on the synthesis of a stabilizing feedback control via lie algebraic methods let the n dimensional system of differential equations dx dt x x t have p in mathbb r n as a rest solution i e x p 0 even in cases when this rest solution is unstable one can often induce a strong stability asymptotic stability by the inclusion of one or more controls e g via a controlled system a dx dt x x uy x where say u leqq 1 lie theory gives a computable sufficient condition to determine when by use of the control u one can steer a full n dimensional neighborhood of p to p by solutions of a this condition is assumed to hold one prefers a feedback control i e that u u x the main result in this paper is an algorithm which determines a modified stabilizing feedback control specifically for given varepsilon 0 one measures the current state q and the algorithm determines u t q 0 leqq t leqq varepsilon such that the solution x cdot u of a initiating from q and corresponding to this control u satisfies distance x varepsilon u p q p in fact iterates are theoretically shown to converge to p numerical examples computed via a simple fortran program are included these substantiate the strong stability achieved via such a modified feedback control

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

area bounds for free boundary minimal surfaces in a geodesic ball in the sphere abstract we extend to higher dimensions earlier sharp bounds for the area of two dimensional free boundary minimal surfaces contained in a geodesic ball of the round sphere this follows work of brendle and fraser schoen in the euclidean case

local stabilization of minimum phase nonlinear systems abstract in this note we briefly review the notion of a nonlinear minimum phase system i e a system which when constrained in such a way as to produce zero output evolves with an asymptotically stable dynamics the main purpose of the paper is to show that any minimum phase nonlinear system can always be locally stabilized by smooth state feedback

volumes of complex analytic subvarieties of hermitian symmetric spaces we give lower bounds of volumes of k dimensional complex analytic subvarieties of certain naturally defined domains in n dimensional complex space forms of constant positive zero or negative holomorphic sectional curvature for each 1 k n the lower bounds are sharp in the sense that these bounds are attained by k dimensional complete totally geodesic complex submanifolds such lower bounds are obtained by constructing singular potential functions corresponding to blow ups of the kahler metrics involved similar lower bounds are also obtained in the case of hermitian symmetric spaces of noncompact type in this case the lower bounds are sharp for those values of k at which the hermitian symmetric space contains k dimensional complete totally geodesic complex submanifolds which are complex hyperbolic spaces of minimum holomorphic sectional curvature

isoperimetric inequalities for complete proper minimal submanifolds in hyperbolic space the classical isoperimetric inequality for a domain sigma subset mathbb r k with smooth boundary partial sigma is displaystyle k k omega k mathrm vol sigma k 1 leq mathrm vol partial sigma k where equality holds if and only if sigma is a ball in mathbb r k

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

a bridge principle for minimal and constant mean curvature submanifolds of rn on demontre un principe de pont qui generalise l etablissement du theoreme de meeks yau la dimension ainsi que la codimension de n 1 et n 2 peuvent etre arbitraires et on ne suppose pas la stabilite on demontre un principe de pont pour des hypersurfaces de courbure moyenne constante

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

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

some combinatorial identities associated with the vandermonde convolution the authors first present a unification and generalization of some combinatorial identities associated with the familiar vandermonde convolution a basic or q extension of this general combinatorial identity is then obtained an interesting open problem relevant to the discussion in this paper is also presented

on trajectory and force tracking control of constrained mobile manipulators with parameter uncertainty this paper studies the trajectory and force tracking control problem of mobile manipulators subject to holonomic and nonholonomic constraints with unknown inertia parameters adaptive controllers are proposed based on a suitable reduced dynamic model the defined reference signals and the mixed tracking errors the proposed controllers not only ensure the entire state of the system to asymptotically converge to the desired trajectory but also ensure the constraint force to asymptotically converge to the desired force a detailed numerical example is presented to illustrate the developed methods

submanifolds isoperimetric inequalities and optimal transportation abstract the aim of this paper is to prove isoperimetric inequalities on submanifolds of the euclidean space using mass transportation methods we obtain a sharp weighted isoperimetric inequality and a nonsharp classical inequality similar to the one obtained in michael and simon 1973 19 the proof relies on the description of a solution of the problem of monge when the initial measure is supported in a submanifold and the final one supported in a linear subspace of the same dimension

bifurcation and chaotic behavior of a discrete singular biological economic system abstract in this paper the complex dynamics of a discrete singular biological economic system are first investigated firstly we establish a discrete singular biological economic system which is based on the discretization of a differential algebraic equations that is described by a ratio dependent predator prey system with harvesting and economic factor then it is shown that the system undergoes flip bifurcation and neimark sacker bifurcation in the interior of r 3 by using the new normal form of discrete singular systems the center manifold theorem and the bifurcation theory as varying the economic profit in some range numerical simulations are presented not only to illustrate our results with the theoretical analysis but also to exhibit the complex dynamic behaviors such as cascades of period doubling bifurcation in orbits of period 2 4 8 and chaotic sets these results reveal far richer dynamics of the discrete model compared with the continuous model the lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors

