nonlinear state feedback for l 1 optimal control this paper considers l1 optimal control problems with full state feedback in contrast to h optimal control previous work has shown that linear l1 optimal controllers can be dynamic and arbitrarily high order however this paper shows that continuous memoryless nonlinear state feedback perform as well as dynamic linear state feedback the derivation which is nonconstructive relies on concepts from viability theory

the nilpotent centralizer method for spectrally arbitrary patterns this paper introduces a new criterion for an n n sign pattern respectively zero nonzero pattern a to be spectrally arbitrary that is to have the property that for each monic real polynomial r x of degree n there exists a matrix with sign pattern respectively zero nonzero pattern a that has r x as its characteristic polynomial to date analytic properties of a certain polynomial map associated with a have been used to prove that a is spectrally arbitrary in this paper we derive a method that uses the algebraic structure of a certain nilpotent matrix to show that a sign pattern respectively zero nonzero pattern is spectrally arbitrary

potentially nilpotent patterns and the nilpotent jacobian method a nonzero pattern is a matrix with entries in 0 a pattern is potentially nilpotent if there is some nilpotent real matrix with nonzero entries in precisely the entries indicated by the pattern we develop ways to construct some potentially nilpotent patterns including some balanced tree patterns we explore the index of some of the nilpotent matrices constructed and observe that some of the balanced trees are spectrally arbitrary using the nilpotent jacobian method inspired by an argument of pereira nilpotent matrices and spectrally arbitrary sign patterns electron j linear algebra 16 2007 235 we uncover a feature of the nilpotent jacobian method in particular we show that if n is the nilpotent matrix employed in this method to show that a pattern is a spectrally arbitrary pattern then n must have full index

the minimal critical sets of refined inertias for 3 3 full sign patterns abstract a sign pattern matrix a is a matrix whose entries are from the set 0 if no entry of a is zero then a is called a full sign pattern the inertia of a real matrix a is the ordered triple n n n 0 in which n n and n 0 are the numbers of its eigenvalues counting multiplicities with positive negative and zero real parts respectively the refined inertia of a is the ordered 4 tuple n n n z 2 n p where n z resp 2 n p is the number of zero resp nonzero pure imaginary eigenvalues of a an n n sign pattern a is an inertially arbitrary pattern iap if it allows all possible inertias similarly a is a refined inertially arbitrary pattern riap if it allows all possible refined inertias a proper subset s of the set of all possible inertias resp refined inertia of real matrices of order n is called a critical set of inertias resp critical set of refined inertias for a family f of sign patterns of order n if for every a f s i a ensures that a is inertially arbitrary resp s ri a ensures that a is refined inertially arbitrary s is called a minimal critical set of inertias resp minimal critical set of refined inertias for f if no proper subset of s is a critical set of inertias resp refined inertias for f in this paper all minimal critical sets of inertias and refined inertias for 3 3 full sign patterns are identified

spectrally and inertially arbitrary sign patterns we introduce some n by n sign patterns which allow for arbitrary spectrum and hence also arbitrary inertia consequently we demonstrate that some known inertially arbitrary patterns are in fact spectrally arbitrary we demonstrate that all inertially arbitrary patterns of order 3 are spectrally arbitrary and classify all spectrally arbitrary patterns of order 3 we illustrate that in general the class of spectrally arbitrary patterns is distinct from the inertially arbitrary patterns and present some observations about inertially arbitrary patterns

on the oscillatory behavior of solutions of second order differential equations a generalized nonlinear second order differential equation is considered oscillation and nonoscillation criteria are given

efficient robust optimization for robust control with constraints this paper proposes an efficient computational technique for the optimal control of linear discrete time systems subject to bounded disturbances with mixed linear constraints on the states and inputs the problem of computing an optimal state feedback control policy given the current state is non convex a recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program while the reparameterized problem is theoretically tractable the number of variables is quadratic in the number of stages or horizon length n and has no apparent exploitable structure leading to computational time of mathcal o n 6 per iteration of an interior point method we focus on the case when the disturbance set is norm bounded or the linear map of a hypercube and the cost function involves the minimization of a quadratic cost here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem that is the policy optimization problem may be decomposed into a set of coupled finite horizon control problems this decomposition can then be formulated as a highly structured quadratic program solvable by primal dual interior point methods in which each iteration requires mathcal o n 3 time this cubic iteration time can be guaranteed using a riccati based block factorization technique which is standard in discrete time optimal control numerical results are presented using a standard sparse primal dual interior point solver that illustrate the efficiency of this approach

