techniques for determining the minimum rank of a small graph abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ij th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise minimum rank is a difficult parameter to compute however there are now a number of known reduction techniques and bounds that can be programmed on a computer we have developed a program using the open source mathematics software sage to implement several techniques we have also established several additional strategies for computation of minimum rank these techniques have been used to determine the minimum ranks of all graphs of order 7

the minimum rank of symmetric matrices described by a graph a survey abstract the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i j is nonzero whenever i j is an edge in g and is zero otherwise this paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues

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

feedback linearization and stabilization of second order non holonomic chained systems this paper presents a theoretical framework for non regular feedback linearization and stabilization of second order non holonomic chained systems by giving a new criterion for the problem of non smooth non regular feedback linearization it is proved that second order chained systems are non regular static state feedback linearizable a discontinuous control law is obtained based on linear system theory and the inversion technique the design mechanism is generalized to higher order non holonomic chained systems simulation studies are carried out to show the effectiveness of the approach

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

comments on adaptive global stabilization of nonholonomic systems with strong nonlinear drifts systems control letters 46 2002 195 205 in the paper to be commented there is a technical problem inherent in the input based switching and it is highlighted here to avoid possible failure when the scheme is used in practice

estimation of states and parameters for linear systems with nonlinearly parameterized perturbations we consider systems that can be described by a linear part with a nonlinear perturbation where the perturbation is parameterized by a vector of unknown constant parameters under a set of technical assumptions about the perturbation and its relationship to the outputs we present a modular design technique for estimating the system states and the unknown parameters the design consists of a high gain observer that estimates the states of the system together with the full perturbation and a parameter estimator constructed by the designer to invert a nonlinear equation we illustrate the technique on a simulated dc motor with friction

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

nonconvergence of formal integrals ii improved estimates for the optimal order of truncation we investigate the asymptotic properties of formal integral series in the neighbourhood of an elliptic equilibrium in nonlinear 2 dof hamiltonian systems in particular we study the dependence of the optimal order of truncation nopt on the distance from the elliptic equilibrium by numerical and analytical means the function nopt determines the region of nekhoroshev stability of the orbits and the time of practical stability we find that the function nopt decreases by abrupt steps the decrease is roughly approximated with an average power law nopt o a with a 1 we find an analytical explanation of this behaviour by investigating the accumulation of small divisors in both the normal form algorithm via lie series and in the direct construction of first integrals precisely we find that the series exhibit an apparent radius of convergence that tends to zero by abrupt steps as the order of the series tends to infinity our results agree with those obtained by servizi g et al 1983 phys lett a 95 11 for a conservative map of the plane moreover our analytical considerations allow us to explain the results of our previous paper contopoulos g et al 2003 j phys a math gen 36 8639 including in particular the different behaviour observed for low order and higher order resonances

computation of minimal rank and path cover number for certain graphs for a given undirected graph g the minimum rank of g is defined to be the smallest possible rank over all real symmetric matrices a whose i j th entry is non zero whenever i j and i j is an edge in g the path cover number of g is the minimum number of vertexdisjoint paths occurring as induced subgraphs of g that cover all the vertices of g for trees the relationship between minimum rank and path cover number is completely understood however for non trees only sporadic results are known we derive formulae for the minimum rank and path cover number for graphs obtained from edge sums and formulae for minimum rank of vertex sums of graphs in addition we examine previously identified special types of vertices and attempt to unify the theory behind them

dynamics in the center manifold of the collinear points of the restricted three body problem this paper focuses on the dynamics near the collinear equilibrium points l1 2 3 of the spatial restricted three body problem rtbp it is well known that the linear behavior of these three points is of the type center center saddle to obtain an accurate description of the dynamics in an extended neighborhood of those points two different but complementary strategies are used first the hamiltonian of the problem is expanded in power series around the equilibrium point then a partial normal form scheme is applied in order to uncouple up to high order the hyperbolic directions from the elliptic ones skipping the remainder we have that the truncated hamiltonian has an invariant manifold tangent to the central directions of the linear part the restriction of the hamiltonian to this manifold is the so called reduction to the center manifold the study of the dynamics of this reduced hamiltonian now with only two degrees of freedom gives a qualitative description of the phase space near the equilibrium point finally a lindstedt poincare procedure is applied to explicitly compute the invariant tori contained in the center manifold these tori are obtained as the fourier series of the corresponding solutions the frequencies being a power expansion of some parameters amplitudes this allows for an accurate quantitative description of these regions in particular the well known halo orbits are obtained

