nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

controllability and stability analysis of planar snake robot locomotion this paper contributes to the understanding of snake robot locomotion by employing nonlinear system analysis tools for investigating fundamental properties of snake robot dynamics the paper has five contributions 1 a partially feedback linearized model of a planar snake robot influenced by viscous ground friction is developed 2 a stabilizability analysis is presented proving that any asymptotically stabilizing control law for a planar snake robot to an equilibrium point must be time varying 3 a controllability analysis is presented proving that planar snake robots are not controllable when the viscous ground friction is isotropic but that a snake robot becomes strongly accessible when the viscous ground friction is anisotropic the analysis also shows that the snake robot does not satisfy sufficient conditions for small time local controllability stlc 4 an analysis of snake locomotion is presented that easily explains how anisotropic viscous ground friction enables snake robots to locomote forward on a planar surface the explanation is based on a simple mapping from link velocities normal to the direction of motion into propulsive forces in the direction of motion 5 a controller for straight line path following control of snake robots is proposed and a poincare map is investigated to prove that the resulting state variables of the snake robot except for the position in the forward direction trace out an exponentially stable periodic orbit

scale sensitive dimensions uniform convergence and learnability learnability in valiant s pac learning model has been shown to be strongly related to the existence of uniform laws of large numbers these laws define a distribution free convergence property of means to expectations uniformly over classes of random variables classes of real valued functions enjoying such a property are also known as uniform glivenko cantelli classes in this paper we prove through a generalization of sauer s lemma that may be interesting in its own right a new characterization of uniform glivenko cantelli classes our characterization yields dudley gine and zinn s previous characterization as a corollary furthermore it is the first based on a gine and zinn s previous characterization as a corollary furthermore it is the first based on a simple combinatorial quantity generalizing the vapnik chervonenkis dimension we apply this result to obtain the weakest combinatorial condition known to imply pac learnability in the statistical regression or agnostic framework furthermore we find a characterization of learnability in the probabilistic concept model solving an open problem posed by kearns and schapire these results show that the accuracy parameter plays a crucial role in determining the effective complexity of the learner s hypothesis class

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

scattering for the yang mills equations we construct wave and scattering operators for the yang mills equations on minkowski space m 0 r 4 sufficiently regular solutions of the yang mills equations on m 0 are known to extend uniquely to solutions of the corresponding equations on the universal cover of its conformal compactification m rxs 3

rate of convergence of empirical measures and costs in controlled markov chains and transient optimality the purpose of this paper is two fold first bounds on the rate of convergence of empirical measures in controlled markov chains are obtained under some recurrence conditions these include bounds obtained through large deviations and central limit theorem arguments these results are then applied to optimal control problems bounds on the rate of convergence of the empirical measures that are uniform over different sets of policies are derived resulting in bounds on the rate of convergence of the costs finally new optimal control problems that involve not only average cost criteria but also measures on the transient behavior of the cost namely the rate of convergence are introduced and applied to a problem in telecommunications the solution to these problems rely on the bounds introduced in previous sections

on the analysis of nonlinear nilpotent switched systems using the hall sussmann system abstract we consider an open problem on the stability of nonlinear nilpotent switched systems posed by daniel liberzon partial solutions to this problem were obtained as corollaries of global nice reachability results for nilpotent control systems the global structure is crucial in establishing stability we show that a nice reachability analysis may be reduced to the reachability analysis of a specific canonical system the nilpotent hall sussmann system furthermore local nice reachability properties for this specific system imply global nice reachability for general nilpotent systems we derive several new results revealing the elegant lie algebraic structure of the nilpotent hall sussmann system

scattering and complete integrability in the massive ϑ4 theory abstract the wave and scattering operators for the equation m 2 3 0 with m 0 on four dimensional minkowski space are analytic on a neighborhood of the space of finite energy cauchy data l 2 1 r 3 l 2 r 3 this allows the construction of infinitely many independent conserved quantities with vanishing poisson brackets and implies that the massive 4 theory is completely integrable at low energy

a class of infinite sums and integrals in this paper closed form sums are given for various slowly convergent infinite series which arise essentially from the differentiation of dirichlet l series some associated integrations are also considered a small number of the results appear in standard tables but most seem to be new

