solving linear parabolic rough partial differential equations abstract we study linear rough partial differential equations in the setting of friz and hairer springer 2014 chapter 12 more precisely we consider a linear parabolic partial differential equation driven by a deterministic rough path w of holder regularity with 1 3 1 2 based on a stochastic representation of the solution of the rough partial differential equation we propose a regression monte carlo algorithm for spatio temporal approximation of the solution we provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space finally we present a simulation study showing the applicability of the proposed algorithm

brief on controllability and trajectory tracking of a kinematic vehicle model this paper presents some further results concerning the issues of controllability and trajectory tracking regarding a front wheel drive vehicle kinematic model a simple procedure for computing an open loop control strategy that transfers the system between given initial and final states is presented in particular the input function is computed by means of a set of linear algebraic equations the resulting motion planning procedure allows us to present a control scheme for solving the trajectory a time parameterized reference signal tracking problem various applications of the approach in forward and backward motions are considered and simulation results are presented

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 phragmen lindelof alternative result for the navier stokes equations for steady compressible viscous flow abstract in this paper the authors consider the navier stokes equations for steady compressible viscous flow in three dimensional cylindrical domain a differential inequality for appropriate energy associated with the solutions of the navier stokes isentropic flow in semi infinite pipe is derived from which the authors show a phragmen lindelof alternative result i e the solutions for steady compressible viscous n s flow problem either grow or decay exponentially as the distance from the entry section tends to infinity in the decay case the authors indicate how to bound explicitly the total energy in terms of data

an approximation algorithm for nonholonomic systems in siam j control optim 37 1997 to appear limiting process of control affine systems with holder continuous inputs submitted we have studied the limiting behavior of trajectories of control affine systems sigma dot x sum k 1 m u k f k x generated by a sequence u j subseteq l 1 0 t rr m where the f k are smooth vector fields on a smooth manifold m we have shown that under very general conditions the trajectories of sigma generated by the u j converge to trajectories of an extended system of sigma of the form sigma ext dot x sum k 1 r v kf k x where f k k 1 ldots m are the same as in sigma and f m 1 ldots f r are lie brackets of f 1 ldots f m in this paper we will apply these convergence results to solve the inverse problem i e given any trajectory gamma of an extended system sigma ext find trajectories of sigma that converge to gamma uniformly this is done by means of a universal construction that only involves the knowledge of the v k k 1 ldots r and the structure of the lie brackets in sigma ext but does not depend on the manifold m and the vector fields f 1 ldots f m these results can be applied to approximately track an arbitrary smooth path in m for controllable systems sigma which in particular gives an alternative approach to the motion planning problem for nonholonomic systems

flows and stochastic taylor series in ito calculus for general stochastic systems driven by continuous semimartingales an explicit formula for the logarithm of the ito flow map is given the computation relies on the lift to quasi shuffle algebras of formulas involving products of ito integrals of semimartingales whereas the chen strichartz formula computing the logarithm of the stratonovich flow map is classically expanded as a formal sum indexed by permutations the analogous formula in ito calculus is naturally indexed by surjections this reflects the change of algebraic background involved in the transition between the two integration theories lastly we extend our formula for the quasi shuffle chen strichartz series for the logarithm of the flow map to the non commutative case for linear matrix valued sdes driven by arbitrary semimartingales we obtain a similar formula

harmonic analysis as spectral theory of laplacians by writing the fourier inversion formula on euclidean space in polar coordinates we obtain f x 0 fgl x dx where fgl x 2fgl x which is the spectral theory of the laplacian how do properties of f relate to properties of the family of eigenfunctions fgl answers are provided for the following spaces l2 s s d in odd dimensions analogous results are obtained for harmonic analysis on hyperbolic space constant curvature semi riemannian spaces the heisenberg group and for differential forms on hyperbolic space for f l2 there is a plancherel formula f 22 limt o 1t bt z f x 2dx d where bt z denotes the ball of radius t about an arbitrary point z which is independent of the dimension and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted for certain semi riemannian symmetric spaces we obtain a paley wiener theorem that explains the role of discrete series in producing functions of compact support and also involve certain non unitary representations that contain the discrete series representations for the heisenberg group the plancherel formula is of a different nature requiring that the eigenfunctions be almost periodic in one variable

