global stabilization for a class of switched nonlinear feedforward systems abstract the problem of global stabilization for a class of switched nonlinear feedforward systems under arbitrary switchings is investigated in this paper based on the integrator forwarding technique and the common lyapunov function method we design bounded state feedback controllers of individual subsystems to guarantee asymptotic stability of the closed loop system a common coordinate transformation of all subsystems is exploited to avoid individual coordinate transformations for subsystems that are required when applying the forwarding recursive design scheme an example is provided to demonstrate the effectiveness of the proposed design method

nonlinear control of feedforward systems with bounded signals the stabilization problem for a class of nonlinear feedforward systems is solved using bounded control it is shown that when the lower subsystem of the cascade is input to state stable and the upper subsystem not exponentially unstable global asymptotic stability can be achieved via a simple static feedback having bounded amplitude that requires knowledge of the upper part of the state only this is made possible by invoking the bounded real lemma and a generalization of the small gain theorem thus stabilization is achieved with typical saturation functions saturations of constant sign or quantized control moreover the problem of asymptotic stabilization of a stable linear system with bounded outputs is solved by means of dynamic feedback finally a new class of stabilizing control laws for a chain of integrators with input saturation is proposed some robustness issues are also addressed and the theory is illustrated with examples on the stabilization of physical systems

brief note on the variation of constants formula for fuzzy differential equations this note gives a theory of state transition matrices for linear systems of fuzzy differential equations this is used to give a fuzzy version of the classical variation of constants formula a simple example of a time independent control system is used to illustrate the methods while similar problems to the crisp case arise for time dependent systems in time independent cases the calculations are elementary solutions of eigenvalue eigenvector problems in particular for nonnegative or nonpositive matrices the problems at each level set can easily be solved in matlab to give the level sets of the fuzzy solution

first order linear fuzzy differential equations under generalized differentiability first order linear fuzzy differential equations are investigated we interpret a fuzzy differential equation by using the strongly generalized differentiability concept because under this interpretation we may obtain solutions which have a decreasing length of their support which means a decreasing uncertainty in several applications the behaviour of these solutions better reflects the behaviour of some real world systems derivatives of the h difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations using different versions of the variation of constants formula some examples show the rich behaviour of the solutions obtained

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

smooth output feedback stabilization of planar systems without controllable observable linearization this note considers the problem of global stabilization by output feedback for a family of planar systems whose jacobian linearization is neither controllable nor observable the problem cannot be dealt with by existing output feedback design methods most of them are based on the separation principle under appropriate growth conditions we propose an output feedback control scheme that does not rely on the separation principle and achieves global asymptotic stabilization the novelty of our control scheme lies in the explicit design of a dynamic output compensator which combines a nonlinear gain observer design and the technique of adding a power integrator as a consequence an interesting global stabilization result by output feedback can be obtained for feedback linearizable systems in a triangular form which turns out to be new even in the two dimensional case

global stabilization of high order nonlinear time delay systems by state feedback abstract we consider the problem of global stabilization by state feedback for a class of high order nonlinear systems with time delay by developing a novel dynamic gain based backstepping approach a state feedback controller independent of the time delay is explicitly constructed with the help of appropriate lyapunov krasovskii functionals the precise knowledge even the upper bound of the time delay is not required it is proved that the states of the nonlinear time delay systems can be regulated to the origin while all the closed loop signals are globally bounded finally both physical and academic examples are given to illustrate the applications of the proposed scheme

fractional calculus for interval valued functions abstract we use a generalization of the hukuhara difference for closed intervals on the real line to develop a theory of the fractional calculus for interval valued functions the properties of riemann liouville fractional integral riemann liouville fractional derivative and caputo fractional derivative for interval valued functions are investigated several examples are presented to illustrate the concepts and results

the absolutely continuous spectrum of jacobi matrices i explore some consequences of a groundbreaking result of brei messer and pearson on the absolutely continuous spectrum of one dimensional schrodinger operators these include an oracle theorem that predicts the potential and rather general results on the approach to certain limit potentials in particular we prove a denisov rakhmanov type theorem for the general nite gap case the main theme is the following it is extremely dicult to produce abso lutely continuous spectrum in one space dimension and thus its existence has strong implications

