regional pole placement of multivariable systems under control structure constraints many controller realizations are structurally constrained some typical examples are static output feedback constant gain feedback for multiple operating points of a system two controller feedback and decentralized feedback a general class of problems of regional pole placement of multivariable systems with such control structure constraints is considered and a unified numerical method is given to solve them first a problem in this class is converted to a problem of solving a system of equalities and inequalities this system is then solved by using a modified homotopy method

integral invariants and limit sets of planar vector fields we show that if a planar system of differential equations admits an inverse integrating factor v defined in a neighborhood of a singular point with exactly one zero eigenvalue then v vanishes along any separatrix of the singular point additionally we prove that if k is a compact or limit set that contains a regular point or has an elliptic or parabolic sector if not and if v is defined on a neighborhood of k then v vanishes at at least one point of k and on all of k if v is real analytic or morse

a procedure for simultaneously stabilizing a collection of single input linear systems using non linear state feedback control abstract this paper considers the problem of simultaneously stabilizing a finite collection of single input linear systems this stabilization is achieved using a single non linear state feedback controller the stability of the resulting closed loop system is established using a collection of quadratic lyapunov functions the main result of this paper is a sufficient condition for simultaneous stabilizability furthermore when this sufficient condition is satisfied the paper gives a formula for constructing the stabilizing feedback control law the paper includes an example in which the stabilization procedure is applied to the stabilization of an f4e fighter aircraft

gain optimization under control structure and stability region constraints in this paper we consider the problem of optimizing the frobenious norm of linear static feedback gains of a system with control structure and stability region constraints we consider four important control structure constraints these are i static output feedback ii constant gain feedback for multiple operating conditions iii two controller feedback and iv decentralized feedback we obtain a nonlinear programming formulation for the optimization problem and present an effective numerical approach for solving it we include four examples

general relaxation methods for initial value problems with application to multistep schemes recently an approach known as relaxation has been developed for preserving the correct evolution of a functional in the numerical solution of initial value problems using runge kutta methods we generalize this approach to multistep methods including all general linear methods of order two or higher and many other classes of schemes we prove the existence of a valid relaxation parameter and high order accuracy of the resulting method in the context of general equations including but not limited to conservative or dissipative systems the theory is illustrated with several numerical examples

the inverse integrating factor and the poincare map this work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit we show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit when the regular orbit is a limit cycle we can determine its associated poincare return map in terms of the inverse integrating factor in particular we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it we also apply this result to study the homoclinic loop bifurcation we only consider homoclinic loops whose critical point is a hyperbolic saddle and whose poincare return map is not the identity a local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable that is the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method

linear conservation laws for odes abstract some physical systems described by odes have quantities that are conserved as the system evolves runge kutta formulas and linear multistep methods preserve linear conservation laws automatically a code for the integration of odes involves a number of algorithms in addition to the basic formula it is shown here that popular codes preserve linear conservation laws if they are used properly an application is made to differential algebraic equations of index 2 arising in the solution of the navier stokes equation by the method of lines

positive numerical integration methods for chemical kinetic systems abstract chemical kinetics conserves mass and renders nonnegative solutions a good numerical simulation would ideally produce mass balanced positive concentration vectors many time stepping methods are mass conservative however unconditional positivity restricts the order of a traditional method to one the projection method presented in this paper ensures mass conservation and positivity first a numerical approximation is computed with one step of a mass preserving traditional scheme if there are negative components the nearest vector in the reaction simplex is found by solving a quadratic optimization problem this vector is shown to better approximate the true solution a simpler version involves just one projection step and stabilizes the reaction simplex this technique works best when the underlying time stepping scheme favors positivity projected methods are more accurate than clipping and allow larger time steps for kinetic systems which are unstable outside the positive quadrant

quadratic representations for groups of lie type over fields of characteristic two abstract suppose k is a field of characteristic two g is a group of lie type over k and v is an irreducible kg module by the steinberg tensor product theorem v i ivi where each vi is an algebraic conjugate of a restricted kg module if g contains a quadratically acting fours group then i 2 if i 2 or if i 1 and some restrictions are imposed on the fours group then a list of the possible restricted modules is able to be determined in all cases the restricted modules are fundamental modules and in many cases the majority of these are ruled out

some new results on darboux integrable differential systems abstract we deal with complex planar differential systems having a darboux first integral h we present a definition of remarkable values and remarkable curves associated to h and characterize the existence of a polynomial inverse integrating factor for these systems furthermore we study the relation between the characteristic polynomial f and the inverse integrating factors of the system and show the importance of the numerator of the exponential factor of h in the construction of f

constraint preserving integrators for general nonlinear higher index daes in the last few years there has been considerable research on numerical methods for differential algebraic equations daes f x x t 0 where f x is identically singular the index provides one measure of the singularity of a dae most of the numerical analysis literature on daes to date has dealt with daes with indices no larger than three even in this case the systems were often assumed to have a special structure recently a numerical method was proposed that could in principle be used to integrate general unstructured higher index solvable daes however that method did not preserve constraints this paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index daes

