resolutions of fat point ideals involving eight general points of p2 abstract the main result theorem 1 provides an algorithm for determining the minimal free resolution of fat point subschemes of p 2 involving up to eight general points of arbitrary multiplicities the resolutions obtained hold for any algebraically closed field independent of the characteristic the algorithm works by giving a formula in nice cases and a reduction to the nice cases otherwise the algorithm which does not involve grobner bases is very fast partial information is also obtained in certain cases with n 8

local and global controllability for nonlinear systems a technique to study local and global controllability properties for a wide class of nonlinear systems is presented in addition to an extensive study of systems on r2 we propose as an application of our method a new criterion for global controllability of systems with polynomial drift term defined on rn in the case of linear systems the criterion reduces to the classical kalman condition a study of local controllability at the equilibrium point of the drift term of systems defined on r2 enables us not only to rederive a local controllability result derived by hermes 4 5 but also to extend this result when no assumptions are made on the boundedness of the controls

optimal control of neutral functional differential inclusions this paper deals with optimal control problems for dynamical systems governed by constrained functional differential inclusions of neutral type such control systems contain time delays not only in state variables but also in velocity variables which make them essentially more complicated than delay differential or differential difference inclusions our main goal is to derive necessary optimality conditions for general optimal control problems governed by neutral functional differential inclusions with endpoint constraints while some results are available for smooth control systems governed by neutral functional differential equations we are not familiar with any results for neutral functional differential inclusions even with smooth cost functionals in the absence of endpoint constraints developing the method of discrete approximations which is certainly of independent interest and employing advanced tools of generalized differentiation we conduct a variational analysis of neutral functional d

comments with reply on the controllability of planar bilinear systems by d e koditschek and k s narendra in the above titled study ibid vol ac 30 p 87 89 jan 1985 two classes of bilinear systems in the plane are examined and it is claimed that necessary and sufficient conditions for complete controllability of both classes are derived the commenter shows that the conditions for one of these classes are only sufficient a counterexample is given the authors briefly review their work and concede that the relevant theorem is incorrect

necessary and sufficient condition for global controllability of planar affine nonlinear systems in this correspondence we will investigate the global controllability of general planar affine nonlinear systems and present a simple necessary and sufficient condition for global controllability the analysis is carried out in detail which is based on some basic facts in planar topology such as the jordan curve theorem and in the geometrical theory of ordinary differential equations including the poincare bendixson theorem furthermore we will also discuss the global controllability for higher dimensional affine nonlinear system with a triangular like structure

necessary conditions controllability and the value function for differential difference inclusions on deduit des conditions necessaires pour la commande optimale d un systeme contraint par des inclusions differentielles aux differences

state constrained optimal control problems with time delays abstract in a recent related paper necessary conditions in the form of a maximum principle were derived for optimal control problems with time delays in both state and control variables different versions of the necessary conditions covered fixed end time problems and under additional hypotheses free end time problems these conditions improved on previous conditions in the following respects they provided the first fully non smooth pontryagin maximum principle for problems involving delays in both state and control variables only special cases of which were previously available they provide a strong version of the weierstrass condition for general problems with possibly non commensurate control delays whereas the earlier literature does so only under structural assumptions about the dynamic constraint they also provided a new two sided generalized transversality condition associated with the optimal end time this paper provides an extension of the pontryagin maximum principle of the earlier paper for time delay systems to allow for the presence of a unilateral state constraint the new results fully recover the necessary conditions of the earlier paper when the state constraint is absent and therefore retain all their advantages but in a setting of greater generality

necessary optimality conditions for differential difference inclusions with state constraints abstract we consider the mayer optimal control problem with dynamics given by a nonconvex differential difference inclusion whose trajectories are constrained to a closed set necessary optimality conditions in the form of the maximum principle are obtained

on the global controllability of nonlinear systems fountains recurrence and applications to hamiltonian systems in this paper a form of open local accessibility for nonlinear control systems is introduced called the continuous fountain condition subject to the conditions that i the states of a system are continuous fountains and ii one of various recurrence conditions holds it is established that the system state space is globally controllable it is shown that these controllability results imply the controllability of certain subsets of the state space of hamiltonian control systems called energy slices the relations between the notion of a fountain and i local accessibility and ii the full lie algebra rank condition for control affine systems are investigated finally the results in this paper are shown to have application to hierarchical hybrid control theory in that they give conditions for a finite analytic partition to satisfy the so called in block controllability property this is illustrated with a linear mass spring system example

second order necessary conditions for differential difference inclusion problems abstract we study an optimal control problem given by differential difference inclusions with end point constraints an approach concerning second order optimality conditions is proposed