viscosity solutions of the bellman equation for exit time optimal control problems with vanishing lagrangians we study the hamilton jacobi bellman equation for undiscounted exit time optimal control problems for fully nonlinear systems and fully nonlinear singular lagrangians using the dynamic programming approach we prove a local uniqueness theorem characterizing the value functions for these problems as the unique viscosity solutions of the corresponding hamilton jacobi bellman equations that satisfy appropriate boundary conditions the novelty of this theorem is in the relaxed hypotheses on the lower bound on the lagrangian and the very general assumptions on the target set as a corollary we show that the value function for the fuller problem is the unique viscosity solution of the corresponding hamilton jacobi bellman equation that vanishes at the origin and satisfies certain growth conditions this implies as special cases first that the value function of this problem is the unique proper viscosity solution of the corresponding hamilton jacobi bellman equation in the class of all functions which are continuous in the plane and null at the origin and second that this value function is the unique viscosity solution of that equation in a class which includes functions which are not bounded below we also apply our results to the degenerate eikonal equation of geometric optics and to the shape from shading equations in image processing our theorem also applies to problems with noncompact targets and unbounded control sets whose lagrangians take negative values

empirical dynamic programming we propose empirical dynamic programming algorithms for markov decision processes in these algorithms the exact expectation in the bellman operator in classical value iteration is replaced by an empirical estimate to get empirical value iteration evi policy evaluation and policy improvement in classical policy iteration are also replaced by simulation to get empirical policy iteration epi thus these empirical dynamic programming algorithms involve iteration of a random operator the empirical bellman operator we introduce notions of probabilistic fixed points for such random monotone operators we develop a stochastic dominance framework for convergence analysis of such operators we then use this to give sample complexity bounds for both evi and epi we then provide various variations and extensions to asynchronous empirical dynamic programming the minimax empirical dynamic program and show how this can also be used to solve the dynamic newsvendor problem preliminary experimental results

on the positive output controllability of linear time invariant systems this paper considers the output controllability of autonomous linear control systems that are subject to non negative input constraints based on the evaluation of the geometric properties of the system necessary and sufficient conditions are proposed for the positive output controllability of continuous linear time invariant systems to aid in the practical evaluation of positive output controllability additional sufficient conditions are derived for which efficient numerical techniques exist these conditions are evaluated over a set of numerical examples which support the theoretical results

analyticity of the scattering operator for the nonlinear klein gordon equation with cubic nonlinearity the wave and scattering operators for the equation left square m 2 right varphi lambda varphi 2 0 withm 0 and 0 on four dimensional minkowski space are analytic on the space of finite energy cauchy data i e l21 r3 l2 r3

on the riemann property of angular lattice sums and the one dimensional limit of two dimensional lattice sums we consider a general class of two dimensional lattice sums consisting of complex powers s of inverse quadratic functions we consider two cases one where the quadratic function is negative definite and another more restricted case where it is positive definite in the former we use a representation due to h kober and consider the limit u where the lattice becomes ever more elongated along one period direction the one dimensional limit in the latter we use an explicit evaluation of the sum due to zucker and robertson in either case we show that the one dimensional limit of the sum is given in terms of 2 s if re s 1 2 and either 2 s 1 or 2 2 s if re s 1 2 in either case this leads to a riemann property of these sums in the one dimensional limit their zeros must lie on the critical line re s 1 2 we also comment on a class of sums that involve complex powers of the distance to points in a two dimensional square lattice and trigonometric functions of their angle we show that certain of these sums can have their zeros on the critical line but not in a neighbourhood of it others are identically zero on it while still others have no zeros on it

on the empirical state action frequencies in markov decision processes under general policies we consider the empirical state action frequencies and the empirical reward in weakly communicating finite state markov decision processes under general policies we define a certain polytope and establish that every element of this polytope is the limit of the empirical frequency vector under some policy in a strong sense furthermore we show that the probability of exceeding a given distance between the empirical frequency vector and the polytope decays exponentially with time under every policy we provide similar results for vector valued empirical rewards

exponential stabilization of an underactuated autonomous surface vessel this paper studies the problem of controlling the planar position and orientation of an autonomous surface vessel using two independent thrusters it is first shown that although the system is not asymptotically stabilizable to a given configuration using a time invariant continuous feedback it is strongly accessible and small time locally controllable at any equilibrium time invariant discontinuous feedback laws are then constructed to asymptotically stabilize the system to the desired configuration with exponential convergence rates a simulation example is included to demonstrate the results