sets of refined inertias of zero nonzero patterns abstract a set h n of refined inertias for zero nonzero patterns is introduced that is analogous to the set h n previously considered for sign patterns for n 3 and 4 a complete characterization of irreducible zero nonzero patterns that allow or require h n is given and each zero nonzero pattern that allows h n has a signing that allows h n in contrast for n 5 a family of irreducible zero nonzero patterns is given that allows h n but for which no one signing allows h n

spectrally arbitrary zero nonzero patterns and field extensions abstract an n n matrix pattern is said to be spectrally arbitrary over a field f provided for every monic polynomial p t of degree n with coefficients from f there exists a matrix with entries from f in the given pattern that has characteristic polynomial p t let e f k be an extension of fields it is natural to ask whether a pattern that is spectrally arbitrary over f must also be spectrally arbitrary over e or k in this article it is shown that if f is dense in k and k is a complete metric space then any spectrally arbitrary or relaxed spectrally arbitrary pattern over f is relaxed spectrally arbitrary over k it is also established that if e is an algebraically closed subfield of a field f then any spectrally arbitrary pattern over f is spectrally arbitrary over e the 2n conjecture and the superpattern conjecture are explored over fields other than the real numbers in particular examples are provided to show that the superpattern conjecture is false over the field with 3 elements

inertially arbitrary 2r 1 diagonal sign patterns the set of all eigenvalues counting multiplicities of a matrix a is denoted by a and the inertia of a is the ordered triple i a i a i a i0 a in which i a i a and i0 a are the numbers of eigenvalues with positive negative and zero real parts respectively an n n sign pattern s sij has sij 0 and the qualitative class of s is q s a aij mn r sign aij sijforalli j the inertia of s is the set of ordered triples i s i a a q s an n n sign pattern s is an inertially arbitrary pattern iap if n1 n2 n3 i s for each nonnegative triple n1 n2 n3 with n1 n2 n3 n consider the n n 2r 1 diagonal sign pattern sn r with positive entry i j for 1 j i r 1 or i j n negative entry i j for 1 i j r 1 or i j 1 and zero entry otherwise j h drew et al proved that sn 2 is an iap for 2 n 7 and conjectured that sn 2 is an iap for n 8 gao et al proved that sn n is an iap for n 2 in this paper it is proved that n k k 0 i sn r for 0 k n 2 r n and that sn n 1 is an iap for n 3

spectrally arbitrary patterns over rings with unity abstract a zero nonzero pattern a is a matrix with entries from the set 0 a square pattern a is spectrally arbitrary over r a commutative ring with unity if for each n th degree monic polynomial f x r x there exists a matrix a over r with pattern a such that the characteristic polynomial p a x f x a pattern a is relaxed spectrally arbitrary over r if for each n th degree monic polynomial f x r x there exists a matrix a over r with either pattern a or a subpattern of a such that the characteristic polynomial p a x f x we evaluate how the structure of rings affects how we determine if a pattern is spectrally arbitrary we consider whether a pattern a that is spectrally arbitrary over r is spectrally arbitrary or relaxed spectrally arbitrary over another commutative ring s with unity in particular we establish that a pattern that is spectrally arbitrary over z is relaxed spectrally arbitrary over z m for all m z and spectrally arbitrary over q similarly a pattern that is spectrally arbitrary over the p adic integers is spectrally arbitrary over the p adic numbers

on nonlinear state feedback controllers for persistent disturbance rejection abstract an l 1 optimal linear time invariant lti compensator may have an order significantly higher than that of the plant in the state feedback case there has been recent work exploring the use of nonlinear static feedback controllers which provide near optimal performance in the case of d 12 square and invertible we show that it is enough to have the nonlinearity on a particular substate which can be used to significantly simplify the design problem

the potentially stable tree sign patterns for dimensions less than five the problem of characterizing potentially stable sign pattern matrices remains unsolved even for tree patterns we present here in tabular form all irreducible n by n potentially stable sign patterns whose undirected graphs are trees for n 5 in hopes that this may be useful to others in formulating and evaluating conjectures on the subject some background and a descriptio of methodology which includes some new theoretical ideas are also included