perturbation theory and singular perturbations for input to state multistable systems on manifolds we consider the notion of input to state multistability which generalizes input to state stability to nonlinear systems evolving on riemannian manifolds and possessing a finite number of compact globally attractive invariant sets and in addition satisfies a specific condition of acyclicity we prove that a parameterized family of dynamical systems whose solutions converge to those of a limiting system inherits such input to state multistability property from the limiting system in a semiglobal practical fashion a similar result is also established for singular perturbation models whose boundary layer subsystem is uniformly asymptotically stable and whose reduced subsystem is input to state multistable known results in the theory of perturbations singular perturbations averaging and highly oscillatory control systems are here generalized to the multistable setting by replacing the classical asymptotic stability requirement of a single invariant set with attractivity and acyclicity of a decomposable invariant one

analysis of chemical kinetic systems over the entire parameter space i the sal nikov thermokinetic oscillator in this series of papers we re examine using recently developed techniques some chemical kinetic models that have appeared in the literature with a view to obtaining a complete description of all the qualitatively distinct behaviour that the system can exhibit each of the schemes is describable by two coupled ordinary differential equations and contain at most three independent parameters we find that even with these relatively simple chemical schemes there are regions of parameter space in which the systems display behaviour not previously found quite often these regions are small and it seems unlikely that they would be found via classical methods in part i of the series we consider one of the thermally coupled kinetic oscillator models studied by sal nikov he showed that there is a region in parameter space in which the system would be in a state of undamped oscillations because the relevant phase portrait consists of an unstable steady state surrounded by a stable limit cycle our analysis has revealed two further regions in which the phase portraits contain respectively two limit cycles of opposite stability enclosing a stable steady state and three limit cycles of alternating stability surrounding an unstable steady state this latter region is extremely small so much so that it could be reasonably neglected in any predictions made from the model

decroissance exponentielle du noyau de la chaleur sur la diagonale i we give examples based upon large deviation s theory where the heat kernel of a degenerate diffusion has an exponential decay over the diagonal using malliavin calculus we give conditions for a more generalized heat kernel to have an exponential decay over the diagonal we give lower bound in some particular case by using the bismut s condition

on a computationally simple form of the generalized campbell baker hausdorff dynkin formula abstract in this paper a computationally simple form of the generalized campbell baker hausdorff dynkin formula gcbhd is given simplifications arise from both combinatorial explicit as well as algorithmic implicit arguments on generating the ph hall basis in a special form and introducing a specific structure representation of the vector field vector fields appearing in gcbhd are automatically transformed into the reduced ph hall basis the formula can be exploited in nonholonomic motion planning what is illustrated with examples

convergence of the exponential lie series improved results on convergence of the magnus expansion are reported consequences are discussed for exponential perturbation theory construction of numerical methods control theory or in general for problems where time ordered integrals are involved

trajectory based local approximations of ordinary differential equations the present paper introduces a new definition of local approximation for ordinary differential equations locally around an equilibrium point this definition generalizes the well known linear and homogeneous approximations the approach is based on approximating trajectories near the origin this concept of local approximation is applied to the study of local uniform asymptotic stability leading to alternative proofs for and extensions of several existing stability results

a software package for lie algebraic computations the paper presents a computer algebra package that facilitates lie algebraic symbolic computations required in the solution of a variety of problems such as the solution of right invariant differential equations evolving on lie groups lie theory is a powerful tool helpful in the analysis and design of modern nonlinear control laws nonlinear filters and the study of particle dynamics the practical application of lie theory often results in highly complex symbolic expressions that are difficult to handle efficiently without the aid of a computer software tool the aim of the package is to facilitate and encourage further research relying on lie algebraic computations

practical stabilization of driftless systems on lie groups the transverse function approach a general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional lie groups is presented the approach is based on the concept of bounded transverse functions the existence of which is equivalent to the system s controllability its outcome is the practical stabilization of any trajectory i e not necessarily a solution of the control system in the state space the possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted via a dynamic extension to a system on a lie group illustrative examples are given