a patchy dynamic programming scheme for a class of hamilton jacobi bellman equations in this paper we present a new algorithm for the solution of hamilton jacobi bellman equations related to optimal control problems the key idea is to divide the domain of computation into subdomains which are shaped by the optimal dynamics of the underlying control problem this can result in a rather complex geometrical subdivision but it has the advantage that every subdomain is invariant with respect to the optimal dynamics and then the solution can be computed independently in each subdomain the features of this dynamics dependent domain decomposition can be exploited to speed up the computation and for an efficient parallelization since the classical transmission conditions at the boundaries of the subdomains can be avoided for their properties the subdomains are patches in the sense introduced by ancona and bressan esaim control optim calc var 4 1999 pp 445 471 several examples in two and three dimensions illustrate the properties of the new method

stabilizability of nonholonomic control systems abstract we consider here the problem of stabilization of nonlinear hamiltonian control systems with nonholonomic constraints such systems are of interest in various applications including the control of robotic systems we discuss the formulation of hamiltonian systems with nonholonomic constraints in particular considering the differences between the dirac theory of constraints the classical theory of nonholonomic constraints and weber s theory of generalized nonholonomic constraints stabilization of systems with these constraints is discussed using the theory of hamiltonian control systems and a reduced stabilization criterion is given

isoholonomic problems and some applications we study the problem of finding the shortest loops with a given holonomy we show that the solutions are the trajectories of particles in yang mills potentials theorem 4 or equivalently the projections of kaluza klein geodesics theorem 2 applications to quantum mechanics berry s phase sect 3 and the optimal control of deformable bodies sect 6 are touched upon

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

the isoperimetric inequality for minimal surfaces in a riemannian manifold it is proved that every minimal surface with one or two boundary components in a simply connected riemannian manifold with sectional curvature bounded above by a nonpositive constant k satisfies the sharp isoperimetric inequality 4 a l ka here equality holds if and only if the minimal surface is a geodesic disk in a surface of constant gaussian curvature k let d be a domain in a simply connected surface of constant gaussian curvature k the area a of d and the perimeter l satisfy the isoperimetric inequality 4 a l ka 1 where equality holds if and only if d is a geodesic disk the case k 0 was proved by steiner in 1842 s k 0 by bernstein in 1905 b and k 0 by schmidt in 1940 sc letm be a simply connected riemannian manifold of constant sectional curvaturek the isoperimetric inequality 1 holds for any domain on a totally geodesic surface in m since a totally geodesic surface is minimal in m it has been naturally conjectured that 1 should hold for every minimal surface in m the first result of this nature is due to carleman c who showed in 1921 that 1 holds for a simply connected domain on a minimal surface in rn so far 1 has been proved only for minimal surfaces with one or two boundary components in rn lsy ch and in hn cg1 in this paper we consider minimal surfaces in a simply connected riemannian manifold n of varying sectional curvature suppose the sectional curvature of n is bounded above by a constant k we prove that 1 holds for minimal surfaces with one or two boundary components in n when k 0 also with no restrictions on the topology of minimal surfaces in n and on the sign of k we obtain a weaker isoperimetric inequality 2 a l ka the gauss equation says that the gaussian curvature of a minimal surface is at most the sectional curvature of the ambient space accordingly it should be mentioned that for the case of simply connected surfaces with gaussian curvature bounded above by k 1 was proved for k 0 by weil w in 1926 and by beckenbach and rado br in 1933 and for k 6 0 by bol bo in 1941 1 the laplacian of functions of distance the distance function on a riemannian manifold m being the simplest geometric function on m implicitly supported in part by garc kosef 981 0104 019 2 bsri

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

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

control problems in super articulated mechanical systems control of super articulated mechanical systems sams controlled mechanical system in which the dimension of the configuration space exceeds the dimension of the control input space is studied as a starting point a graphical characterization of general sams is developed in which a so called control flow diagram cfd is constructed to represent the interaction forces among the degrees of freedom of the system three types of structure namely the chain structure the tree structure and isolated points are identified degree of complexity of the system is defined based on the structure of the system a systematic backstepping type controller design is proposed for systems with a chain structure to achieve global asymptotic stability and trajectory tracking such a controller design itself illustrates the sufficient conditions for the existence of smooth feedback control such that the systems with a chain structure can be globally asymptotically stabilized the connection of the authors control algorithm with existing results is discussed

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

complete minimal varieties in hyperbolic space it is a basic problem in the subject of minimal varieties to determine the class of complete minimal and area minimizing varieties m k in a given riemannian manifold n besides being of their own intrinsic interest the cases r of euclidean space and mn sn of the sphere are fundamental in understanding the local structure of minimal varieties due to the existence of tangent cones to general minimal varieties there has been much study of the complete areaminimizing varieties in these cases in particular relating to the bernstein conjecture and minimizing cones e g l2 f1 nevertheless many deep problems remain unsolved this paper is centered around the question of existence of complete areaminimizing varieties in hyperbolic space h here by way of contrast to the situation in ir n or s we find a comparatively complete solution to the problem of existence in fact the main result of this paper exhibits a wealth of complete area minimizing varieties in h in all dimensions and codimensions more precisely we have