asymptotically stabilizing feedback controls abstract the problem considered is the existence and construction of an asymptotically stabilizing feedback control for a rest solution of an n dimensional single input affine control system the method is to construct high order homogeneous approximations of the vector fields defining the control system which then describe an approximating control system such that an asymptotically stabilizing control of the latter system is a local asymptotically stabilizing control of the original system unlike linear approximations there are many possible nonlinear homogeneous approximations these are constructed from lie algebra filtrations obtained by assigning various weights to the defining vector fields one seeks a weighting hence an approximating system which satisfies the various known necessary conditions for the existence of an asymptotically stabilizing feedback control and for which such a control can be constructed as the solution to a homogeneous hamilton jacobi bellman equation associated with an optimization regulator problem

steering for a class of dynamic nonholonomic systems we derive control algorithms for a class of dynamic nonholonomic steering problems characterized as mechanical systems with nonholonomic constraints and symmetries we use a simplified geometric framework in developing two types of control algorithms for such systems the first targets local configuration controllability using a perturbation approach to establish results similar to steering using sinusoids the second method builds on this approach by utilizing kinematic steering algorithms for mobile robots in generating more nonlocal motion plans for this class of systems

controllability of a class of underactuated mechanical systems with symmetry in this paper we develop results based on geometric mechanics to study the controllability of a class of controlled under actuated left invariant mechanical systems on lie groups we exploit the invariance of the controlled nonlinear dynamics to the group action symmetry to derive a set of reduced dynamics for the system we first present sufficient conditions for the controllability of the reduced dynamics we prove conditions boundedness of coadjoint orbits and existence of a radially unbounded lyapunov function under which the drift vector field of the reduced system is weakly positively poisson stable wpps the wpps nature of the drift vector field along with the lie algebra rank condition is used to show controllability of the reduced system we then extend these results to the unreduced dynamics considering separately the cases when the symmetry group is compact and noncompact in the noncompact case under further assumptions of equilibrium controllability sufficient conditions for controllability of the unreduced dynamics are derived jetpuck hovercraft and underwater vehicles are used as mechanical systems to motivate our work and illustrate our theoretical results

further results on the bellman equation for optimal control problems with exit times and nonnegative lagrangians abstract in a series of papers we proved theorems characterizing the value function in exit time optimal control as the unique viscosity solution of the corresponding bellman equation that satisfies appropriate side conditions the results applied to problems which satisfy a positivity condition on the integral of the lagrangian this positive integral condition assigned a positive cost for remaining outside the target on any interval of positive length in this note we prove a new theorem which characterizes the exit time value function as the unique bounded from below viscosity solution of the bellman equation that vanishes on the target the theorem applies to problems satisfying an asymptotic condition on the trajectories including cases where the positive integral condition is not satisfied our results are based on an extended version of barb lat s lemma we apply the theorem to variants of the fuller problem and other examples where the lagrangian is degenerate

scattering and the geometry of the solution manifold of ƒ λƒ3 0 abstract the wave transforms for the equation 3 0 on r 4 can be represented as maps from the hilbert space h s of finite energy cauchy data for the equation 1 3 0 on r 4 s 3 to hilbert spaces h c of finite energy goursat data we show that the wave transforms and associated scattering transform are diffeomorphisms the action of the group g so 2 4 on solutions of the latter equation gives rise to actions of g as diffeomorphisms of h s and h c which are intertwined by the wave and scattering transforms moreover the spaces h s and h c have symplectic structures relative to which the group g and the wave and scattering transforms act as symplectomorphisms these symplectic structures extend to flat kahler structures that are invariant under the scaling extended poincare group p

configuration controllability of mechanical systems underactuated by one control we investigate local configuration controllability for mechanical control systems within the affine connection formalism we rely on previous results on controllability and series expansions for the evolution of mechanical systems starting from rest extending the work by lewis for the single input case we are able to characterize local configuration controllability for systems with n degrees of freedom and n 1 input forces