l sub 2 state feedback control with a prescribed rate of exponential convergence in this paper we consider the l sub 1 state feedback problem with an internal stability constraint in particular we establish the connection between controlled invariant contractive sets and static control laws that achieve a level of l sub 1 performance as well as a desired unforced rate of convergence we outline two algorithms for computing controlled invariant contractive sets the first is a modification of standard recursive techniques used in the literature whereas the second is based on dynamic games and involves solving an appropriate discrete isaacs recursion the latter approach results in a min max characterization of l sub 1 state feedback controllers we point out that the isaacs recursion provides a one shot as opposed to iterative computation of the optimal l sub 1 performance

on the potential stability of star sign pattern matrices abstract a sign pattern matrix a is called potentially stable if there exists a real matrix b q a such that b is stable a combinatorially symmetric sign pattern matrix a is called tree star sign pattern if its graph g a is a tree star in this paper a characterization for a star sign pattern matrix being potentially stable is presented

minimization of the worst case peak to peak gain via dynamic programming state feedback case considers the problem of designing a controller that minimizes the worst case peak to peak gain of a closed loop system in particular we concentrate on the case where the controller has access to the state of a linear plant and it possibly knows the maximal disturbance input amplitude we apply the principle of optimality and derive a dynamic programming formulation of the optimization problem under mild assumptions we show that at each step of the dynamic program the cost to go has the form of a gauge function and can be recursively determined through simple transformations we study both the finite horizon and the infinite horizon case under different information structures the proposed approach allows us to encompass and improve earlier results based on viability theory in particular we present a computational scheme alternative to the standard bisection algorithm or gamma iteration that allows us to compute the exact value of the worst case peak to peak gain for any finite horizon we show that the sequence of finite horizon optimal costs converges as the length of the horizon goes to infinity to the infinite horizon optimal cost the sequence of such optimal costs converges from below to the optimal performance for the infinite horizon problem we also show the existence of an optimal state feedback strategy that is globally exponentially stabilizing and derive suboptimal globally exponentially stabilizing strategies from the solutions of finite horizon problems

optimization of the l sup spl infin induced norm under full state feedback this paper considers the l sup spl infin optimal control problem i e minimization of the effects of disturbances as measured by the l sup spl infin induced norm earlier work showed that even in the case of full state feedback optimal and near optimal linear controllers may be dynamic and of arbitrarily high order however previous work by the author derived the existence of near optimal nonlinear controllers which are static this paper presents a constructive algorithm for such nonlinear controllers the main idea is to construct a certain subset of the state space such that achieving disturbance rejection is equivalent to restricting the state dynamics to this set a construction of this subset requires the solution of several finite linear programs with the number of variables being at least the number of states but less than the number of states plus the number of controls the concept of controlled invariance plays a central role throughout

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

set valued observers and optimal disturbance rejection a set valued observer also called guaranteed state estimator produces a set of possible states based on output measurements and models of exogenous signals we consider the guaranteed state estimation problem for linear time varying systems with a priori magnitude bounds on exogenous signals we provide an algorithm to propagate the set of possible states based on output measurements and show that the centers of these sets provide optimal estimates in an l sup spl infin induced norm sense we then consider the utility of set valued observers for disturbance rejection with output feedback and derive the following general separation structure an optimal controller can consist of a set valued observer followed by a static nonlinear function on the observed set of possible states a general construction of this function is provided in the scalar control case furthermore in the special case of full control i e the number of control inputs equals the number of states optimal output feedback controllers can take the form of an optimal estimate of the full state feedback controller

potentially nilpotent and spectrally arbitrary even cycle sign patterns abstract an n n sign pattern s n is potentially nilpotent if there is a real matrix having sign pattern s n and characteristic polynomial x n a new family of sign patterns c n with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern c n these nilpotent matrices are used together with a jacobian argument to show that c n is spectrally arbitrary i e there is a real matrix having sign pattern c n and characteristic polynomial x n i 1 n 1 i i x n i for any real i some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given

spectrally arbitrary patterns reducibility and the 2n conjecture for n 5 a sign pattern z a matrix whose entries are elements of 0 is spectrally arbitrary if for any self conjugate spectrum there is a real matrix with sign pattern z having the given spectrum spectrally arbitrary sign patterns were introduced in 5 where it was incorrectly stated that if a sign pattern z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern then z is a spectrally arbitrary sign pattern and it was conjectured that the converse is true as well we present counterexamples to both of these statements in 2 it was conjectured that any n n spectrally arbitrary sign pattern must have at least 2n nonzero entries we establish that this conjecture is true for 5 5 sign patterns we also establish analogous results for nonzero patterns