extremal approximately convex functions and estimating the size of convex hulls abstract a real valued function f defined on a convex set k is an approximately convex function iff it satisfies f x y 2 f x f y 2 1 a thorough study of approximately convex functions is made the principal results are a sharp universal upper bound for lower semi continuous approximately convex functions that vanish on the vertices of a simplex and an explicit description of the unique largest bounded approximately convex function e vanishing on the vertices of a simplex a set a in a normed space is an approximately convex set iff for all a b a the distance of the midpoint a b 2 to a is 1 the bounds on approximately convex functions are used to show that in rn with the euclidean norm for any approximately convex set a any point z of the convex hull of a is at a distance of at most log2 n 1 1 n 1 2 log2 n 1 from a examples are given to show this is the sharp bound bounds for general norms on rn are also given

adaptive regulation of high order lower triangular systems an adding a power integrator technique abstract in this paper a new adaptive control technique called adding a power integrator is proposed for a class of linearly parametrized high order nonlinear systems that are of a lower triangular structure but neither feedback linearizable nor affine in the control input using the adaptive adding a power integrator technique as a basic design tool we show how to explicitly construct in a recursive manner a smooth adaptive controller that solves the problem of adaptive regulation with global stability the development of the adding a power integrator technique leads to a significant relaxation on the system nonlinearities such as feedback linearizability and affineness which are known to be two standard hypotheses made in the existing nonlinear adaptive control schemes

global robust controllability of the triangular integro differential volterra systems a solution of the global controllability problem for a class of nonlinear control systems of the volterra integro differential equations is presented it is proven that there exists a family of continuous controls that solve the global controllability problem for this class the constructed controls depend continuously on the initial and the terminal states it makes possible to prove the global controllability of the uniformly bounded perturbations of these systems under the global lipschitz condition for the unperturbed system with respect to the states and the controls

global finite time stabilisation for a class of stochastic high order time varying nonlinear systems abstractin this paper the problem of global finite time stabilisation by state feedback is investigated for a class of stochastic high order nonlinear systems with time varying nonlinearities based on the generalised stochastic lyapunov theorem on finite time stability and by skillfully using the method of adding a power integrator a continuous state feedback controller is successfully constructed to guarantee that the closed loop system is globally finite time stable in probability finally several simulation examples are provided to illustrate the effectiveness of the proposed method

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

adding one power integrator a tool for global stabilization of high order lower triangular systems a new feedback design tool called adding a power integrator is introduced and used to solve the problem of global robust stabilization for a significant class of uncertain nonlinear systems that are of a lower triangular form but neither necessarily feedback linearizable fully or partially nor affine in the control input and therefore cannot be dealt with via conventional approaches sufficient conditions are presented under which a globally stabilizing smooth state feedback control law can be explicitly constructed using repeatedly the adding one power integrator technique this novel design technique permits removal of the feedback linearizability and linearity of the control input which have been two common assumptions so far in the literature of nonlinear control

a stacking theorem approach for fuzzy differential equations abstract this paper presents new existence results for fuzzy differential equations our analysis combines the stacking theorem with new solution set results for ordinary differential equations

solving interval valued fractional initial value problems under caputo gh fractional differentiability abstract in this paper interval valued fractional differential equations ifdes under caputo generalized hukuhara differentiability are introduced we present existence and uniqueness results for ifdes with a krasnoselskii krein type condition the solution to interval valued fractional initial value problems under caputo type interval valued fractional derivatives by a modified fractional euler method mfem and a modified adams bashforth moulton method mabmm is also presented

fuzzy fractional functional integral and differential equations this paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle moreover the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives finally the solution to fuzzy fractional functional initial value problem under caputo type fuzzy fractional derivatives by a modified fractional euler method mfem is presented the method is illustrated by solving some examples

on periodic solutions to first order linear fuzzy differential equations under differential inclusions approach in this work we study some properties of the solutions to first order linear fuzzy differential equations under differential inclusions approach we show that the differential inclusions approach generally adapts better to some properties of the solutions to ordinary differential equations sufficient conditions for the solutions under this approach to be periodic and bounded are presented we also give some examples to show the applicability of these results