state steering by learning for a class of nonlinear control systems abstract the problem of steering the state of nonlinearly perturbed linear systems by learning is investigated a family of algorithms which compute the steering control by means of successive trials on the real plant is presented convergence in the face of a class of nonlinear plant perturbations is proven state feedback linearizable systems are shown to be addressable by the presented algorithms two examples illustrate the applicability of algorithms

wick products of the free bose field we consider wick products of the free bose field in the abstract context of a complex hilbert space h equipped with a self adjoint operator a satisfying a ei for some e 0 let k w be the free bose field over h and let h d a let h and k denote d a and d h respectively given their natural frechet topologies then for any 1 n h the wick product 1 n is constructed as a continuous sesquilinear form on k characterized by a generalization of the heisenberg commutation relations as an application we treat pointwise products of the free scalar field and its derivatives on r m m an arbitrary complete riemannian manifold for example if ck0 r m for large enough k then r m p p n corresponds to an operator with domain k if in addition is real valued n 2 and m is compact then this operator is essentially self adjoint

algebraic structure of classical field theory i kinematics and linearized dynamics for real scalar fields we describe the elements of a novel structural approach to classical field theory inspired by recent developments in perturbative algebraic quantum field theory this approach is local and focuses mainly on the observables over field configurations given by certain spaces of functionals which are studied here in depth the analysis of such functionals is characterized by a combination of geometric analytic and algebraic elements which 1 make our approach closer to quantum field theory 2 allow for a rigorous analytic refinement of many computational formulae from the functional formulation of classical field theory and 3 provide a new pathway towards understanding dynamics particular attention will be paid to aspects related to nonlinear hyperbolic partial differential equations and their linearizations

controllability of linear systems on generalized heisenberg groups this paper is devoted to the study of global controllability of linear systems on higher dimensional heisenberg groups hn necessary conditions and sufficient ones are provided they are compared to the known necessary and sufficient conditions on h1 stated in dath and jouan 2016 controllability of linear systems on low dimensional nilpotent and solvable lie groups journal of dynamical and control systems 22 2 207 225 the notion of decoupled systems is introduced in order to get examples and counter examples and more precise controllability criteria are stated for them

high order variations for families of vector fields sufficient conditions involving lie brackets of arbitrarily high order are obtained for local controllability of families of vector fields after providing a general framework for the generation of high order control variations we propose a specific method for generating such variations the theory is applied to a number of nontrivial examples

conserved quantities for the yang mills equations abstract there are infinitely many gauge invariant conserved quantities for sufficiently regular solutions of the yang mills equations on 4 dimensional minkowski space for example those that extend to c2 classical solutions on the universal cover of the conformal compactification which may be identified with the einstein universe r s3 these conserved quantities are expressed in terms of the in or out fields constructed via the formulation of scattering as a global goursat problem on the einstein universe

controllability is harder to decide than accessibility this article compares the difficulties of deciding controllability and accessibility these are standard properties of control systems but complete algebraic characterizations of controllability have proved elusive the article shows in particular that for subsystems of bilinear systems accessibility can be decided in polynomial time but controllability is np hard

the master ward identity and generalized schwinger dyson equation in classical field theory in the framework of perturbative quantum field theory a new universal renormalization condition called master ward identity was recently proposed by one of us m d in a joint paper with f m boas the main aim of the present paper is to get a better understanding of the master ward identity by analyzing its meaning in classical field theory it turns out that it is the most general identity for classical local fields which follows from the field equations it is equivalent to a generalization of the schwinger dyson equation and is closely related to the quantum action principle of lowenstein and lam as a byproduct we give a self contained treatment of peierls manifestly covariant definition of the poisson bracket

on scattering and everywhere defined scattering operators for nonlinear klein gordon equations asymptotic properties of solutions of the nonlinear klein gordon equation t2u u m2u f u 0 nlkg u 0 tu 0 are investigated which are inherited from the corresponding solutions v of the linear klein gordon equation t2v v m2v 0v 0 tv 0 kg in particular the finiteness of time integrals in lq over r of certain sobolevnorms in space of the solution is proved to be such a hereditary property together with a device by w a strauss and a weak decay result for the kg due to r s strichartz this is used to prove that under suitable restrictions on the nonlinearity the scattering operator for the nlkg is defined on all of l21 l2 for n 3

