asymptotic tracking of a reference state for systems with a feedforward structure we consider the problem of asymptotic tracking for a system which can be written in a feedforward form the data of a bounded reference state trajectory is assumed our solution relies on a lyapunov construction the time varying state feedbacks obtained are bounded and ensure the global uniform asymptotic stability when the signals are periodic and the local exponential stability of the reference state trajectory of the closed loop system

an optimal control approach to robust tracking of linear systems in our early work we show that one way to solve a robust control problem of an uncertain system is to translate the robust control problem into an optimal control problem if the system is linear then the optimal control problem becomes a linear quadratic regulator lqr problem which can be solved by solving an algebraic riccati equation in this article we extend the optimal control approach to robust tracking of linear systems we assume that the control objective is not simply to drive the state to zero but rather to track a non zero reference signal we assume that the reference signal to be tracked is a polynomial function of time we first investigated the tracking problem under the conditions that all state variables are available for feedback and show that the robust tracking problem can be solved by solving an algebraic riccati equation because the state feedback is not always available in practice we also investigated the output feedback we show that if we place the poles of the observer

limit boundary value problems for first order delay differential equations where f r x r r g r r r 0 co are continuous and g t i t for t l 0 g t 00 as t co for a given a 2 0 let ci inf g f a continuous function x ci a r is said to be a solution of lbvp e a p if x is continuously differentiable on a co and satisfies e for t 2 a and x t p as t co a continuous function defined on some right half line is said to be a solution of the terminal value problem e 1 1 tvp e j for short if there is a to 2 0 such that it is a solution of lbvp e 1 i the solution of lbvp e a j3 is said to be not unique if there are two solutions xi and x of lbvp e a p such that xi t x2 t for t 1 a and is unique otherwise aftabizadeh and lakshmikantham i have established the theory of terminal differential inequalities existence of extremal solutions of tvp e p with g t t it is difficult to establish the parallel theory for lbvp e 0 p with g t s t the difficulties arise from the delay i g t to see this we consider the following lbvp

on a nonlinear multivariable servomechanism problem abstract multivariable nonlinear plants with both measured and unmeasured vector disturbance signals are considered output feedback dynamic control laws are designed for the problem of tracking a constant or slowly varying reference input while rejecting constant or slowly varying disturbances

convergence results in a well known delayed predator prey system abstract in this paper we provide a detailed and explicit procedure of obtaining some regions of attraction for the positive steady state assumed to exist of a well known lotka volterra type predator prey system with a single discrete delay our procedure requires the delay length to be small a detailed example is presented the method used here is to construct a proper liapunov functional in a restricted region

global analysis of a system of predator prey equations we study a class of ode s modelling the interaction of one predator and one prey the prey is assumed to be asocial in the sense that there is a threshold below which it will die even in the absence of predators knowledge of the separatrices of the flow and the local nature of the equilibrium points yields an analogue of the kolmogorov theorem a more detailed study of the direction of hopf bifurcation and uniqueness of limit cycles together with recent work on their continuation allows us to exhibit a very rich dynamics for both this system as well as for the more classical case of a prey without threshold we briefly discuss the implications of this work for ecology

the dimension free structure of nonhomogeneous random matrices let x be a symmetric random matrix with independent but non identically distributed centered gaussian entries we show that begin aligned mathbf e vert x vert s p asymp mathbf e bigg bigg sum i bigg sum j x ij 2 bigg p 2 bigg 1 p bigg end aligned for any 2 le p le infty where s p denotes the p schatten class and the constants are universal the right hand side admits an explicit expression in terms of the variances of the matrix entries this settles in the case p infty a conjecture of the first author and provides a complete characterization of the class of infinite matrices with independent gaussian entries that define bounded operators on ell 2 along the way we obtain optimal dimension free bounds on the moments mathbf e vert x vert s p p 1 p that are of independent interest we develop further extensions to non symmetric matrices and to nonasymptotic moment and norm estimates for matrices with non gaussian entries that arise for example in the study of random graphs and in applied mathematics

an l p theory of sparse graph convergence ii ld convergence quotients and right convergence we extend the lplp theory of sparse graph limits which was introduced in a companion paper by analyzing different notions of convergence under suitable restrictions on node weights we prove the equivalence of metric convergence quotient convergence microcanonical ground state energy convergence microcanonical free energy convergence and large deviation convergence our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree while the proofs require new techniques based on uniform upper regularity examples to which our theory applies include stochastic block models power law graphs and sparse versions of ww random graphs

tracking control of a class of non linear systems with applications to cruise control of air breathing hypersonic vehicles a new tracking control method for general non linear systems is proposed a virtual controller and some command references are introduced to asymptotically stabilise the system of the tracking error dynamics then the actual controller and command references are derived by solving a system of linear algebraic equations compared with other tracking control methods in the literature the tracking controller design in this paper is simple because it needs only to solve a system of linear algebraic equations the boundedness of the tracking controller and command references is guaranteed by the solvability of the terminal value problem tvp of an ordinary differential equation for non linear systems with minimum phase properties the tvp is automatically solvable a numerical example shows that the tracking control method is still available for some systems with non minimum phase properties to enhance the robustness of the tracking controller a non linear disturbance observer ndo is introduced to esti

