affine and bilinear systems on lie groups abstract in this paper we study affine and bilinear systems on lie groups we show that there is an intrinsic connection between the solutions of both systems such relation allows us to obtain some preliminary controllability results of affine systems on compact and solvable lie groups we also show that the controllability property of bilinear systems is very restricted and may only be achieved if the state space g is an euclidean space

elliptic problems with nonlinearities indefinite in sign abstract let r n n 3 be a bounded open set with smooth boundary and r we study the dirichlet problem formula with 1 q p and h k l nonnegative functions we prove existence nonexistence and multiplicity results depending on and according to the integrability properties of the ratio k p 1 h q 1

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

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

existence and bifurcation behavior of positive solutions for a class of kirchhoff type problems abstract we study the existence of positive solutions for the nonlinear kirchhoff type equation a b u 2 d x u u u 3 with dirichlet boundary condition where r are two real parameters we provide a description of a two dimensional set in the plane which correspond to the existence of positive solutions for the above kirchhoff type equation

needle variations that cannot be summed this article analyzes sets of higher order tangent vectors to reachable sets of analytic control systems affine in the control both small time local output controllability and small time local controllability about a nonstationary reference trajectory are considered in a series of purposefully constructed examples it is shown that the cones generated by needle variations may fail to be convex the examples demonstrate that the usual technical condition that needle variations must be movable is essential to guarantee desirable convexity properties moreover new doubts are cast on the structural stability of controllability properties as apparently higher order perturbations can reverse the lack of controllability of lower order nilpotent approximating systems thereby providing new insights about the ultimate question of whether controllability is finitely determined

multiple positive solutions to a class of kirchhoff equation on r3 with indefinite nonlinearity abstract we study the existence and multiplicity of positive solutions to a class of kirchhoff equation 1 b r 3 u x 2 d x u x u x v x u x p 2 u x k x u x u x u h 1 r 3 when v x is sign changing in r 3 we prove that there is 0 such that the kirchhoff equation possesses at least one positive solution for 0 1 and at least two positive solutions for 1 1 here 1 is the first eigenvalue of u u k x u u h 1 r 3 a positive eigenfunction corresponding to 1 is denoted by e 1 an interesting phenomena is that the term b r 3 u 2 d x u can be used to compete with the indefinite nonlinearity and we do not assume the condition r 3 v x e 1 p d x 0 which has been shown to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with indefinite nonlinearity alama and tarantello 1993

2d systems feedback compensation an approach based on commutative linear transformations abstract algebraic properties of a pair of commutative matrices associated with an ideal in r z 1 z 2 are exploited for characterizing the closed loop polynomial variety of a 2d system also algorithms are given to find under what constraints the closed loop variety can be assigned and to compute the mfd of a compensator

controllability of linear systems on lie groups with finite semisimple center this paper studies controllability for a given linear system sigma on a connected lie group g by taking into consideration the eigenvalues of an associated derivation mathcal d if we assume that the lie group g has finite center and for some tau 0 the identity element of g is an interior point of its reachable set at time tau then the system is controllable if mathcal d has only eigenvalues with zero real part

existence of solutions for a higher order kirchhoff type problem with exponential critical growth abstract a higher order kirchhoff type equation r 2 m m u 2 0 m 1 a x u 2 d x m u 0 m 1 1 a x u f x u x h x in r 2 m is considered in this paper we assume that the nonlinearity of the equation has exponential critical growth and prove that for a positive which is small enough there are two distinct nontrivial solutions to the equation when 0 we also prove that the equation has a nontrivial mountain pass type solution

controllability along a trajectory a variational approach this paper unifies and improves most sufficient conditions of local controllability both along a trajectory and at a point this is accomplished by defining high order variations that can be continuously summed the peculiar property of these variations is that they may be generated by thin conditions as relations in the lie algebra associated with a control system this property leads to a variational interpretation of sussmann type sufficient conditions of local controllability neutralization of obstructions and it allows the use of different weights for neutralization

controllability properties of affine systems this paper deals with transitivity controllability of affine families of vector fields on a finite dimensional vector space v in particular we focus on affine families whose corresponding families of linear fields are transitive on v 0 and which in addition have no fixed points in v we show that such families are necessarily transitive on v and we also show that they remain transitive under small perturbations in general however affine families need not remain transitive under small perturbations for example small affine perturbations of transitive linear systems are not necessarily transitive since any affine system mathcal f naturally defines a system mathcal f r of right invariant vector fields on the semi direct product of v with gl v we also investigate transitivity properties mathcal f r our result is that if mathcal f is an affine family satisfying the preceding conditions then mathcal f r generates the full lie algebra on the semi direct product

lie brackets and local controllability a sufficient condition for scalar input systems we develop a general formalism based on exponential lie series which can be used to study control variations we apply the formalism to prove a result conjectured by h hermes that gives a sufficient condition for small time local controllability for single input systems

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

normal local controllability of order one this paper gives a sufficient condition of normal local controllability at a point x0 for a family of analytical vector fields the families considered are of the following where is a bounded subset of r the condition is of the first order that is it depends only on the values of the fields and their first derivatives at x0 it is proved that this condition is the most general condition of this type in fact if this condition is not satisfied there are families of vector fields which at the first order coincide with the given family and which are not normally locally controllable at x0