sign patterns requiring a unique inertia abstract a sign pattern requires a unique inertia if every real matrix in the sign pattern class has the same inertia several sufficient or necessary conditions are given for a sign pattern to require a unique inertia it is proved that a sign pattern requires a unique inertia if and only if it requires a unique refined inertia all sign patterns of orders 2 and 3 that require a unique inertia are characterized if the underlying graph of a sign pattern is a tree then it is shown that any skew symmetric sign pattern requires a unique inertia and a linear time algorithm is given to determine whether or not a symmetric sign pattern requires a unique inertia

bordering for spectrally arbitrary sign patterns abstract we develop a matrix bordering technique that can be applied to an irreducible spectrally arbitrary sign pattern to construct a higher order spectrally arbitrary sign pattern this technique generalizes a recently developed triangle extension method we describe recursive constructions of spectrally arbitrary patterns using our bordering technique and show that a slight variation of this technique can be used to construct inertially arbitrary sign patterns

inertially arbitrary sign patterns with no nilpotent realization abstract an n by n sign pattern s is inertially arbitrary if each ordered triple n1 n2 n3 of nonnegative integers with n1 n2 n3 n is the inertia of some real matrix in q s the sign pattern class of s if every real monic polynomial of degree n having a positive coefficient of xn 2 is the characteristic polynomial of some matrix in q s then it is shown that s is inertially arbitrary a new family of irreducible sign patterns g 2 k 1 k 2 is presented and proved to be inertially arbitrary but not potentially nilpotent and thus not spectrally arbitrary the well known nilpotent jacobian method cannot be used to prove that g 2 k 1 is inertially arbitrary since g 2 k 1 has no nilpotent realization in order to prove that q g 2 k 1 allows each inertia with n3 1 a realization of g 2 k 1 with only zero eigenvalues except for a conjugate pair of pure imaginary eigenvalues is identified and used with the implicit function theorem matrices in q g 2 k 1 with inertias having n3 0 are constructed by a recursive procedure from those of lower order some properties of the coefficients of the characteristic polynomial of an arbitrary matrix having certain fixed inertias are derived and are used to show that g 5 and g 7 are minimal inertially arbitrary sign patterns

the minimum number of nonzeros in a spectrally arbitrary ray pattern abstract a ray pattern a is spectrally arbitrary if given any monic polynomial r x of order n with complex coefficients there exists b q r a such that the characteristic polynomial of b is r x in 1 the authors first presented that a family of n n irreducible ray patterns with exactly 3n nonzeros is spectrally arbitrary then they proved that every n n irreducible spectrally arbitrary ray pattern has at least 3 n 1 nonzeros so as pointed in 1 the minimum number of nonzeros in an n n irreducible spectrally arbitrary ray pattern is either 3n or 3 n 1 in this paper we provide an irreducible spectrally arbitrary ray pattern of order n with 3 n 1 nonzeros so the minimum number of nonzeros in an n n irreducible spectrally arbitrary ray pattern is 3 n 1

spectrally arbitrary star sign patterns abstract an n n sign pattern s n is spectrally arbitrary if for any given real monic polynomial g x of degree n there is a real matrix having sign pattern s n and characteristic polynomial g x all n n star sign patterns that are spectrally arbitrary and all minimal such patterns are characterized this subsequently leads to an explicit characterization of all n n star sign patterns that are potentially nilpotent it is shown that any super pattern of a spectrally arbitrary star sign pattern is also spectrally arbitrary

spectrally arbitrary pattern extensions abstract a matrix pattern is often either a sign pattern with entries in 0 or more simply a nonzero pattern with entries in 0 a matrix pattern a is spectrally arbitrary if for any choice of a real matrix spectrum there is a real matrix having the pattern a and the chosen spectrum we describe a graphical technique a triangle extension for constructing spectrally arbitrary patterns out of some known lower order spectrally arbitrary patterns these methods provide a new way of viewing some known spectrally arbitrary patterns as well as providing many new families of spectrally arbitrary patterns we also demonstrate how the technique can be applied to certain inertially arbitrary patterns to obtain larger inertially arbitrary patterns we then provide an additional extension method for zero nonzero patterns patterns with entries in 0

on the spectra of close to schwarz matrices several inverse eigenvalue problems for tridiagonal matrices are discussed in particular it is shown that for every set s of n complex numbers which is symmetric with respect to the real axis there exists a symmetric in modulus tridiagonal n n matrix a which has a nonnegative super diagonal a nonpositive subdiagonal all diagonal elements zero except for a11 which is nonnegative and ann which is nonpositive and with spectrum s 2002 elsevier science inc all rights reserved