sparse exchangeable graphs and their limits via graphon processes in a recent paper caron and fox suggest a probabilistic model for sparse graphs which are exchangeable when associating each vertex with a time parameter in r here we show that by generalizing the classical definition of graphons as functions over probability spaces to functions over finite measure spaces we can model a large family of exchangeable graphs including the caron fox graphs and the traditional exchangeable dense graphs as special cases explicitly modelling the underlying space of features by a finite measure space s s and the connection probabilities by an integrable function w s s 0 1 we construct a random family gt t 0 of growing graphs such that the vertices of gt are given by a poisson point process on s with intensity t with two points x y of the point process connected with probability w x y we call such a random family a graphon process we prove that a graphon process has convergent subgraph frequencies with possibly infinite limits and that in the natural extension of the cut metric to our setting the sequence converges to the generating graphon we also show that the underlying graphon is identifiable only as an equivalence class over graphons with cut distance zero more generally we study metric convergence for arbitrary not necessarily random sequences of graphs and show that a sequence of graphs has a convergent subsequence if and only if it has a subsequence satisfying a property we call uniform regularity of tails finally we prove that every graphon is equivalent to a graphon on r equipped with lebesgue measure

effect of cross diffusion on the stationary problem of a predator prey system with a protection zone abstract this paper is concerned with the steady state problem of a predator prey cross diffusion system with herd effect allee effect and a protection zone some sufficient conditions for the existence of positive steady state solutions are given our proof is based on the local and global bifurcation theory and some a priori estimates some limiting behavior of positive steady states with respect to the allee effect constant or the diffusion coefficient d are discussed

nonlinear large deviations abstract we present a general technique for computing large deviations of nonlinear functions of independent bernoulli random variables the method is applied to compute the large deviation rate functions for subgraph counts in sparse random graphs previous technology based on szemeredi s regularity lemma works only for dense graphs applications are also made to exponential random graphs and three term arithmetic progressions in random sets of integers

a new look at independence the concentration of measure phenomenon in product spaces is a farreaching abstract generalization of the classical exponential inequalities for sums of independent random variables we attempt to explain in the simplest possible terms the basic concepts underlying this phenomenon the basic method to prove concentration inequalities and the meaning of several of the most useful inequalities

three limit cycles in a leslie gower predator prey model with additive allee effect in this work a bidimensional continuous time differential equations system is analyzed which is derived of leslie type predator prey schemes by considering a nonmonotonic functional response and allee effect on population prey for the system obtained we describe the bifurcation diagram of limit cycles that appears in the first quadrant the only quadrant of interest for the sake of realism we show that under certain conditions over the parameters the system allows the existence of three limit cycles the first two cycles are infinitesimal ones generated by hopf bifurcation the third one arises from a homoclinic bifurcation furthermore we give conditions over the parameters such that the model allows long term extinction or survival of both populations in particular the presence of a weak allee effect does not imply extinction of populations necessarily for our model

a michaelis menten type food chain model with strong allee effect on the prey dynamical behaviours of a tritrophic food chain model with strong allee effect in the prey are studied in this paper positivity and boundedness of the system are discussed some global results on extinction of the species are derived stability analysis of the equilibrium points is presented the effect of discrete time delay is studied where the delay may be regarded as the gestation period of the superpredator numerical simulations are carried out to validate our analytical findings implications of our analytical and numerical findings are discussed critically

effect of time delay on a food chain model this paper aims to study the effect of discrete time delay on a tritrophic food chain model with holling type ii functional responses dynamical behaviours such as boundedness stability persistence and bifurcation of the model are studied our analytical findings are illustrated through computer simulation biological implications of our analytical findings are addressed critically

large deviations of subgraph counts for sparse erdős renyi graphs abstract for any fixed simple graph h v e and any fixed u 0 we establish the leading order of the exponential rate function for the probability that the number of copies of h in the erd s renyi graph g n p exceeds its expectation by a factor 1 u assuming n h p 1 with h 1 2 where 1 is the maximum degree of h this improves on a previous result of chatterjee and the second author who obtained h c e for a constant c 0 moreover for the case of cycle counts we can take as large as 1 2 we additionally obtain the sharp upper tail for schatten norms of the adjacency matrix as well as the sharp lower tail for counts of graphs for which sidorenko s conjecture holds as a key step we establish quantitative versions of szemeredi s regularity lemma and the counting lemma suitable for the analysis of random graphs in the large deviations regime

upper tails and independence polynomials in random graphs abstract the upper tail problem in the erd s renyi random graph g g n p asks to estimate the probability that the number of copies of a graph h in g exceeds its expectation by a factor 1 chatterjee and dembo showed that in the sparse regime of p 0 as n with p n for an explicit h 0 this problem reduces to a natural variational problem on weighted graphs which was thereafter asymptotically solved by two of the authors in the case where h is a clique here we extend the latter work to any fixed graph h and determine a function c h such that for p as above and any fixed 0 the upper tail probability is exp c h o 1 n 2 p log 1 p where is the maximum degree of h as it turns out the leading order constant in the large deviation rate function c h is governed by the independence polynomial of h defined as p h x i h k x k where i h k is the number of independent sets of size k in h for instance if h is a regular graph on m vertices then c h is the minimum between 1 2 2 m and the unique positive solution of p h x 1

an l p theory of sparse graph convergence i limits sparse random graph models and power law distributions we introduce and develop a theory of limits for sequences of sparse graphs based on lp graphons which generalizes both the existing l1 theory of dense graph limits and its extension by bollob as and riordan to sparse graphs without dense spots in doing so we replace the no dense spots hypothesis with weaker assumptions which allow us to analyze graphs with power law degree distributions this gives the rst broadly applicable limit theory for sparse graphs with unbounded average degrees in this paper we lay the foundations of the lp theory of graphons characterize convergence and develop corresponding random graph models while we prove the equivalence of several alternative metrics in a companion paper