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

bound state solutions of kirchhoff type problems with critical exponent abstract in the present paper we consider the following kirchhoff type problems a b r 3 u 2 d x u v x u u 5 in r 3 u d 1 2 r 3 where a 0 b 0 and v l 3 2 r 3 is a given nonnegative function if v 3 2 is suitable small we prove that this problem has at least one bound state solution

multiplicity of positive solutions for some quasilinear elliptic equation in rnwith critical sobolev exponent consider the equation pu g x u p 2u f x u p 2u 1 in rn where 1 0 be the principal eigenvalue of pu g x u p 2uinrn rng x u p 0 2 withu 1 0 the associated eigenfunction we prove i equation 1 has at least one positive solution if 0 1 ii suppose rn f x u 1 p 1such that 1 has at least one positive solution for 1 0 moreover ifp 2 there exists 0 1such that 1 has at least two positive solutions for 1 0

existence and concentration of positive solutions for kirchhoff type problems with a steep well potential abstract the existence and concentration of positive solutions for a class of kirchhoff type problems with a nonnegative steep well potential in r 3 are obtained via variational methods

controllability of linear systems on solvable lie groups linear systems on lie groups are a natural generalization of linear systems on euclidian spaces for such systems this paper studies controllability by taking into consideration the eigenvalues of an associated derivation mathcal d when the state space is a solvable connected lie group controllability of the system is guaranteed if the reachable set of the neutral element is open and the derivation mathcal d has only pure imaginary eigenvalues for bounded systems on nilpotent lie groups such conditions are also necessary

singular perturbed kirchhoff type problem with critical exponent abstract in the present paper we consider the following kirchhoff type problem with critical exponent e 2 a e b r 3 u 2 d x u v x u f u in r 3 u h 1 r 3 by variational methods we show the existence of the positive solutions concentrating around global minima of the potential v x as e 0 we do not need the monotonicity of the function f 3

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

a note on output feedback stabilizability of multivariable 2d systems abstract the output feedback stabilizability conditions of 2d systems are expressed in term of structural properties of a pair of commuting linear transformations an algorithm is given for obtaining a stable closed loop 2d characteristic polynomial

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

dynamic regulation of 2d systems a state space approach abstract the possibilities of modifying the dynamical behavior of 2d state space models by output feedback compensation are investigated and a complete characterization of the closed loop polynomial varieties is given it turns out that plant hidden modes and rank singularities of the transfer function are the unique constraints we have to cope with in the compensator synthesis the proof of this result is based on algebraic manipulations of 2d mfds and on a coprime realization algorithm

multiple solutions for n kirchhoff type problems with critical exponential growth in rn abstract in this paper we consider a class of nonlocal n kirchhoff type problems involving a nonlinearity term having critical exponential growth by applying variational methods together with trudinger moser inequality in whole r n we obtain the existence of at least two positive solutions

existence and concentration of sign changing solutions to kirchhoff type system with hartree type nonlinearity abstract in this paper we discuss the existence and the concentration of sign changing solutions to a class of kirchhoff type systems with hartree type nonlinearity in r 3 by the minimization argument on the sign changing nehari manifold and a quantitative deformation lemma we prove that the system has a sign changing solution moreover concentration behaviors of sign changing solutions are obtained when the coefficient of the potential function tends to infinity specially our results cover general schrodinger equations kirchhoff equations and schrodinger poisson systems

control systems which generate decomposable lie algebras let x y be real analytic vector fields on an n manifold m ad x y x y denote the lie product inductively ad k 1 x y x ad k x y and 1 ad v x y v 0 the attainable set at time t denoted download full size image t for the control system 1 x x x u t y x x 0 p u t 1 is contained in the integral manifold through p of the lie algebra l 1 generated by 1 l 1 is said to be decomposable at p if essentially when dim l 1 p n the solution of 1 can be written as exp tx exp f 1 t u v 1 exp f n t u v n p with v 1 v n l 1 this is true for example at any p if l 1 is nilpotent the f i t u are shown to satisfy a system on n of the form 2 x i u t g i t x 1 x i 1 x i 0 0 i 1 n where if l 1 is nilpotent each g i is a polynomial in x 1 x i 1 in this case reachability properties of 2 for controls being countably additive measures impluses reduce to the study of algebraic equations homogeneity allows these to be considered on projective space n giving local results by bezout s theorem similar results hold for controls in the unit ball of krener s results on approximating systems of the form 1 by systems which generate nilpotent lie algebras can then be used to remove the condition that l 1 be nilpotent the method applies to show that if m denotes the set of all products of k elements of 1 with k m and dim span m 1 p dim span m for all odd m then exp tx p int download full size image t for all t 0

a high order test for optimality of bang bang controls for control systems of the form dot x x x sum i 1 m y i x u i a strengthened version of the classical pontryagin maximum principle is proved the necessary condition for optimality given here is obtained using functional analytic techniques and quite general high order perturbations of the reference control as shown by an example this test is particularly effective when applied to bang bang controls a case where other high order tests do not provide additional information