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

spectrally arbitrary patterns abstract an n n sign pattern matrix s is an inertially arbitrary pattern iap if each nonnegative triple n 1 n 2 n 3 with n 1 n 2 n 3 n is the inertia of a matrix with sign pattern s analogously s is a spectrally arbitrary pattern sap if for any given real monic polynomial r x of order n there is a matrix with sign pattern s and characteristic polynomial r x focusing on tree sign patterns consider the n n tridiagonal sign pattern t n with each superdiagonal entry positive each subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other entry zero it is conjectured that t n is an iap by constructing matrices a n with pattern t n it is proved that t n allows any inertia with n 3 0 1 2 n 1 n for all n 2 this leads to a proof of the conjecture for n 5 the truth of the conjecture is extended to n 7 by showing the stronger result that t n is a sap the proof of this latter statement involves finding a matrix a n with pattern t n that is nilpotent further questions about patterns that are saps and iaps are considered

oscillation criteria for second order nonlinear perturbed differential equations abstract some criteria are obtained which guarantee that all solutions of the second order nonlinear perturbed differential equations r t y t p t y t q t y t h t y t y t are oscillatory

spectrally arbitrary ray patterns abstract an n n ray pattern a is said to be spectrally arbitrary if for every monic nth degree polynomial f x with coefficients from c there is a matrix in the pattern class of a such that its characteristic polynomial is f x in this article the authors extend the nilpotent jacobi method for sign patterns to ray patterns establishing a means to show that an irreducible ray pattern and all its superpatterns are spectrally arbitrary they use this method to establish that a particular family of n n irreducible ray patterns with exactly 3 n nonzeros is spectrally arbitrary they then show that every n n irreducible spectrally arbitrary ray pattern has at least 3 n 1 nonzeros

inertia sets of two classes of symmetric sign patterns abstract a sign pattern is a matrix whose entries are from the set 0 for a symmetric sign pattern a of order n the inertia set of a is the set of inertias of all real symmetric matrices with the same sign pattern as a the purpose of this paper is to characterize the inertia sets of symmetric star and nonnegative symmetric tridiagonal sign patterns

persistent disturbance rejection via static state feedback in contrast with spl hscr sub spl infin and spl hscr sub 2 control theories the problem of persistent disturbance rejection l sup 1 optimal control leads to dynamic controllers even when the states of the plant are available for feedback using viability theory shamma showed 1993 in a nonconstructive way that in the state feedback case the same performance achieved by any dynamic linear time invariant controller can be achieved using memoryless nonlinear state feedback in this paper we give an alternative constructive proof of these results for discrete and continuous time systems the main result of the paper shows that in both cases the l sup 1 norm achieved by any stabilizing state feedback linear dynamic controller can be also achieved using a memoryless variable structure 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

sparse inertially arbitrary patterns an n by n sign pattern a is a matrix with entries in 0 an n by n nonzero pattern a is a matrix with entries in 0 where represents a nonzero entry a pattern a is inertially arbitrary if for every set of nonnegative integers n1 n2 n3 with n1 n2 n3 n there is a real matrix with pattern a having inertia n1 n2 n3 we explore how the inertia of a matrix relates to the signs of the coefficients of its characteristic polynomial and describe the inertias allowed by certain sets of polynomials this information is useful for describing the inertia of a pattern and can help show a pattern is inertially arbitrary britz et al t britz j j mcdonald d d olesky p van den driessche minimal spectrally arbitrary sign patterns siam j matrix anal appl 26 2004 257 271 conjectured that irreducible spectrally arbitrary patterns must have at least 2n nonzero entries we demonstrate that irreducible inertially arbitrary patterns can have less than 2n nonzero entries