an energy momentum map for the time reversal symmetric 1 1 resonance with z 2 z 2 symmetry abstract we present a general analysis of the bifurcation sequences of periodic orbits in general position of a family of reversible 1 1 resonant hamiltonian normal forms invariant under z 2 z 2 symmetry the rich structure of these classical systems is investigated both with a singularity theory approach and geometric methods the geometric approach readily allows to find an energy momentum map describing the phase space structure of each member of the family and a catastrophe map that captures its global features quadrature formulas for the actions periods and rotation number are also provided

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

the dynamics around the collinear equilibrium points of the rtbp this paper is devoted to the analysis of an extended neighborhood of the collinear equilibrium points of the restricted three body problem the analysis is done using numerical tools for the determination of periodic orbits and invariant 2d tori all the relevant information of the neutrally stable behavior of the dynamics in the vicinity of the three libration points is given the results in this paper extend those given in physica d 132 1999 189 since they avoid the convergence restrictions of the semi analytical approach used in this reference

adaptive output feedback stabilization for nonholonomic systems with strong nonlinear drifts abstract this paper investigates the design of an output feedback adaptive stabilization controller for a nonholonomic chained system with strong nonlinear drifts including modeled nonlinear dynamics unmodeled dynamics and dynamics modeled with unknown parameters also the virtual control directions of the system are unknown the purpose is to design a nonlinear output feedback switching controller such that the closed loop system is globally asymptotically stable a novel observer and estimator are introduced for states and parameter estimates respectively a constructive procedure of design for an output feedback adaptive controller is given by using the integrator backstepping approach and based on the proposed observer and parameter estimator an example is given to show the effectiveness of the proposed scheme

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 unifying framework for the semiglobal stabilization of nonlinear uncertain systems via measurement feedback we study the problem of semiglobally stabilizing an uncertain nonlinear system consisting of a linear nominal system perturbed by either nonlinearities or model uncertainties our approach relies on well known h sub spl infin linear control tools and allows one to recover and improve in the unifying framework of a semiglobal separation result existing results on the semiglobal stabilization via output feedback in particular we discuss the case of uncorrupted outputs input and output nonlinearities or model uncertainties which may include for example practical situations such as backlash hysteresis and saturations the key feature of our design procedure is given by the choice of two continuous functions the first one is instrumental in constructing a stabilizing controller the second one arises in the candidate lyapunov function for the closed loop system relying on our main theorem we give general tools for achieving large regions of attraction via bounded measurement feedback for a wide class of nonlinear uncertain interconnected systems

zero forcing sets and the minimum rank of graphs the minimum rank of a simple graph g is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry for i 6 j is nonzero whenever i j is an edge in g and is zero otherwise this paper introduces a new graph parameter z g that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs often enabling computation of the minimum rank

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

on minimum rank and zero forcing sets of a graph abstract for a graph g on n vertices and a field f the minimum rank of g over f written as mr f g is the smallest possible rank over all n n symmetric matrices over f whose i j th entry for i j is nonzero whenever ij is an edge in g and is zero otherwise the maximum nullity of g over f is m f g n mr f g the minimum rank problem of a graph g is to determine mr f g or equivalently m f g this problem has received considerable attention over the years in f barioli w barrett s butler s m cioab d cvetkovic s m fallat c godsil w haemers l hogben r mikkelson s narayan o pryporova i sciriha w so d stevanovic h van der holst k v meulen a w wehe aim minimum rank special graphs work group zero forcing sets and the minimum rank of graphs linear algebra appl 428 2008 1628 1648 a new graph parameter z g the zero forcing number was introduced to bound m f g from above the authors posted an attractive question what is the class of graphs g for which z g m f g for some field f this paper focuses on exploring the above question