a sufficient condition for local controllability of nonlinear systems along closed orbits a computable sufficient condition for determining local controllability of a differential nonlinear control system along a reference closed orbit is presented comparisons to other results on local controllability along a reference trajectory are made

on nagata s conjecture abstract this paper gives an improved lower bound on the degrees d such that for general points p 1 p n p 2 and m 0 there is a plane curve of degree d vanishing at each p i with multiplicity at least m

some results on rational surfaces and fano varieties this is the publisher s version also available electronically from http www degruyter com view j crll 2001 2001 issue 538 crll 2001 068 crll 2001 068 xml

a connection between the maximum principle and dynamic programming for constrained control problems we consider the mayer optimal control problem with dynamics given by a nonconvex differential inclusion whose trajectories are constrained to a closed set and obtain necessary optimality conditions in the form of the maximum principle together with a relation between the costate and the value function this additional relation is applied in turn to show that the maximum principle is nondegenerate we also provide a sufficient condition for the normality of the maximum break principle to derive these results we use convex linearizations of differential inclusions and convex linearizations of constraints along optimal trajectories then duality theory of convex analysis is applied to derive necessary conditions for optimality in this way we extend the known relations between the maximum principle and dynamic programming from the unconstrained problems to the constrained case

a maximum principle for generalized control systems on definit un systeme de commande generalise comme une collection de fonctions f t x definies pour t x 0 1 r n dans r n et satisfaisant les hypotheses de caratheodory dans t x on etablit un theoreme donnant une condition necessaire pour la trajectoire d un tel systeme d appartenance a la frontiere de l ensemble atteignable on applique ce theoreme a des cas particuliers

global external linearization of nonlinear systems via feedback this note presents necessary conditions and sufficient conditions for an affine nonlinear system to be globally feedback equivalent to a controllable linear system over an open subset v of rn when v equals rn necessary and sufficient conditions are obtained

configuration types and cubic surfaces this paper is a sequel to the paper e guardo b harbourne resolutions of ideals of six fat points in p2 j algebra 318 2 2007 619 640 we relate the matroid notion of a combinatorial geometry to a generalization which we call a configuration type configuration types arise when one classifies the hilbert functions and graded betti numbers for fat point subschemes supported at n 8 essentially distinct points of the projective plane each type gives rise to a surface x obtained by blowing up the points we classify those types such that n 6 and kx is nef the surfaces obtained are precisely the desingularizations of the normal cubic surfaces by classifying configuration types we recover in all characteristics the classification of normal cubic surfaces which is well known in characteristic 0 j w bruce c t c wall on the classification of cubic surfaces j london math soc 2 19 2 1979 245 256 as an application of our classification of configuration types we obtain a numerical procedure for determining the hilbert function and graded betti numbers for the ideal of any fat point subscheme z m1p1 m6p6 such that the points pi are essentially distinct and kx is nef given only the configuration type of the points p1 p6 and the coefficients mi

contingent cones to reachable sets of control systems high order necessary conditions for optimality for an optimal control problem are studied via properties of contingent cones to reachable sets along the optimal trajectory it is shown that the adjoint vector of pontryagin s maximum principle is normal to the set of variations of reachable sets results are applied to study optimal control problems for dynamical systems described by 1 closed loop control systems 2 nonlinear implicit systems 3 differential inclusions 4 control systems with jumps

the controllability of planar bilinear systems recent results concerning the controllability of planar bilinear systems are summarized the results are then used to characterize the controllability behavior of the homogeneous and general bilinear system in r 2

controllability in codimension one abstract local and global controlability of analytic affine control systems with an arbitrary number of controls are studied assuming strong accessibility of and codimension one of the lie algebra t generated by the input vector fields i e dim t p n 1 at every p m controls are assumed to have no a priori bound from the study of the set h where a necessary condition for local controllability at a point is verified and assuming some transversality relations an easy to verify geometric condition is proved h is a submanifold and is locally controllable at every point of h outside a codimension one submanifold a geometric sufficient condition for global controllability on simply connected manifolds is then obtained if every leaf of t intersects the manifold h and some transversality relations including those involved in the local condition are verified is globally controllable

the hierarchical lattices of a finite machine the concept of dynamical consistency is introduced to define a high level transition function for an arbitrary partition of the state space of a finite input state machine m the existence of a dynamically consistent transition from a partition member xi to a member xj corresponds to the existence for every state x in xi of some x dependent control sequence which takes that state directly into xj this paper analyses what are termed in block controllable partition machines and between block controllable partition machines and establishes the existence of the associated in block controllable hierarchical lattice ibcp m it is shown that any partition machine in ibcp m is between block controllable if and only if the base machine m is controllable this setting allows the specification of state to state control trajectories of m to be decomposed into high level and low level components