geometric conditions for local controllability abstract small time local controllability of a c1 affine control system with a generic drift vector field and n 1 linearly independent input vector fields on an n dimensional connected c1 manifold m is studied showing that a necessary condition namely the drift vector field belonging to the convex hull of the control vector fields at the given point is also sufficient on an open dense subset h1 of the set of points where it is verified local controllability at the points in the interior of h 1 is also considered when the input vector fields generate an involutive distribution and a criterion based on the type of contract of h with the hyperplane spanned by the input vector fields is proved an example shows a situation where the sufficient condition for small time local controllability of h sussmann siam j control option 25 1987 158 194 cannot be applied but in which the method now presented distinguishes the possible cases

on the relation between local controllability and stabilizability for a class of nonlinear systems the problem of local stabilizability of locally controllable nonlinear systems is considered it is well known that contrary to the linear case local controllability does not necessarily imply stabilizability a class of nonlinear systems for which local controllability implies local asymptotic stabilizability using continuous static state feedback is described as for this class of systems the well known hermes controllability condition is necessary and sufficient for local controllability

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

proximal analysis and the minimal time function under general hypotheses on the target set s and the dynamics of the system we show that the minimal time function ts cdot is a proximal solution to the hamilton jacobi equation uniqueness results are obtained with two different kinds of boundary conditions a new propagation result is proven and as an application we give necessary and sufficient conditions for ts cdot to be lipschitz continuous near s a petrov type modulus condition is also shown to be sufficient for continuity of ts cdot near s

analyticity of scattering for the ϕ 4 theory we consider scattering for the equation m2 3 0 on four dimensional minkowski space form 0 one to one and onto wave operatorsw h h are known to exist for all 0 whereh denotes the hilbert space of finite energy cauchy data we prove that the maps u w u and u w 1 u are continuous from 0 h toh and extend to real analytic functions from an open neighborhood of 0 h 0 h to the hilbert spaceh 1 of cauchy data with poincare invariant norm form 0 wave operatorsw are known to exist as diffeomorphisms ofh for all 0 where hereh denotes the hilbert space of finite einstein energy cauchy data in this case we prove that the maps u w u and u w 1 u extend to real analytic functions from a neighborhood of 0 h h toh

analysis in space time bundles i general considerations and the scalar bundle abstract mathematical features involved in the systematic development of elementary particle theory on an alternative cosmos space time are presented bundles representative of physical fields are studied in the unique variant of minkowski space m 0 enjoying similar properties of causality and symmetry the universal cosmos m is the universal cover of the causal compactification of m 0 the bundles studied are induced from representations of the scale extended poincare group which forms the isotropy group in m of the universal cover g s u 2 2 of the connected component of the group of all causal transformations on m discrete symmetries and higher dimensional cases are also discussed the primary focus is on the temporal evolution especially stability involving positivity of the energy wave equations implicative of finite propagation velocity and the unitarity and or composition series of associated actions of g general spin bundles on m are treated parallelized and correlated with bundles on m 0 associated covariant wave equations and the spectral resolution of fundamental quantum numbers are studied in detail in the scalar case

asymptotic model matching for nonlinear systems this paper investigates the problem of designing a compensating control law for a square invertible nonlinear plant so that the response of the closed loop system asymptotically matches that of a prescribed driven nonlinear model a set of necessary conditions for achieving asymptotic model matching is established one of these involves the stability properties for subdynamics which are common to every model matching closed loop these subdynamics called the fixed dynamics are intrinsically characterized based on these results a new set of sufficient conditions for achieving asymptotic model matching is given the interrelation between the minimum phase and vector relative degree properties of a plant and a matchable model are studied

local controllability for families of diffeomorphisms abstract we study groups and semigroups which are generated by analytic families of diffeomorphisms the central notion is that of local controllability of a family of diffeomorphisms at a given point of the state manifold which generalizes the familiar notion of local controllability of control systems with continuous as well as discrete time lie theory methods are used we systematically exploit the so called fast switching variations and properties of the jet spaces of curves on the state manifold

simulation based optimization of markov decision processes an empirical process theory approach we generalize and build on the pac learning framework for markov decision processes developed in jain and varaiya 2006 we consider the reward function to depend on both the state and the action both the state and action spaces can potentially be countably infinite we obtain an estimate for the value function of a markov decision process which assigns to each policy its expected discounted reward this expected reward can be estimated as the empirical average of the reward over many independent simulation runs we derive bounds on the number of runs needed for the convergence of the empirical average to the expected reward uniformly for a class of policies in terms of the v c or pseudo dimension of the policy class we then propose a framework to obtain an e optimal policy from simulation we provide sample complexity of such an approach