stability analysis of systems with impulse effects the authors establish conditions for the uniform stability and the uniform asymptotic stability of equilibria of systems with impulse effects described by systems of nonlinear time varying ordinary differential equations under mild additional assumptions these sufficient conditions turn out to be necessary as well for the case of time invariant systems with impulse effects described by nonlinear ordinary differential equations the above results are used to establish conditions under which the uniform asymptotic stability of equilibria can be deduced from the linearizations of such systems

adaptive control of continuous time systems with convex concave parametrization this paper deals with adaptive control of a class of nonlinear dynamic systems with nonlinear parametrization in this class the state variables are assumed to be accessible and the nonlinearity in the parameters is assumed to be either convex or concave by introducing a tuning function and an adaptive law based on a min max strategy it is shown that such a class of dynamic systems can be adaptively controlled in a stable manner global boundedness of the overall adaptive system and tracking to within a desired precision are established with the new adaptive controller the proposed controller is applied to a precise positioning problem in the presence of nonlinearly parametrized friction dynamics it is shown that the controller leads to position errors and friction compensation that are orders of magnitude better than those based on estimation of linear parametrizations the fact that the new parameter estimation strategy used is distinct from the traditionally used gradient schemes permits the expansion of the scope of adaptive control which has been restricted hitherto in most cases to systems with linear parametrization

adaptive disturbance attenuation for generalized high order uncertain nonlinear systems abstract this paper investigates the problem of adaptive disturbance attenuation for a class of generalized high order uncertain nonlinear systems the control strategy is on the basis of continuous domination and delicate adaptive technique and it can cope with serious coexistence among uncertainties including time varying control coefficients which have unknown upper and lower bounds nonlinear parameters and external disturbances adaptive state feedback controller is one dimensional irrespective of the number of unknown parameters and its performance is evaluated in terms of l 2 l 2 p gain

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

on the minimum rank of the third power of a starlike tree abstract for a simple graph g of order n let a be a real n n symmetric matrix whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise the minimum rank of g is the smallest possible rank over all such symmetric matrices the jth power of a graph g is the graph g j v f where u v f if and only if there is a walk of length j from u to v in 2007 brualdi hogben and shader reported a conjecture that if t is not the star k 1 n 1 then mr t 3 mr t 2 1 in this paper we construct a class of starlike trees such that mr t 3 mr t 2 which give a negative answer to this conjecture

a switching algorithm for global exponential stabilization of uncertain chained systems this note deals with chained form systems with strongly nonlinear unmodeled dynamics and external disturbances the objective is to design a robust nonlinear state feedback law such that the closed loop system is globally kexponentially stable we propose a novel switching control strategy involving the use of input state scaling and integrator backstepping the new features of our controllers include the ability to achieve lyapunov stability exponential convergence and robustness to a set of uncertain drift terms

an upper bound for the minimum rank of a graph abstract for a graph g of order n the minimum rank of g is defined to be the smallest possible rank over all real symmetric n n matrices a whose i j th entry for i j is nonzero whenever i j is an edge in g and is zero otherwise we prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs including all bipartite graphs and conjecture this bound is true over for all graphs and prove a related bound for all zero nonzero patterns of not necessarily symmetric matrices most of the results are valid for matrices over any infinite field but need not be true for matrices over finite fields

global stabilisation of high order nonlinear systems with multiple time delays this paper focuses on the stabilisation for a class of high order nonlinear systems with multiple time delays growth restriction on system nonlinearities is further relaxed design procedures of a continuous controller are provided by the method of adding a power integrator and the stability of the resulting closed loop system is rigorously proven with the help of the elegant choice of a lyapunov krasovskii functional finally a simulation example is provided to demonstrate the validness of the proposed approach

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

adaptive global stabilization of nonholonomic systems with strong nonlinear drifts a new input to state scaling scheme is first introduced to transform a class of nonholonomic systems in a chained form with strong nonlinear drifts and unknown constant parameters into a strict feedback form the backstepping technique is then applied to design a global adaptive stabilization controller a switching strategy based on the control input magnitude rather than the time is derived to get around the phenomenon of uncontrollability simulation examples validate the effectiveness of the proposed controller