low rank perturbations and the spectrum of a tridiagonal sign pattern abstract the n by n tridiagonal sign pattern t n has every superdiagonal entry positive every subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other diagonal entry zero inertia and spectral results for matrices a n having the sign pattern t n are proved using new techniques on low rank perturbations it is also shown by using maple that for 8 n 16 t n allows any spectrum these results extend those previously in the literature and strengthen the conjecture that t n allows any spectrum for all values of n in addition bounds on the algebraic multiplicity of an eigenvalue of a low rank perturbation of a general matrix are obtained

on static feedback for the l 1 and other optimal control problems although e 1 optimal linear state feedback controllers are known to be dynamic it has been shown that suboptimal performance arbitrarily closed to optimal can be achieved by using a static non linear feedback law in this paper this fact is established by using a novel approach which shows that the result is a natural consequence of elementary state space theory the approach is motivated by recent works in active vision systems which have considered a state feedback problem tightly connected with e 1 optimization this problem which has independent interest is discussed in some detail the new formulation of the problem provides additional insight in e 1 state feedback in particular it leads naturally to some extensions which do not follow in a straightforward manner from previous works on the subject

internal behaviour provided by nonlinear l1 optimal controllers abstract an l 1 optimal linear time invariant lti compensator may have an order significantly higher than that of the plant even when the state is measurable recently there has been work exploring the use of nonlinear static feedback controllers which provide near optimal performance here we consider a class of nonlinear state feedback controllers and derive superposition like bounds on both the plant state and the controlled output in the event that the plant initial condition the disturbance and the noise are all non zero

potentially nilpotent sign pattern matrices abstract a sign pattern is said to be potentially nilpotent if it allows nilpotence in this paper a number of qualitative necessary or sufficient conditions for a sign pattern to allow nilpotence are established the sign patterns that allow nilpotence of index 2 are investigated for orders up to three potentially nilpotent sign patterns are characterized potentially nilpotent tree sign patterns are also explored

oscillation and nonoscillation criteria for second order nonlinear differential equations i abstract a second order nonlinear differential equation is considered oscillation and nonoscillation criteria are given

qualitative stability of linear systems abstract this paper provides a finitely computable graph theoretic answer to the following question concerning linear dynamical systems when given only the signs of entries or 0 in a real square matrix a can one be certain that all positive trajectories of the system ax are bounded matrices having such sign patterns are called sign quasistable with bounded replaced by convergent to the origin the matrices are called sign stable and were fully described in earlier papers however when a s digraph has several strong components so that the system is actually a hierarchy of subsystems and when some of those subsystems fail to be sign stable the recognition of sign quasistability is a very delicate matter by means of certain graph color tests it is possible to identify the system variables that are capable for some choice of matrix entry magnitudes and initial conditions of emitting nonzero constant

sign patterns allowing nilpotence of index 3 suppose p is a property referring to a real matrix we say that a sign pattern a allows p if there exists at least one matrix with the same sign pattern as a that has the property p in this paper we study sign patterns allowing nilpotence of index 3 four methods for constructing sign patterns that allow nilpotence of index 3 are obtained all tree sign patterns that allow nilpotence of index 3 are characterized sign patterns of order 3 that allow nilpotence are identified

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

worst case analysis of nonlinear systems the authors work out a framework for evaluating the performance of a continuous time nonlinear system when this is quantified as the maximal value at an output port under bounded disturbances the disturbance problem this is useful in computing gain functions and l sub spl infin induced norms which are often used to characterize performance and robustness of feedback systems the approach is variational and relies on the theory of viscosity solutions of hamilton jacobi equations convergence of euler approximation schemes via discrete dynamic programming is established the authors also provide an algorithm to compute upper bounds for value functions differences between the disturbance problem and the optimal control problem are noted and a proof of convergence of approximation schemes for the control problem is given case studies are presented which assess the robustness of a feedback system and the quality of trajectory tracking in the presence of structured uncertainty

allow problems concerning spectral properties of sign pattern matrices a survey abstract an n n sign pattern matrix has entries in 0 this paper surveys the following problems concerning spectral properties of sign pattern matrices sign patterns that allow all possible spectra spectrally arbitrary sign patterns sign patterns that allow all inertias inertially arbitrary sign patterns sign patterns that allow nilpotency potentially nilpotent sign patterns and sign patterns that allow stability potentially stable sign patterns relationships between these four classes of sign patterns are given and several open problems are identified