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

reachable sets and controllability of bilinear time invariant systems a qualitative approach this paper is concerned with a qualitative method for the study of reachable sets and controllability of bilinear time invariant systems a nonlinear decomposition technique which reduces the solution of the problem to the analysis of the qualitative behavior of a finite number of linear uncontrolled systems is introduced special consideration is given to the homogeneous bilinear systems with commutative matrices two and three dimensional examples are discussed

motion planning for control affine systems satisfying low order controllability conditions abstractthis paper is devoted to the motion planning problem for control affine systems by using trigonometric polynomials as control functions the class of systems under consideration satisfies the controllability rank condition with the lie brackets up to the second order the approach proposed here allows to reduce a point to point control problem to solving a system of algebraic equations the local solvability of that system is proved and formulas for the parameters of control functions are presented our local and global control design schemes are illustrated by several examples

a new theoretical approach to absorbing layers we consider the construction of absorbing boundary layers using asymptotic expansions the small parameter being the width of the layer this allows us to define the order of the layer we compute layers of various orders and analyze their stability properties finally we perform some numerical tests to evaluate the practical utility of our layers

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

series expansions for the evolution of mechanical control systems this paper presents a series expansion that describes the evolution of a mechanical system starting at rest and subject to a time varying external force mechanical systems are presented as second order systems on a configuration manifold via the notion of affine connections the series expansion is derived by exploiting the homogeneity property of mechanical systems and the variations of constant formula a convergence analysis is obtained using some analytic functions and combinatorial analysis results this expansion provides a rigorous means of analyzing locomotion gaits in robotics and lays the foundation for the design of motion control algorithms for a large class of underactuated mechanical systems

energy of static electron lattices a new approach to evaluating the static lattice energy of any given wigner lattice is proposed the method is much simpler than the traditional one and the results are much faster to evaluate the new approach is applied to two dimensional triangular lattices where it is shown that the triangular lattice is more stable than the square lattice three dimensional hexagonal lattices are also investigated in addition the authors place many of their considerations on a rigorous mathematical footing

low order controllability and kinematic reductions for affine connection control systems controllability and kinematic modeling notions are investigated for a class of mechanical control systems first low order controllability results are given for the class of mechanical control systems second a precise connection is made between those mechanical systems which are dynamic i e have forces as inputs and those which are kinematic i e have velocities as inputs interestingly and surprisingly these two subjects are characterized and linked by a certain intrinsic vector valued quadratic form that can be associated to an affine connection control system

analyticity of the scattering operator for semilinear dispersive equations we present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic and to compute the coefficients of its taylor series at any point we illustrate this method in the case of the schrodinger equation with power like nonlinearity or with hartree type nonlinearity and in the case of the wave and klein gordon equations with power nonlinearity finally we discuss the link of this approach with inverse scattering and with complete integrability

simulation based uniform value function estimates of markov decision processes the value function of a markov decision process mdp assigns to each policy its expected discounted reward this expected reward can be estimated as the empirical average of the reward over many independent simulation runs we derive bounds on the number of runs needed for the uniform convergence of the empirical average to the expected reward for a class of policies in terms of the vapnik chervonenkis or p dimension of the policy class further we show through a counterexample that whether we get uniform convergence or not for an mdp depends on the simulation method used uniform convergence results are also obtained for the average reward case for partially observed markov decision processes and can be easily extended to markov games the results can be viewed as a contribution to empirical process theory and as an extension of the probably approximately correct pac learning theory for partially observable mdps and markov games

some logarithmic lattice sums in this paper we consider certain lattice sums which arise when a system of line charges set in a compensating jelly of opposite charge interact via the two dimensional logarithmic potential a lattice limit discontinuity phenomenon similar to that discovered by borwein et al 1989 in the three dimensional case is explored and a high precision asymptotic method is described for the numerical computation of two dimensional lattice limits

