable in addition the necessary and sufficient condition that characterizes the binary control vectors to ensure the controllability of the system is derived

observability through a matrix weighted graph observability of an array of identical linear time invariant systems with incommensurable output matrices is studied where an array is called observable when identically zero relative outputs imply synchronized solutions for the individual systems it is shown that the observability of an array is equivalent to the connectivity of its interconnection graph whose edges are assigned matrix weights moreover to better understand the relative behavior of distant units pairwise observability that concerns with the synchronization of a certain pair of individual systems in the array is studied this milder version of observability is shown to be closely related to certain connectivity properties of the interconnection graph as well pairwise observability is also analyzed using the circuit theoretic tool effective conductance the observability of a certain pair of units is proved to be equivalent to the nonsingularity of the matrix valued effective conductance between the associated pair of nodes of a resistive network with matrix valued parameters whose node admittance matrix is the laplacian of the array s interconnection graph

upper and lower bounds for controllable subspaces of networks of diffusively coupled agents this technical note studies the controllability of diffusively coupled networks where some agents called leaders are under the influence of external control inputs first we consider networks where agents have general linear dynamics then we turn our attention to infer network controllability from its underlying graph topology to do this we consider networks with agents having single integrator dynamics for such networks we provide lower and upper bounds for the controllable subspaces in terms of the distance partitions and the maximal almost equitable partitions respectively we also provide an algorithm for computing the maximal almost equitable partition for a given graph and a set of leaders

the inverse sturm liouville problem ii soit q q 0 2 l r 2 0 1 le probleme de sturm liouville y q x y y 0 x 1 y 0 cos y 0 sin 0 y 1 cos y 1 sin 0 a un spectre discret de valeurs propres simples 0 q 1 q avec des fonctions propres h 0 x q h 1 x q on fixe p dans 0 2 l r 2 0 1 et soit m p q 0 2 l r 2 0 1 n q n p n 0 l espace de tous les points q de meme spectre que p on etudie m p comme une sous variete de 0 2 l r 2 0 1

degree maximal graphs are laplacian integral abstract an integer sequence d d 1 d 2 d n is graphic if there is a graph whose degree sequence is d a graph g is maximal if its degree sequence is majorized by no other graphic sequence the laplacian matrix of g is l g d g a g where d g is the diagonal matrix of vertex degrees and a g is the 0 1 adjacency matrix the article contains an explicit algorithmic construction of all maximal graphs from which it follows e g that apart from 0 the laplacian spectrum of a maximal graph is the conjugate of its degree sequence

bounded d0l languages for an alphabet a and a morphism h a a the set of words w such that the dol language l a h w is a bounded language is shown to be b where b is an effectively computable subset of a consequently boundedness is decidable for dol languages the result depends on the authors recent results on periodic dol languages interpretation of the result for polynomially bounded dol languages is also considered

coordination and geometric optimization via distributed dynamical systems this paper discusses dynamical systems for disk covering and sphere packing problems we present facility location functions from geometric optimization and characterize their differentiable properties we design and analyze a collection of distributed control laws that are related to nonsmooth gradient systems the resulting dynamical systems promise to be of use in coordination problems for networked robots in this setting the distributed control laws correspond to local interactions between the robots the technical approach relies on concepts from computational geometry nonsmooth analysis and the dynamical system approach to algorithms

constructing a controllable graph under edge constraints abstract this paper is concerned with the problem of constructing a controllable graph subject to some practical edge constraints specifically suppose the total amount of vertices and the upper bounds on the graph diameter or on the vertex degree are given we consider the problem of exploring a class of feasible graphs that satisfy the constraints using the hybrid of a path graph and an antiregular graph we propose a simple and systematic method to generate a class of controllable graphs whose diameters or degrees cover the full possible ranges the method to select the control vector to ensure the controllability of the combined graph is also proposed numerical examples are provided to demonstrate our results

regularity of sets of initial strings of periodic d0l systems for an alphabet a and a morphism h a a the set of words w such that the d0l system a h w is periodic is shown to be a noncounting regular language use is made of the author s recent investigations of periodic d0l systems