continuous global stabilisation of high order time delay nonlinear systems this paper studies the global stabilisation problem for a class of high order nonlinear systems with multiple time delays a distinct property of the systems to be investigated is that powers on the upper bound restrictions of nonlinearities are allowed to take values on a continuous interval by introducing the sign functions and using the generalised method of adding a power integrator this paper successfully designs a continuous global state feedback controller independent of time delays moreover a novel lyapunov krasovskii functional is constructed to prove the globally asymptotic stability of the resulting closed loop system

quadratic forms and solutions of the schrodinger equation the spectral theory for the schr dinger equation on the half line is treated through an analysis of the asymptotics of quadratic forms in a pair of solutions solutions of the second and third order differential equations for these forms are derived in the case of the second order de milne s equation it is shown that a single solution leads to the determination of the singular spectrum this generalizes previous results which applied only to isolated points of the discrete spectrum for the absolutely continuous spectrum it is shown that a single solution allows one not only to locate the spectrum but also to determine the spectral density function explicitly

numerical solution of fuzzy differential equations by predictor corrector method in this paper three numerical methods to solve the fuzzy ordinary differential equations are discussed these methods are adams bashforth adams moulton and predictor corrector predictor corrector is obtained by combining adams bashforth and adams moulton methods convergence and stability of the proposed methods are also proved in detail in addition these methods are illustrated by solving two fuzzy cauchy problems

backstepping with bounded feedbacks an extension of the backstepping approach is proposed it allows to globally asymptotically stabilize by bounded feedbacks families of nonlinear control systems explicit expressions of control laws and lyapunov functions are given

further results on global adaptive stabilisation for a class of uncertain stochastic nonlinear systems this paper considers the global adaptive stabilisation for a class of uncertain stochastic nonlinear systems comparing with the existing relevant literature the systems under investigation allow more serious unknowns uncertainties more importantly by remarkably reducing the dimension of dynamic compensator for system unknowns from n to 2 and considerably diminishing the overparametrisation in the existing literature this paper proposes an improved design scheme for adaptive state feedback controllers it is turned out that with the designed controller in loop all the closed loop system states are globally bounded and the original system states converge to the origin both in the sense of probability one a simulation example is also given to demonstrate the effectiveness of the theoretical results

hyers ulam stability of a first order partial differential equation abstract we prove that the existence of a global prime integral leads in appropriate conditions to the hyers ulam stability of a linear partial differential equation of first order

an explicit formula of bounded feedback stabilizers for feedforward systems abstract this work constitutes continuation of a recent result of the same authors which provides an explicit formula for bounded feedback stabilizers for a wide class of triangular nonlinear systems in the present paper the previous result is extended for nonlinear systems having feedforward structure and whose linearization at the equilibrium is in general uncontrollable

initial and boundary value problems for fuzzy differential equations we prove a wintner type result for fuzzy ivps and a superlinear type result for fuzzy bvps both results rely on a generalized schauder theorem in metric spaces

a characterization of hyers ulam stability of first order linear differential operators abstract let x be a complex banach space and h r c a continuous function let t h c 1 r x c r x be the linear differential operator defined by thu u hu we give a necessary and sufficient condition in order that the operator th has the hyers ulam stability

immersion and invariance a new tool for stabilization and adaptive control of nonlinear systems a new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented the method relies upon the notions of system immersion and manifold invariance and in principle does not require the knowledge of a control lyapunov function the construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the invariant output zeroing manifold and its friend the method is well suited in situations where we know a stabilizing controller of a nominal reduced order model which we would like to robustify with respect to higher order dynamics this is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one we also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective our construction does not invoke certainty equivalence nor requires a linear parameterization furthermore viewed from a lyapunov perspective it provides a procedure to add cross terms between the parameter estimates and the plant states finally it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form yielding new stabilizing control laws we illustrate the method with several academic and practical examples including a mechanical system with flexibility modes an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application

robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization this paper investigates the problem of robust output feedback stabilization for a family of uncertain nonlinear systems with uncontrollable unobservable linearization to achieve global robust stabilization via smooth output feedback we introduce a rescaling transformation with an appropriate dilation which turns out to be very effective in dealing with uncertainty of the system using this rescaling technique combined with the nonseparation principle based design method we develop a robust output feedback control scheme for uncertain nonlinear systems in the p normal form under a homogeneous growth condition the construction of smooth state feedback controllers and homogeneous observers uses only the knowledge of the bounding homogeneous system rather than the uncertain system itself the robust output feedback design approach is then extended to a class of uncertain cascade systems beyond a strict triangular structure examples are provided to illustrate the results of the paper