scattering and complete integrability in conformally invariant nonlinear theories conformally invariant nonlinear wave equations in four dimensions corresponding to multicomponent massless scalar fields with a quartic interaction are studied it is proved that the scattering operator s on the space h of finite einstein energy cauchy data has infinitely many fixed points as well as periodic points of all orders there are also h such that sn is almost periodic but not periodic and h such that sn is not almost periodic it is also proved that h admits no conformally invariant kahler metrics but infinitely many distinct kahler metrics invariant under the poincare group and scale transformations moreover it is proved that time evolution for these nonlinear wave equations is completely integrable on the space h

minimum time with bounded energy minimum energy with bounded time necessary and sufficient conditions for the regularity of the minimum time function and minimum energy function for a control system with controls in l p 0 infty bbb r m and p ge 1 are given in terms of topological properties of the reachable sets in particular standard local controllability assumptions are sufficient to yield the continuity of both value functions for linear systems and p ge1 and the holder continuity of the minimum time function for nonlinear systems and p 1

a polynomial approach to discrete time non linear system controllability this paper uses a polynomial approach to present a necessary and sufficient condition for local controllability of single input single output siso discrete time non linear systems the condition is presented in terms of common factors of a non commutative polynomial expression this extends the result on controllability of linear systems and the recent result on controllability of continuous time non linear systems to expose controllability of discrete time non linear systems in the input output framework the controllability condition yields a finite step computing procedure for examining controllability of discrete time input output non linear systems

local equilibrium controllability of multibody systems controlled via shape change we study local equilibrium controllability of shape controlled multibody systems the multibody systems are defined on a trivial principal fiber bundle by a lagrangian that characterizes the base body motion and shape dynamics a potential dependent on an advected parameter e g uniform gravitational potential is considered this potential breaks base body symmetries but a symmetry subgroup is assumed to exist symmetric product formulas are derived and important properties are obtained for symmetric products of horizontal shape control vector fields and a potential vector field that is dependent on an advected parameter based on these properties sufficient conditions for local equilibrium controllability and local fiber equilibrium controllability are developed these results are applied to two classes of shape controlled multibody systems in a uniform gravitational field multibody attitude systems and neutrally buoyant multibody underwater vehicles

controllability distributions and systems approximations a geometric approach given a nonlinear system we determine a relation at an equilibrium between controllability distributions defined for a nonlinear system and a taylor series approximation of it the value of such a relation is appreciated if we recall that the solvability conditions as well as the solutions to some control synthesis problems can be stated in terms of geometric concepts like controlled invariant controllability distributions the relation between these distributions at the equilibrium will help us to decide when the solvability conditions of this kind of problems are equivalent for the nonlinear system and its approximation

the global goursat problem and scattering for nonlinear wave equations abstract the goursat problem for nonlinear scalar equations on the einstein universe m with finite energy datum has a unique global solution in the positive energy sobolev controllable case such equations include those of the form h 0 where h denotes a hamiltonian that is a fourth order polynomial bounded below in components of the multicomponent scalar section in particular the conformally invariant equation 1 3 0 0 is included in the higher dimensional analog r s n to the einstein universe the same result holds under the stronger conditions on h required for sobolev controllability irrespective of energy positivity there is a unique local in time solution for arbitrary finite energy goursat datum for all n 3 establishing evolution from the given lightcone to any sufficiently close lightcone these results show the existence of wave operators in the sense of scattering theory and their continuity in the einstein energy metric for positive energy equations of the indicated type they also permit the comprehensive reduction of scattering theory for conformally invariant wave equations in minkowski space m 0 to the goursat problem in m in particular any solution of the equation arising from a nonnegative conformally invariant biquadratic interaction lagrangian on multicomponent scalar sections having finite einstein energy at any one time is asymptotic to solutions of the corresponding multicomponent free wave equation as the minkowski time x 0 thus given a finite einstein energy solution of the equation 3 0 on m 0 0 there exist unique solutions of the free wave equation which approach in the minkowski energy norm as x 0 and every finite einstein energy solution of the free wave equation is of the form or for a unique solution of the nonlinear equation this generalizes in part in maximality sharp form earlier results of strauss for this equation

the complexity of deciding controllability abstract it is shown that the problem of deciding whether a control system is small time locally controllable or not is np hard this is an extension of a recent paper by sontag to single input systems and to bounded controls