stabilization of daes and invariant manifolds many methods have been proposed for the stabilization of higher index differential algebraic equations daes such methods often involve constraint differentiation and problem stabilization thus obtaining a stabilized index reduction a popular method is baumgarte stabilization but the choice of parameters to make it robust is unclear in practice here we explain why the baumgarte method may run into trouble we then show how to improve it we further develop a unifying theory for stabilization methods which includes many of the various techniques proposed in the literature our approach is to i consider stabilization of odes with invariants ii discretize the stabilizing term in a simple way generally different from the ode discretization and iii use orthogonal projections whenever possible the best methods thus obtained are related to methods of coordinate projection we discuss them and make concrete algorithmic suggestions

brief paper a continuation approach to eigenvalue assignment this paper describes a continuation approach to eigenvalue assignment the method is a homotopy technique which embeds the control problem into a parameterized family of control problems this parameterization describes a continuous deformation of a system with the desired spectrum into the original system based on this deformation a differential equation is constructed whose solution trajectory has an end point which is a constant output feedback matrix assigning the desired spectrum to the original system the derivation of this differential equation and conditions which guarantee the existence of a solution are given also two examples of its numerical implementation are detailed

some remarks on inverse jacobi multipliers around hopf singularities abstract we study several properties of inverse jacobi multipliers v around hopf singularities of analytic vector fields x in r n which are relevant to the study of the local bifurcation of periodic orbits when n 3 and the singularity is a saddle focus we show that i any two locally smooth and non flat linearly independent inverse jacobi multipliers have the same taylor expansion ii any smooth and non flat v has associated exactly one smooth center manifold w c of x such that w c v 1 0 we also study whether the properties of the vanishing set v 1 0 proved in the 3 dimensional case remain valid when n 4

recent progress in extrapolation methods for ordinary differential equations the paper is intended to give a survey on the state of the art of extrapolation methods for initial value problems in ordinary differential equation systems all basic discretizations which are su

state space design of low order stabilizers an algorithm is presented for stabilizing a linear multivariable system with a controller of fixed dynamic order this is an output feedback stabilization problem the algorithm attempts to solve this by a sequence of approximate pole assignment problems the approximation is driven by the optimization of an objective function consisting of a weighted sum of the condition number of the closed loop eigenvectors and the norm of the difference between the computed and actual equivalent state feedback controls

a class of non integrable systems admitting an inverse integrating factor abstract we study the existence of an inverse integrating factor for a class of systems in general non integrable whose lowest degree quasi homogeneous term is a hamiltonian system and its hamiltonian function only has simple factors over c x y

on the continuous realization of iterative processes many important mathematical problems are solved by iterative methods many of these iterative schemes may be regarded as the discrete realization of certain continuous dynamical systems this paper summarizes some of the recent developments in the continuous realization of several popular basic iterative methods

a homotopy method for eigenvalue assignment using decentralized state feedback this paper addresses the following problem given an interconnected system m composed of n subsystems of the form a i b i k i i 1 n a i b i a controllable pair and where the off diagonal blocks of m lie in the image of the appropriate b i then is it possible to arbitrarily assign the characteristic polynomial of m by a suitable selection of the characteristic polynomials of a i b i k i moreover is it possible to compute the appropriate characteristic polynomials of the a i b i k i or equivalently construct the k i needed to do so the first question is answered by constructing a mapping f r n rightarrow r n which maps a prescribed set of n of the feedback gains elements of k i i 1 n to the n coefficients of the characteristic polynomial of m the question then becomes given a p in r n does f x p have a solution the answer is found by constructing a homotopy h r n x o 1 rightarrow r n where h x 1 f x and h x 0 is some trivial function degree theory is then applied to guarantee that there exists an x t such that h x t t p for all t in 0 1 the parameterized sard s theorem is then utilized to prove that with probability 1 x t is a smooth curve and hence can be followed numerically from x 0 to x 1 by the solution of a differential equation davidenko s method

divergence of c1 vector fields and nontrivial minimal sets on 2 manifolds abstract we prove a bendixson dulac type criterion for the nonexistence of nontrivial compact minimal sets of c 1 vector fields on orientable 2 manifolds as a corollary we get that the divergence with respect to any volume 2 form of such a vector field must vanish at some point of any nontrivial compact minimal set we also prove that all the nontrivial compact minimal sets of a c 1 vector field on an orientable 2 manifold are contained in the vanishing set of any inverse integrating factor from this we get that if a c 1 vector field on an orientable 2 manifold has a nontrivial compact minimal set then an infinitesimal symmetry is inessential on the minimal set

a second order pseudo transient method for steady state problems this article gives a second pseudo transient method for a special system of nonlinear equations which arises from chemical reaction rate equations this method uses a special second order rosenbrock method as the discrete difference scheme which satisfies a linear conservation law moreover it adaptively adjusts the time step in inverse proportion to an arithmetic mean of the current residual and the previous residual at every iteration step for a singular system of nonlinear equations under some standard assumptions local convergence of the new method is addressed finally some promise numerical results are also reported