stabilization of nonlinear systems in the plane abstract it is shown that every small time locally controllable system in the plane can locally be asymptotically stabilized by employing locally holder continuous feedback laws as essentially was conjectured by e sontag an explicit algorithm for the construction of such feedback laws is given

a generalised homogeneous solution for global stabilisation of a class of non smooth upper triangular systems this paper investigates the problem of using small controls to globally stabilise a class of upper triangular systems including non smooth systems a generalised definition of homogeneity with new weights is employed to relax the existing restrictions imposed on the nonlinearities and enable the design of non smooth stabilisers by developing a more delicate design approach small state feedback controllers are recursively constructed in a bottom to top fashion to globally stabilise the upper triangular systems in addition a number of typical examples are discussed in order to illustrate that the proposed result not only encompasses several existing works under the same topic but also solves the global stabilisation problem of more general upper triangular systems using simpler controller structures

instability of set differential equations abstract this paper is devoted to the instability of set differential equations sdes using the geometric inequalities of brunn minkowski and a d aleksandrov we propose new methods for constructing lyapunov functions in combination with the known methods of stability theory the lyapunov s direct method the comparison method and the vector function method we establish conditions for the collapse of the solutions of the sdes estimates of the collapse time of solutions are also obtained examples of sdes in spaces of dimension 2 and 3 illustrating general theorems are given

brief paper decentralized control design of interconnected chains of integrators a case study we develop a constructive decentralized control design procedure for a class of systems that may be loosely described as chained integrators which are dynamically coupled the design method is inspired by nested saturation control ideas and formulated by applying the singular perturbation theory we demonstrate that the proposed design provides a lyapunov function for an associated closed loop system from which semi global stability may be deduced using the proposed idea we design a semi globally stabilizing control law for a four degree of freedom spherical inverted pendulum

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

asymptotic stabilization of a class of smooth two dimensional systems this paper studies the asymptotic stabilizability of two dimensional control systems the class under consideration includes c infty systems that satisfy a certain genericity assumption and all real analytic systems necessary and sufficient conditions for feedback stabilization using continuous feedback and a sufficient condition for c 1 feedback stabilization are given this latter condition is given in terms of an inequality involving two indices if the direction of the inequality is changed an obstruction to c infty feedback stabilizability is obtained a subclass of polynomial systems is also studied and given complete necessary and sufficient conditions for global asymptotic stabilization using c 1 feedback

global stabilization of stochastic nonlinear systems via c 1 and c infty controllers the problem of designing c 1 or c infty controllers for a class of stochastic nonlinear systems snss in lower triangular form is studied in this note by using the well known backstepping method the concept of homogeneity with monotone degrees hwmd and the sign function approach we construct a c 1 state feedback controller recursively meanwhile by employing a polynomial lyapunov function with sign functions we prove that the solution to snss is globally asymptotically stable in probability furthermore based on the concept of hwmd it shows that c 1 controllers for a class of three dimensional snss can be modified to c infty controllers under certain conditions finally two simulation examples are given to demonstrate the effectiveness of the proposed controllers

viability theory and fuzzy differential equations new results are presented which guarantee the existence of viable solutions to differential inclusions we show that a viable solution exists if i a tangency condition is satisfied and ii there exists a maximal solution to an appropriate differential equation in addition we discuss the topological structure of the solution set these results will then be used with the stacking theorem to establish new existence criteria for fuzzy differential equations if there exists a maximal solution for an appropriate set of differential equations

nilpotent approximations of control systems and distributions a constructive method is given to approximate the vector fields in a nonlinear control system which is linear in the controls by a system of similar form and on the same state space with the describing vector fields of the approximating system generating a lie algebra which has a certain relevant sub algebra nilpotent given a k dimensional distribution on r n say locally defined near zero the method leads to an approximating k distribution which has nilpotent basis agrees with the original distribution at zero and each derived distribution of the approximating distribution and original distribution also agree at zero