green s conjecture for curves on rational surfaces with an anticanonical pencil green s conjecture is proved for smooth curves c lying on a rational surface s with an anticanonical pencil under some mild hypotheses on the line bundle l mathcal o s c constancy of clifford dimension clifford index and gonality of curves in the linear system vert l vert is also obtained

birational morphisms of rational surfaces abstract a fairly complete characterization on anticanonical rational surfaces of the divisor classes whose sections define birational morphisms is given most results are proven over an algebraically closed field of arbitrary characteristic

further results on global controllability of planar nonlinear systems the global controllability of a class of planar affine nonlinear systems with perturbation is investigated it has been proven that the systems can still hold their global controllability under the perturbation which can be dominated by a function of state and vanishes when control input is zero

the maximum principle for an optimal solution to a differential inclusion with end points constraints we derive pontryagin s maximum principle for a general optimal control problem using the set valued version of variational equation we achieve this aim by exploiting an adequate differential calculus of set valued maps furthermore the calmness condition is replaced by a surjectivity condition involving reachable sets of the set valued linearization of the initial control problem duality then provides both the adjoint differential inclusion and the maximum principle

local controllability of nonlinear systems abstract the local controllability of control systems with an arbitrary number of controls is considered first on an open set and then at a given point necessary conditions are derived concerning the lie algebra t generated by the input vector fields and applied to gradient systems the main results are a geometric sufficient condition for local controllability at a point and an equivalent condition based on the computation of lie brackets assuming t to be n 1 dimensional

controllability of spacecraft systems in a central gravitational field the configuration space for rigid spacecraft systems in a central gravitational field can be modeled by so 3 spl times ir sup 3 where the special orthogonal group so 3 represents the attitude dynamics and ir sup 3 is for the orbital motion the attitude dynamics of the spacecraft system is affected by the orbital elements through the well known gravity gradient torque on the other hand the effects of attitude orbit coupling can also possibly be used to alter orbital motions by controlling the attitude this controllability property is the primary issue of this paper the control systems for spacecraft with either reaction wheels or gas jets being used as attitude controllers are proven in this study to be controllable rigorously establishing these results necessitates starting with the formal definitions of controllability and poisson stability it is then shown that if the drift vector field of the system is weakly positively poisson stable and the lie algebra rank condition is satisfied controllability can be concluded the hamiltonian structure of the spacecraft model provides a natural route of verifying the property of weakly positive poisson stability accordingly the controllability is obtained after confirming the lie algebra rank condition developing a methodology in deriving lie brackets in the tangent space of t so 3 spl times ir sup 3 i e the second tangent bundle is thus deemed necessary a general formula is offered for the computation of lie brackets of second tangent vector fields in tt so 3 sup m spl times ir sup n in light of its importance in the fields of mechanics robotics optimal control and nonlinear control among others with these tools the controllability results can be proved the analysis in this paper gives some insight into the attitude orbit coupling effects and may potentially lead towards new techniques in designing controllers for large spacecraft systems

on the controllability of piecewise linear hypersurface systems this paper investigates the global controllability of piecewise linear hypersurface systems which are defined as control systems that are subject to affine dynamics on each of the components of a finite polyhedral partition various new tools are developed for the study of the problem including a classification of the facets of the polyhedra in the partition necessary and sufficient conditions for complete controllability are obtained via the study of a suitably defined controllability connection matrix of polyhedra

n dimensional controllability with n 1 controls let m be a connected real analytic n dimensional manifold f g 1 g n 1 be complete real analytic vector fields on m which are linearly independent at some point of m and u 1 u n 1 be real valued controls consider the controllability of the system dot x t f x t sum min i 1 max n 1 u i t g i x t x 0 x 0 in m necessary and sufficient conditions are given so that this system is controllable on any simply connected domain d contained in m on which g 1 g n 1 are linearly independent these conditions depend on the computation of lie brackets at those points where f g 1 g n 1 are linearly dependent

reachability of a set of facets for linear affine systems with n 1 inputs this note provides new necessary and sufficient conditions for an n dimensional linear affine system with n 1 inputs to reach an exit facet or set of exit facets of a simplex the conditions reduce the original np hard necessary and sufficient conditions to a set of at most n lp problems

local controllability along a reference trajectory abstract local controllability along a reference trajectory of analytic and smooth affine single input control systems without any a priori bounds on the control is considered and a geometric proof of a version of the hermes conjecture is given