output feedback adaptive stabilization control design for non holonomic systems with strong non linear drifts this paper investigates the problem of output feedback adaptive stabilization control design for non holonomic chained systems with strong non linear drifts including modelled non linear dynamics unmodelled dynamics and those modelled but with unknown parameters an observer and an estimator are introduced for state and parameter estimates respectively by using the integrator backstepping approach and based on the observer and parameter estimator a constructive design procedure for output feedback adaptive stabilization control is given it is shown that under some conditions the control design ensures the closed loop system is globally asymptotically stable when there is no non linear drift in the first subsystem and semiglobally asymptotically stable otherwise an example is given to show the effectiveness of the theory

adaptive dynamic surface control for mimo nonlinear time varying systems with prescribed tracking performance in this paper an adaptive dynamic surface control scheme is proposed for a class of multi input multi output mimo nonlinear time varying systems by fusing a bound estimation approach a smooth function and a time varying matrix factorisation the obstacle caused by unknown time varying parameters is circumvented the proposed scheme is free of the problem of explosion of complexity and needs only one updated parameter at each design step moreover all tracking errors can converge to predefined arbitrarily small residual sets with a prescribed convergence rate and maximum overshoot such features result in a simple adaptive controller which can be easily implemented in applications with less computational burden and satisfactory tracking performance simulation results are presented to illustrate the effectiveness of the proposed scheme

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

minimum rank problems abstract a graph describes the zero nonzero pattern of a family of matrices with the type of graph undirected or directed simple or allowing loops determining what type of matrices symmetric or not necessarily symmetric diagonal entries free or constrained are described by the graph the minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family the determination of maximum nullity is equivalent this problem has been solved for simple trees p m nylen minimum rank matrices with prescribed graph linear algebra appl 248 1996 303 316 c r johnson a leal duarte the maximum multiplicity of an eigenvalue in a matrix whose graph is a tree linear and multilinear algebra 46 1999 139 144 trees allowing loops l m dealba t l hardy i r hentzel l hogben a wangsness minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns linear algebra appl 418 2006 389 415 and directed trees allowing loops f barioli s fallat d hershkowitz h t hall l hogben h van der holst b shader on the minimum rank of not necessarily symmetric matrices a preliminary study electron j linear algebra 18 2000 126 145 we survey these results from a unified perspective and solve the minimum rank problem for simple directed trees

adaptive control of nonlinearly parameterized systems the smooth feedback case studies global adaptive control of nonlinearly parameterized systems with uncontrollable linearization using a parameter separation technique and the tool of adding a power integrator we develop a feedback domination design approach for the explicit construction of a smooth adaptive controller that solves the problem of global state regulation in contrast to the existing results in the literature a key feature of our adaptive regulator is its minimum order property namely no matter how big the number of unknown parameters is the order of the dynamic compensator is identical to one and is therefore minimal as a consequence global state regulation of feedback linearizable systems with nonlinear parameterization is achieved by one dimensional adaptive controllers without imposing any extra e g convex concave conditions on the unknown parameters

discontinuous control of nonholonomic systems the problem of asymptotic convergence for a class of nonholonomic control systems via discontinuous control is addressed and solved from a new point of view it is shown that control laws which ensures asymptotic exponential convergence of the closed loop system can be easily designed if the system is described in proper coordinates in such coordinates the system is discontinuous hence the problem of local asymptotic stabilization for a class of discontinuous nonholonomic control systems is dealt with and a general procedure to transform a continuous system into a discontinuous one is presented moreover a general strategy to design discontinuous control laws yielding asymptotic convergence for a class of nonholonomic control systems is proposed the enclosed simulation results show the effectiveness of the method

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

adaptive control of a class of nonlinearly parameterized plants a stable adaptive control method is developed for a class of first order nonlinearly parameterized plants the method is based on modified adaptive algorithms that yield a stable system in which the output errors are guaranteed to converge to zero the stability analysis is carried out using suitably constructed lyapunov functions

forbidden minors for the class of graphs g with ξ g 2 abstract for a given simple graph g s g is defined to be the set of real symmetric matrices a whose i j th entry is nonzero whenever i j and ij is an edge in g in f barioli s fallat l hogben a variant on the graph parameters of colin de verdiere implications to the minimum rank of graphs electron j linear algebra 13 2005 387 404 g is defined to be the maximum corank i e nullity among a s g having the strong arnold property is used to study the minimum rank maximum eigenvalue multiplicity problem for g since is minor monotone the graphs g such that g k can be described by a finite set of forbidden minors we determine the forbidden minors for g 2 and present an application of this characterization to computation of minimum rank among matrices in s g