controllability of multiagent networks with antagonistic interactions this paper addresses the controllability of a class of antagonistic multiagent networks with both positive and negative edges all the agents of the multiagent network run a consensus algorithm using a signed laplacian based on the generalized equitable partition we propose a graph theoretic characterization of an upper bound on the controllable subspace then we provide a necessary condition for the controllability of the system and give an algorithm to compute the partition furthermore we prove that for a structurally balanced network the controllability is equivalent to that of the corresponding all positive network if the leaders are chosen from the same vertex set several examples are given to illustrate these results

dynamical properties of pwdol systems the definitions of a piecewise deterministic zero lindenmayer pwd0l scheme and system are given and dynamical properties of such systems are introduced harrison 1994 showed that given an arbitrary finite alphabet a the emptiness problem is undecidable for the class of languages which are intersections of a d0l language and a context sensitive language this result is used to prove that many dynamical properties of pwd0l systems such as finiteness periodicity etc are in general undecidable over a two member context sensitive partition of a the idea of a morphic equivalence relation on a a is finite is defined and the idea of a finite morphic refinement is introduced harrison 1994 showed that every regular language and its complement is refined by a finite partition which is induced by a morphic congruence using this theorem it is shown that dynamical properties such as finiteness and periodicity are in general decidable for rwd0l systems i e a pwd0l system over a finite partition of a made up of all regular languages

periodicity and ultimate periodicity of dol systems a unified presentation of periodicity and ultimate periodicity of d0l systems is given bounds are provided for the period and index of periodic systems and simplified algorithms are obtained by using elementary morphisms it was shown previously harju and linna 1986 pansiot 1986 that if a h w is a d0l system with h w wy then it is decidable whether the word h w is ultimately periodic i e of the form xu in this paper a decision procedure is given to decide if there are an index i and a period p such that hp hi w hi w y and h p hi w is ultimately periodic given a morphism h a a bounds are obtained for the period and index of ultimately periodic systems it is also shown that the set w a h p hi w is ultimately periodic is a constructable regular language

recovering the density of a string from only lowest frequency data a classical inverse problem is can you hear the density of a string clamped at both ends the answer as narrowly stated is negative one has to give an additional data sequence such as the energy in each mode norming constants or information on the string slope at the endpoints or alternatively change the boundary conditions and obtain the new frequencies what if neither of these options is possible one solution recently proposed was to add a known mass density and remeasure the dirichlet spectrum w rundell and p e sacks siam j appl math 73 2013 pp 1020 1037 in this work we take a related approach we are able to add any number of known mass densities but we can only measure the fundamental frequency that is the lowest eigenvalue in each case we will show that our problem can be closely approximated by a related problem for which there is uniqueness and we discuss a suite of reconstruction methods

controllability and observability of grid graphs via reduction and symmetries in this paper we investigate the controllability and observability properties of a family of linear dynamical systems whose structure is induced by the laplacian of a grid graph this analysis is motivated by several applications in network control and estimation quantum computation and discretization of partial differential equations specifically we characterize the structure of the grid eigenvectors by means of suitable decompositions of the graph for each eigenvalue based on its multiplicity and on suitable symmetries of the corresponding eigenvectors we provide necessary and sufficient conditions to characterize all and only the nodes from which the induced dynamical system is controllable observable we discuss the proposed criteria and show through suitable examples how such criteria reduce the complexity of the controllability respectively observability analysis of the grid

a uniqueness theorem for an inverse sturm liouville problem a new uniqueness theorem is established for the inverse sturm liouville problem it is shown that the measurement of a particular eigenvalue for an infinite set of different boundary conditions is sufficient to determine the unknown potential