homogeneity with incremental degrees and global stabilisation of a class of high order upper triangular systems the problem of global asymptotic stabilisation is investigated for a class of upper triangular systems with uncontrollable linearisation a notion of generalised homogeneity is introduced in this article providing a new insight and a deeper perspective on how to categorise nonlinear terms into the higher order and equal order terms in a non traditional way the delicate analysis and new categorisation of the vector field make it possible to encompass more general nonlinearities a globally stabilising controller is recursively constructed by a bottom up step by step procedure containing the equal order terms as its indispensable components it is illustrated that the results obtained in this article incorporate and extend existing results for the global stabilisation of upper triangular systems

on the ulam stability of first order linear fuzzy differential equations under generalized differentiability in this paper we investigate under some suitable conditions and generalized differentiability the ulam stability problems of three variants of first order linear fuzzy differential equations respectively

brief paper feedback stabilization for high order feedforward nonlinear time delay systems this paper investigates the problem of global strong stabilization by state feedback for a family of high order feedforward nonlinear time delay systems the uncertain nonlinearities are assumed to satisfy a polynomial growth assumption with an input or delayed input dependent rate with the help of the appropriate lyapunov krasovskii functionals and a rescaling transformation with a gain to be tuned online by a dynamic equation we propose a dynamic low gain state feedback control scheme a simulation example is given to demonstrate the effectiveness of the proposed design procedure

stabilization of feedforward systems approximated by a non linear chain of integrators we prove that for any odd integer p and any strictly positive integer n feedforward systems which are approximated at the origin by a chain of integrators of degree p and length n can be globally asymptotically stabilized by bounded smooth time invariant state feedbacks our proof is based on the construction of a lyapunov function and the feedback laws we obtain are given by explicit formulas

adaptive stabilisation for a large class of high order uncertain non linear systems the problem of global adaptive stabilisation by state feedback is investigated for a class of high order non linear systems with uncertain control coefficients and zero dynamics first some appropriate unknown parameters are introduced to obtain the updating laws when adapting control design then by the flexible way of combining adding a power integrator with adaptive technique and the idea of changing supply functions the requirement on the uncertain control coefficients is relaxed and a recursive design procedure is successfully developed to achieve a continuous adaptive stabilising controller finally an example is provided to illustrate the correctness of the theoretical results

global robust stabilization of minimum phase nonlinear systems with uncertainty the problem of global stabilization via state feedback is studied for a class of minimum phase nonlinear systems with unknown unmodelled time varying parameters or disturbances belonging to a given compact set sufficient conditions are presented for the existence of a robust stabilizing state feedback controller the controller is constructed explicitly by using a lyapunov based recursive design procedure the proposed feedback control law is a smooth function of the state and requires information of bounding functions on the size of uncertainties in the case where the disturbance signals represent only an unknown constant vector parameter and appear linearly in the system an adaptive controller is developed based on similar arguments achieving global stability with state regulation the main results of this paper incorporate and generalize a number of robust and adaptive control schemes for global stabilization of uncertain feedback linearizable systems without zero dynamics

periodic boundary value problems for impulsive fuzzy differential equations the peculiarity of the hukuhara derivative makes it impossible to find periodic solutions for fuzzy differential equations with the exception of very restrictive situations in this work we consider a boundary value problem associated with an impulsive fuzzy differential equation and approximate the extremal solutions in a fuzzy functional interval using the monotone method fuzzy comparison results are useful in our procedure and the expression of the solution for some impulsive periodic linear differential problems is also provided

almost disturbance decoupling for a class of high order nonlinear systems the problem of almost disturbance decoupling with internal stability add is formulated in terms of an l sub 2 l sub 2p instead of an l sub 2 gain for a class of high order nonlinear systems which consist of a chain of power integrators perturbed by a lower triangular vector field a significant feature of the systems considered in the paper is that they are neither feedback linearizable nor affine in the control input which have been two basic assumptions made in all the existing add nonlinear control schemes using the technique of adding a power integrator we solve the add problem via static smooth state feedback under a set of growth conditions that can be viewed as a high order version of the feedback linearization conditions we also show how to explicitly construct a smooth state feedback controller that attenuates the disturbance s effect on the output to an arbitrary degree of accuracy with internal stability