a continuous feedback approach to global strong stabilization of nonlinear systems we present a continuous feedback approach to the problem of global strong stabilization for genuinely nonlinear systems that may not be stabilized even locally by any smooth feedback we describe conditions under which it is possible to prove while no smooth controllers exist the existence of continuous state feedback control laws that achieve global strong stability gss in the sense of kurzweil 1956 the proof is constructive and carried out by developing a machinery which combines the theory of homogeneous systems with the idea of adding a power integrator for the explicit construction of globally stabilizing continuous controllers we then illustrate by means of examples how this machinery can be used to overcome the topological obstruction caused by smooth feedback and hence resulting in new solutions to a variety of open control problems including global stabilization of an underactuated unstable two degree of freedom mechanical system

adaptive neural control for a class of nonlinear systems with uncertain hysteresis inputs and time varying state delays in this paper adaptive variable structure neural control is investigated for a class of nonlinear systems under the effects of time varying state delays and uncertain hysteresis inputs the unknown time varying delay uncertainties are compensated for using appropriate lyapunov krasovskii functionals in the design and the effect of the uncertain hysteresis with the prandtl ishlinskii pi model representation is also mitigated using the proposed control by utilizing the integral type lyapunov function the closed loop control system is proved to be semi globally uniformly ultimately bounded sguub extensive simulation results demonstrate the effectiveness of the proposed approach

halo orbits around the collinear points of the restricted three body problem abstract we perform an analytical study of the bifurcation of the halo orbits around the collinear points l 1 l 2 l 3 for the circular spatial restricted three body problem following a standard procedure we reduce to the center manifold constructing a normal form adapted to the synchronous resonance introducing a detuning which measures the displacement from the resonance and expanding the energy in series of the detuning we are able to evaluate the energy level at which the bifurcation takes place for arbitrary values of the mass ratio in most cases the analytical results thus obtained are in very good agreement with the numerical expectations providing the bifurcation threshold with good accuracy care must be taken when dealing with l 3 for small values of the mass ratio between the primaries in that case the model of the system is a singular perturbation problem and the normal form method is not particularly suited to evaluate the bifurcation threshold

switched adaptive control of switched nonlinearly parameterized systems with unstable subsystems this paper investigates the problem of adaptive stabilization for a class of switched nonlinearly parameterized systems where the solvability of the adaptive stabilization problem for subsystems is unnecessary a new switched adaptive control technique for the problem studied is set up by exploiting the generalized multiple lyapunov functions method and the parameter separation technique different update laws are designed to reduce the conservativeness caused by adoption of a common update law for all subsystems also the proposed technique permits removal of a common restriction in which the parameterization in the switched systems is restricted to a linear parameterization as an application of the proposed design technique stabilization for a class of non triangular switched systems with nonlinear parameterization is achieved by design of adaptive controllers since no subsystem of such a switched system is assumed to be adaptively stabilizable the problem under study cannot be handled by the existing methods a two inverted pendulums as a practical example is also provided to demonstrate the effectiveness of the proposed design method

adaptive stabilization of stochastic nonholonomic systems with nonlinear parameterization this paper investigates the problem of adaptive stabilization by state feedback for a class of stochastic nonholonomic systems in chained form with nonlinear parameterization a parameter separation technique is introduced to transform the nonlinear parameterized nonholonomic system into a linear like parameterized one then using input state scaling and backstepping techniques an adaptive state feedback controller is obtained based on switching strategy to eliminate the phenomenon of uncontrollability global asymptotic regulation in probability of the closed loop system states are guaranteed finally a simulation example is provided to illustrate the effectiveness of the proposed method

resonance tongues in hill s equations a geometric approach abstract the geometry of resonance tongues is considered in mainly reversible versions of hill s equation close to the classical mathieu case hill s map assigns to each value of the multiparameter the corresponding poincare matrix by an averaging method the geometry of hill s map locally can be understood in terms of cuspoid whitney singularities this adds robustness to the result the algorithmic nature of the averaging method enables a pull back to the resonance tongues of the original system