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

a homotopy approach for stabilizing single input systems with control structure constraints a linear state feedback gain vector used in the control of a single input dynamical system may be constrained because of the way feedback is realized some examples of feedback realizations which impose constraints on the gain vector are static output feedback constant gain feedback for several operating points of a system and two controller feedback we consider a general class of problems of stabilization of single input dynamical systems with such structural constraints and give a numerical method to solve them each of these problems is cast into a problem of solving a system of equalities and inequalities in this formulation the coefficients of the quadratic and linear factors of the closed loop characteristic polynomial are the variables to solve the system of equalities and inequalities a continuous realization of the gradient projection method and a barrier method are used under the homotopy framework our method is illustrated with an example for each class of control structure constraint

on the integrability of two dimensional flows this paper deals with the notion of integrability of flows or vector fields on two dimensional manifolds we consider the following two key points about first integrals 1 they must be defined on the whole domain of definition of the flow or vector field or defined on the complement of some special orbits of the system 2 how are they computed we prove that every local flow on a two dimensional manifold m always has a continuous first integral on each component of m where is the set of all separatrices of we consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two dimensional flow or vector field finally we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus of a non resonant hyperbolic node and of a siegel hyperbolic saddle

brief paper reliable stabilization using a multi controller configuration given a plant p we consider the problem of designing a pair of controllers c 1 and c 2 such that their sum stabilizes p and in addition each of them also stabilizes p should the other one fail this is referred to as the reliable stabilization problem it is shown that every strongly stabilizable plant can be reliably stabilized moreover one of the two controllers can be specified arbitrarily subject only to the constraint that it should be stable the stabilization technique is extended to reliable regulation

convergence results for a coordinate projection method applied to mechanical systems with algebraic constraints the equations of motion of mechanical multibody systems with algebraic constraints are of index 3 and therefore not directly solvable by standard ode or dae methods reducing the index by different

existence of inverse integrating factors and lie symmetries for degenerate planar centers abstract we prove that any possibly degenerate center of an analytic planar system admits a c inverse integrating factor and a c lie symmetry in a neighborhood of the singularity we also show that there exists an analytic degenerate center for which any c inverse integrating factor is flat at the singular point and that a sufficient condition for the existence of an analytic inverse integrating factor is the existence of an analytic first integral

non negative solutions of odes this paper discusses procedures for enforcing non negativity in a range of codes for solving ordinary differential equations odes this codes implement both one step and multistep methods all of which use continuous extensions and have event finding capabilities examples are given

continuation methods theory and applications this paper surveys in a tutorial fashion theoretical and applications aspects of the confirmation method for the solution of large scale system engineering problems the continuation method is motivated and defined the existence of the method is formulated in terms of degree theory a la algebraic topology examples are given throughout the paper and applications to engineering and economic problems are cited and described

design of software for odes we discuss the design of software that is easy to use for simple problems but still capable of solving complicated problems nowadays a design should accommodate related but fundamentally different tasks such as solving daes and ddes in addition to ivps and bvps for odes major issues are illustrated with the matlab ode suite and selected solvers written in fortran 90 and maple

non nested configuration of algebraic limit cycles in quadratic systems this work deals with algebraic limit cycles of planar polynomial differential systems of degree two more concretely we show among other facts that a quadratic vector field cannot possess two non nested algebraic limit cycles contained in different irreducible invariant algebraic curves

on the remarkable values of the rational first integrals of polynomial vector fields abstract the remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by poincare he was mainly interested in their algebraic aspects here we are interested in their dynamic aspects i e how they contribute to the phase portrait of the system to its separatrices to its singular points etc the relationship between remarkable values and dynamics mainly takes place through the inverse integrating factor

irreducible modules and parabolic subgroups summary it seems desirable from the point of view of finite group theorists to develop further the representation theory of finite chevalley groups in the characteristic of their definition say p by focusing attention on the full array of p local subgroups that is the parabolics rather than the usual single borel or cartan subgroup the following observation may be regarded as a generalization to arbitrary parabolic subgroups of the standard result 3 theorem 39 d for a borel subgroup namely that in an irreducible module the subspace fixed by a maximal unipotent subgroup is 1 dimensional and so affords an irreducible module for a levi complement which is an abelian cartan subgroup it seems natural to state the result first for algebraic groups

decentralized eigenvalue assignment in two terminal hvdc systems abstract this paper is concerned with eigenvalue assignment in a two machine infinite bus two terminal hvdc system a continuations algorithm is used to design decentralized state feedback controllers which assign the eigenvalues of the linearized power system to specified locations the decentralized feedbacks take the form of current modulation at the rectifier and voltage modulation at the inverter

on the preservation of invariants by explicit runge kutta methods a new strategy to preserve invariants in the numerical integration of initial value problems with explicit runge kutta methods is presented it is proved that this technique retains the order of the original method has an easy and cheap implementation and can be used in adaptive runge kutta codes some numerical experiments with the classical code of dormand and prince dopri5 4 based on a pair of embedded methods with orders 5 and 4 are presented to show the behavior of the new method for several problems which possess invariants