on the continuum limit of a discrete inverse spectral problem on optimal finite difference grids we consider finite difference approximations of solutions of inverse stur m liouville problems in bounded intervals using three point finite difference sc hemes we discretize the equations on so called optimal grids constructed as follow s for a staggered grid with 2k points we ask that the finite difference operator a k k jacobi matrix and the sturm liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval this requirement determines uniquely the entries in the jacobi matrix which are grid cell averages of the coefficients in the continuum problem if these coefficients are known we can find the grid which we call optimal becau se it gives by design a finite difference operator with a prescribed spectr al measure we focus attention on the inverse problem where neither the coefficients n or the grid are known a key question in inversion is how to parametrize the coefficients i e how to choose the grid it is clear that to be successful this grid must be close to the optimal one which is unknown fortunately as we show here the grid dependence on the unknown coefficients is weak so the inversion can be done on a pr ecomputed grid for an a priori guess of the unknown coefficients this obse rvation leads to a simple yet efficient inversion algorithm which gives coefficien ts that converge pointwise to the true solution as the number k of data points tends to infinity the cornerstone of our convergence proof is showing that op timal grids provide an implicit natural regularization of the inverse problem by givin g reconstructions with uniformly bounded total variation the analysis is based on a novel explicit perturbation analysis of lanczos recursions and on a discrete

on the reachability and observability of path and cycle graphs in this technical note we investigate the reachability and observability properties of a network system running a laplacian based average consensus algorithm when the communication graph is a path or a cycle specifically we provide necessary and sufficient conditions based on simple rules from number theory to characterize all and only the nodes from which the network system is reachable respectively observable interesting immediate corollaries of our results are i a path graph is reachable observable from any single node if and only if the number of nodes of the graph is a power of two n 2i i n and ii a cycle is reachable observable from any pair of nodes if and only if n is a prime number for any set of control observation nodes we provide a closed form expression for the unreachable unobservable eigenvalues and for the eigenvectors of the unreachable unobservable subsystem

a decomposition approach to distributed control of spatially invariant systems this note considers the decentralized control of spatially invariant systems i e systems of homogeneous interacting components the main idea is for individual components to model interactions with neighbors as disturbances that satisfy certain magnitude bounds while simultaneously self imposing symmetric magnitude bounds these magnitude bounds can be interpreted as negotiated levels of interaction among components it turns out that this approach is equivalent to constructing a feedback that is robustly stabilizing with structured uncertainties

zero forcing number constrained matchings and strong structural controllability abstract the zero forcing number is a graph invariant introduced to study the minimum rank of the graph in 2008 aazami proved the np hardness of computing the zero forcing number of a simple undirected graph we complete this np hardness result by showing that the non equivalent problem of computing the zero forcing number of a directed graph allowing loops is also np hard the rest of the paper is devoted to the strong controllability of a networked system this kind of controllability takes into account only the structure of the interconnection graph but not the interconnection strengths along the edges we provide a necessary and sufficient condition in terms of zero forcing sets for the strong controllability of a system whose underlying graph is a directed graph allowing loops moreover we explain how our result differs from a recent related result discovered by monshizadeh et al finally we show how to solve the problem of finding efficiently a minimum size input set for the strong controllability of a self damped system with a tree structure

periodic d0l languages abstract periodic d0l systems and languages are defined and periodicity is shown to be a decidable property of d0l systems the fundamental tools used are recent results on the representation of stationary words head and lando 1986 and the decidability of ultimate periodicity of words harju and linna 1986 pansiot 1986 the relation of d0l periodicity to local catenativity and to n codes is examined

analytical methods for recovering coefficients in differential equations from spectral data in this paper analytical methods for recovering coefficients from spectral data are discussed properties of these methods are given and mathematical models for which these methods might apply are presented