hyers ulam stability of fuzzy fractional volterra integral equations with the kernel ψ function via successive approximation method abstract in this paper we investigative the hyers ulam stability and the hyers ulam rassias stability of fuzzy fractional volterra integral equations ffvies involving the kernel function we propose conditions for the existence uniqueness and stability of ffvie by employing the method of successive approximation to visualize the theoretical results some examples and numerical simulations are given

on the solutions of first order linear impulsive fuzzy differential equations abstract this paper gives a representation of solutions to first order linear impulsive fuzzy differential equations with constant coefficients using a generalized differentiability concept and an impulsive fundamental matrix we present ideas similar to those in linear fuzzy differential equations for impulsive fuzzy cauchy problems finally numerical examples are given to illustrate the theory

a general theory of almost convex functions let m t 0 t m r m 1 t i 0 m i 0 t i 1 be the standard m dimensional simplex and let s c m 1 m then a function h c r with domain a convex set in a real vector space is s almost convex iff for all t 0 t m s and x 0 x m c the inequality h t 0 x 0 t m x m 1 t 0 h x 0 t m h x m holds a detailed study of the properties of s almost convex functions is made if s contains at least one point that is not a vertex then an extremal s almost convex function eg n r is constructed with the properties that it vanishes on the vertices of an and if h n r is any bounded s almost convex function with h e k 0 on the vertices of an then h x e s x for all x an in the special case s 1 m 1 1 m 1 the barycenter of m very explicit formulas are given for eg and s n sup x n e s x these are of interest as e s and s n are extremal in various geometric and analytic inequalities and theorems

on the stabilization of feedforward systems with bounded control abstract for the global asymptotic stabilization of nonlinear controllable in the first approximation cascades consisting of a globally asymptotically locally exponentially stable system driving a stable system a lyapunov like design is proposed this yields bounded control laws where the control magnitude can be chosen arbitrarily large this result provides an alternative to classical forwarding

variation of constant formula for first order fuzzy differential equations in this paper we study first order linear fuzzy differential equations by using the generalized differentiability concept and we present the general form of their solutions we also correct and complete some previous results finally some examples are given to illustrate our results

ulam s type stability of impulsive ordinary differential equations abstract in this paper we introduce four ulam s type stability concepts for impulsive ordinary differential equations by applying the integral inequality of gronwall type for piecewise continuous functions ulam s type stability results for impulsive ordinary differential equations are obtained an example is also provided to illustrate our results

extremal approximately convex functions and the best constants in a theorem of hyers and ulam abstract let n 1 and b 2 a real valued function f defined on the n simplex n is approximately convex with respect to b 1 if f i 1 b t i x i i 1 b t i f x i 1 for all x 1 x b n and all t 1 t b b 1 we determine the extremal function of this type which vanishes on the vertices of n we also prove a stability theorem of hyers ulam type which yields as a special case the best constants in the hyers ulam stability theorem for e convex functions

value distribution and spectral analysis of differential operators the theory of value distribution is applied through a study of boundary value distribution for the weyl titchmarsh m function to the spectral analysis of differential operators on the half line particular consequences are the theoretical underpinning of a numerical approach to spectral analysis of differential operators exhibiting absolutely continuous or singular spectra and new identities for the spectral measure in terms of solutions of the associated differential equation

adding an integrator for the stabilization problem abstract we study the relationship between the following two properties p1 the system x f x y y v is locally asymptotically stabilizable and p2 the system x f x u is locally asymptotically stabilizable where x r n y r dayawansa martin and knowles have proved that these properties are equivalent if the dimension n 1 here using the so called control lyapunov function approach a we propose another more constructive and somewhat simpler proof of dayawansa martin and knowles s result b we show that in general p1 does not imply p2 for dimensions n larger than 1 c we prove that p2 implies p1 if some extra assumptions are added like homogeneity of the system by using the latter result recursively we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

practical output tracking of nonlinear systems with uncontrollable unstable linearization the paper focuses on the problem of having the output of nonlinear systems track a prescribed c sup 1 reference signal it is proved that under mild assumptions global practical output tracking is achievable by smooth state feedback although asymptotic output tracking is usually not possible even locally because the linearized system has uncontrollable modes whose eigenvalues are on the right half plane smooth feedback controllers which solve the problem of global practical output tracking are explicitly constructed based on a modified adding a power integrator approach this feedback design method also leads to solutions to challenging benchmark control problems including practical output tracking of an underactuated unstable two degrees of freedom mechanical system