global transformations of nonlinear systems recent results have established necessary and sufficient conditions for a nonlinear system of the form dot x t f x t u t g x t with f 0 0 to be locally equivalent in a neighborhood of the origin in rnto a controllable linear system we combine these results with several versions of the global inverse function theorem to prove sufficient conditions for the transformation of a nonlinear system to a linear system in doing so we introduce a technique for constructing a transformation under the assumptions that g ldot f dotg ad n 1 f ldotg span an n dimensional space and that g ldot f ldot g ad n 2 f ldotg is an involutive set

on global controllability of planar affine nonlinear systems with a singularity this paper is concerned with the global controllability of planar affine nonlinear systems with a singular point by using some recent developed methods a necessary and sufficient condition is obtained which can also be viewed as a generalization of the corresponding result in the book by nikitin s nikitin global controllability and stabilization of nonlinear systems world scientific publishing co pte ltd singapore 1994

hierarchical hybrid control systems a lattice theoretic formulation the notion of dynamical consistency is extended to hybrid systems so as to define the set of dynamically consistent hybrid partition machines associated with a continuous system s following the formulation of the notions of between block and in block controllable hybrid partition machines the lattice hibc s of hybrid in block controllable partition machines is defined and investigated based on these constructions the definition and properties of hierarchical hybrid control systems are presented together with an example

gd aided iol input output linearisation controller for handling affine form nonlinear system with loose condition on relative degree abstractinput output linearisation iol may encounter a singularity problem when applied to the tracking control of affine form nonlinear system afns which may not have a well defined relative degree the singularity problem has occurred in the area of control for decades in this paper we incorporate the gradient dynamics gd into iol which leads to the gd aided iol method to solve the singularity problem with the proposition of the loose condition on relative degree moreover detailed theoretical analyses on tracking error bound and convergence performance of the corresponding gd aided iol controller are presented simulations and comparisons substantiate that the proposed gd aided iol method is capable of completing the tracking control task and conquering the singularity encountered in the afns

controllability of convex processes the purpose of this paper is to provide several characterizations of controllability of differential inclusions whose right hand sides are convex processes convex processes are the set valued maps whose graphs are convex cones they are the set valued analogues of linear operators such differential inclusions include linear systems where the controls range over a convex cone and not only a vector space the characteristic properties are couched in terms of invariant cones by convex processes or eigenvalues of convex processes or a rank condition we also show that controllability is equivalent to observability of the adjoint inclusion

value function and necessary conditions in optimal control problems for differential difference inclusions this paper deals with the optimal control problems for differential difference inclusions subject to endpoint constraints we follow a twofold goal first we develop a method for estimating the generalized gradients of value function in the problems above second we use these estimates for obtaining necessary optimality conditions the results obtained are expressed in terms of clarke constructions for nonsmooth mappings and sets

linear controllability versus global controllability in this note the relationship between linear controllability and global controllability for a class of two dimensional nonlinear control systems x dot ax b x u has been discussed in detail it is shown that the linear controllability can imply the global controllability under some certain conditions furthermore a counterexample is constructed to answer an open problem in the paper x m hu and c martin linear reachability versus global stabilization ieee transactions on automatic control vol 44 no 6 pp 1303 1305 1999

on the hybrid optimal control problem theory and algorithms a class of hybrid optimal control problems hocp for systems with controlled and autonomous location transitions is formulated and a set of necessary conditions for hybrid system trajectory optimality is presented which together constitute generalizations of the standard maximum principle these are given for the cases of open bounded control value sets and compact control value sets the derivations in the paper employ i classical variational and needle variation techniques and ii a local controllability condition which is used to establish the adjoint and hamiltonian jump conditions in the autonomous switching case employing the hybrid minimum principle hmp necessary conditions a class of general hmp based algorithms for hybrid systems optimization are presented and analyzed for the autonomous switchings case and the controlled switchings case using results from the theory of penalty function methods and ekeland s variational principle the convergence of these algorithms is established under reasonable assumptions the efficacy of the proposed algorithms is illustrated via computational examples

controllability of a planar body with unilateral thrusters investigates the minimal number of unilateral thrusters required for different versions of nonlinear controllability of a planar rigid body for one to three unilateral thrusters one gets a new property with each additional thruster one thruster suffices for small time accessibility on the body s state space tse 2 two thrusters suffice for global controllability on tse 2 and three thrusters suffice for small time local controllability at zero velocity states

necessary optimality conditions for nonconvex differential inclusions with endpoint constraints in this paper we derive necessary optimality conditions for optimization problems defined by non convex differential inclusions with endpoint constraints we do this in terms of parametrizations of the convexified form of the differential inclusion and under additional assumptions in terms of the inclusion itself