graph controllability classes for the laplacian leader follower dynamics in this paper we consider the problem of obtaining graph theoretic characterizations of controllability for the laplacian based leader follower dynamics our developments rely on the notion of graph controllability classes namely the classes of essentially controllable completely uncontrollable and conditionally controllable graphs in addition to the topology of the underlying graph the controllability classes rely on the specification of the control vectors our particular focus is on the set of binary control vectors the choice of binary control vectors is naturally adapted to the laplacian dynamics as it captures the case when the controller is unable to distinguish between the followers and moreover controllability properties are invariant under binary complements we prove that the class of essentially controllable graphs is a strict subset of the class of asymmetric graphs and provide numerical results that suggests that the ratio of essentially controllable graphs to asymmetric graphs increases as the number of vertices increases although graph symmetries play an important role in graph theoretic characterizations of controllability we provide an explicit class of asymmetric graphs that are completely uncontrollable namely the class of block graphs of steiner triple systems we prove that for graphs on four and five vertices a repeated laplacian eigenvalue is a necessary condition for complete uncontrollability but however show through explicit examples that for eight and nine vertices a repeated eigenvalue is not necessary for complete uncontrollability for the case of conditional controllability we give an easily checkable necessary condition that identifies a class of binary control vectors that result in a two dimensional controllable subspace several constructive examples demonstrate our results

controllability of multi agent systems with directed and weighted signed networks abstract this paper studies the controllability of multi agent systems with signed networks which are represented by directed weighted signed graphs the adjacency weights of network depict the property of interactions positive weight means cooperative interaction while negative weight antagonistic interaction first by using switching equivalent transformation the controllability of three kinds of signed networks is studied structurally balanced anti balanced and strictly unbalanced it is shown that the controllability of the structurally balanced network is equivalent to that of the associated underlying network then a graph theoretic necessary condition for controllability is given by virtue of almost equitable partition of directed weighted signed networks furthermore some necessary and sufficient conditions are given for the controllability of generic linear multi agent systems in addition several examples are presented to illustrate the theoretical results

distributed control design for spatially interconnected systems this paper deals with analysis synthesis and implementation of distributed controllers designed for spatially interconnected systems we develop a state space framework for posing problems of this type and focus on systems whose model is spatially discrete in this paper analysis and synthesis results are developed for this class of systems using the l sub 2 induced norm as the performance criterion the results are stated in terms of linear matrix inequalities and are thus readily amenable to computation a special implementation of the resulting controllers is presented which is particularly attractive for distributed operation of the controller several examples are provided to further illustrate the application of the results

the set of strings mapped into a submonoid by iterates of a morphism abstract let s be a finitely generated submonoid of a free monoid a p let h be an endomorphism of a p which maps s into itself let e s be the submonoid of all strings x in a p which are eventually mapped into s by h i e the strings x for which there exists a non negative integer i such that h p i x is in s then e s is a constructable regular language applications of this result are given to the study of periodic behaviors of morphisms

a class of uncontrollable diffusively coupled multiagent systems with multichain topologies we construct systematically a class of uncontrollable diffusively coupled multiagent systems with a single leader and multichain topologies for studying the controllability of diffusively coupled multiagent systems such identified uncontrollable systems serve as counterexamples that prove the need to modify the existing sufficient condition using graph partitioning characterization the uncontrollability of the constructed multichain structures can be preserved when the structures are further augmented to get better connected the paper also provides an algorithm to obtain the minimal leader invariant relaxed equitable partition for the graph associated with any diffusively coupled multiagent system guided by a single leader

protocols design and uncontrollable topologies construction for multi agent networks this note investigates the controllability issues of multi agent systems where each node contains generic linear dynamics first a neighbor based control protocol is proposed under which it is shown that the controllability of a multi agent system is solely decided by its communication topology structure we then further consider the effects of communication topology on the controllability from a graph theory point of view by explicitly constructing topologies the construction exhibits a partition of the designed graph with identified leaders under which the system is shown to be uncontrollable

on two sided infinite fixed points of morphisms let s be a finite alphabet and let h s s be a morphism finite and infinite fixed points of morphismsi e those words w such that h w wplay an important role in formal language theory head characterized the finite fixed points of h and later head and lando characterized the one sided infinite fixed points of h our paper has two main results first we complete the characterization of fixed points of morphisms by describing all two sided infinite fixed points of h for both the pointed and unpointed cases second we completely characterize the solutions to the equation h xy yx in finite words