interconnected nonlinear systems local and global stabilization a family of interconnected nonlinear systems including partially linear systems is studied no assumption on the stability of each uncontrolled subsystem is imposed but asymptotic stabilizability for each of them is assumed the objective of this work is the state feedback stabilization of the origin of such composite systems local and global results are derived using simple techniques such as coordinate and feedback transformations

global stabilization of cascade systems by c sup 0 partial state feedback this paper shows how the nonsmooth but continuous feedback design approach developed recently for global stabilization of nonlinear systems with uncontrollable unstable linearization and the notion and properties of the input to state stability lyapunov function can be effectively coupled resulting in globally stabilizing c sup 0 partial state feedback controllers for a class of cascade systems which may not be smoothly stabilizable even locally

explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization for a class of nonlinear systems having triangular structure we provide explicit formulas for feedback controllers that exhibit global stabilization the global part of our analysis is mainly based on a version of sontag s input to state stability property and constitutes a generalization of previous works of the second author

a necessary and sufficient condition for local asymptotic stability of a class of nonlinear systems in the critical case by the theory of linear differential equations a system described by a chain of integrators with a linear feedback is globally asymptotically stable if and only if the characteristic polynomial is hurwitz which implies that all the coefficients in the linear feedback equation are negative however negative coefficients may not guarantee the local asymptotic stability of the linear system in this paper we reveal that by monotonizing the powers of the integrators the strict negativity of the feedback coefficients is not only necessary but also sufficient for the local asymptotic stability of the system a dual result is also obtained for the dual power integrator systems

asymptotic stabilization of minimum phase nonlinear systems how a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown more precisely conditions under which for every compact set of initial states it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described the theory includes the development of globally defined transformations of the system equations to their global normal form

ulam type stability of first order linear impulsive fuzzy differential equations abstract in this paper we consider ulam stability concepts for first order linear impulsive fuzzy differential equations which generalize standard definitions for impulsive differential equations sufficient conditions are established to guarantee that our equations are ulam stable numerically examples are also provided to illustrate our results

fuzzy differential equations we will concentrate on the second order linear constant coefficient ordinary differential equation for x in interval i i can be 0 t for t 0 or i can be 0 the initial conditions are y 0 0 y 0 1 we assume g is continuous on i

constructive nonsmooth stabilization of triangular systems the problem of local asymptotic continuous feedback stabilization of single input nonlinear systems is considered here the aim is to explicitly construct a continuous asymptotically stabilizing feedback for a class of nonlinear systems that is known to be continuous feedback asymptotically stabilizable but for which the known results do not provide an effective method to compute the stabilizer computer simulations are included to show practical applicability of our approach

decomposition of higher order wright convex functions in this paper we consider higher order wright convex functions and prove that they are representable as the sum of a continuous higher order convex function and a polynomial function

a note on initial value problems for fractional fuzzy differential equations abstract in this note we present some remarks on solutions of fractional fuzzy differential equation in general a fractional fuzzy differential equation and a fractional fuzzy integral equation are not equivalent in this paper we give an appropriate condition so that this equivalence is valid some examples are provided to illustrate the theory

note on numerical solutions of fuzzy differential equations by predictor corrector method in the present note it is shown that the examples presented in a recent paper by allahviranloo et al are incorrect namely the exact solutions proposed by the authors are not solutions of the given fuzzy differential equations fdes the correct exact solutions are also presented here together with some results for characterizing solutions of fdes under hukuhara differentiability by an equivalent system of odes in this way a new direction for the numerical solutions of fdes is proposed

finite time stabilization of high order stochastic nonlinear systems in strict feedback form this paper deals with the problem of finite time stabilization for a class of high order stochastic nonlinear systems in strict feedback form by using ito s formula mathematical induction and backstepping design method a novel state feedback controller is constructed to guarantee that the closed loop high order nonlinear system has a unique solution and the solution of the closed loop high order nonlinear system is finite time stable a systematic design algorithm is developed for the construction of the backstepping controller finally the effectiveness of the state feedback controller is illustrated by two examples