analysis of chemical kinetic systems over the entire parameter space ii isothermal oscillators two simple but similar isothermal kinetic schemes are studied by using recently developed qualitative techniques with the aim of finding all of the qualitatively distinct behaviour that the system can exhibit the first scheme which can be regarded as one of the simplest isothermal models of interest is based on the quadratic autocatalytic step a b rightarrow 2b text rate k q ab where a and b are the concentrations of a and b respectively combined with a nonlinear termination step of the form b rightarrow c text rate k t b 1 rb where r is some parameter the second scheme is a simplified by treating one parameter as a constant version of a model that has been proposed for the oxidation of carbon monoxide co this scheme has a termination step of the same form as above and like the quadratic model the highest nonlinearity apart from this term is quadratic for the quadratic model we find 10 qualitatively distinct phase portraits and with our choice of bifurcation parameter 22 distinct bifurcation diagrams the most complex phase portraits contain 3 steady states and 2 limit cycles or 2 steady states and 3 limit cycles for the co model we find 8 distinct phase portraits but leading to only 5 separate bifurcation diagrams in both models much of the behaviour occurs over only extremely small regions in parameter space

stochastic lie group integrators we present lie group integrators for nonlinear stochastic differential equations with noncommutative vector fields whose solution evolves on a smooth finite dimensional manifold given a lie group action that generates transport along the manifold we pull back the stochastic flow on the manifold to the lie group via the action and subsequently pull back the flow to the corresponding lie algebra via the exponential map we construct an approximation to the stochastic flow in the lie algebra via closed operations and then push back to the lie group and then to the manifold thus ensuring that our approximation lies in the manifold we call such schemes stochastic munthe kaas methods after their deterministic counterparts we also present stochastic lie group integration schemes based on castell gaines methods these involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential lie series they become stochastic lie group integrator schemes if we use munthe kaas methods as the underlying ordinary differential integrator further we show that some castell gaines methods are uniformly more accurate than the corresponding stochastic taylor schemes lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

on inequalities of korn friedrichs and babuska aziz korn s inequalities for integrals of quadratic functionals subject to certain side conditions have played a fundamental role in the development of elasticity theory see e g i 14 the korn inequalities are essential in establishing coerciveness of the differential operators of linear elastostatics and thus form the basis for existence results in that theory 9 among other areas in elasticity where consideration of korn s inequalities arise we cite for example fundamental studies on the mathematical foundations of finite elements 15 17 stability theory 11 and qualitative analyses of solution behavior such as those involved in the investigation of saint venant s principle 18 20 in the latter two areas of application in particular information on the optimal constants appearing in the inequalities the korn constants is of importance thus several investigations have been concerned with evaluating the korn constants for specific domains and with obtaining bounds for these constants 4 6 8 12 13 our purpose in this paper is to draw attention to an equivalence for two dimensional simply connected domains between korn s inequality in the second case see w 2 here and two other inequalities for quadratic functionals which have received comparatively little attention in the literature the first of these is an inequality due apparently to k o frmorichs 2 between the real and imaginary parts of an analytic function or equivalently between a harmonic function and its conjugate defined on a two dimensional domain r thus if w z f ig z xl ix2 is analytic on r friedrichs inequality reads

nonholonomic motion planning steering using sinusoids methods for steering systems with nonholonomic c onstraints between arbitrary configurations are investigated suboptimal trajectories are derived for systems that are not in canonical form systems in which it takes more than one level of bracketing to achieve controllability are considered the trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level a class of systems that can be steered using sinusoids claimed systems is defined conditions under which a class of two input systems can be converted into this form are given

analysis of chemical kinetic systems over the entire parameter space iv the sal nikov oscillator with two temperature dependent reaction rates this paper investigates a model chemical reaction in which a substance undergoes a two stage decay each stage exhibiting arrhenius temperature dependence the system is assumed to be well stirred so it is described by two highly nonlinear ordinary differential equations containing four parameters degenerate bifurcation theory and numerical analysis are used to investigate the behaviour of the system over the whole of physical parameter space a large number of degenerate bifurcations are identified and located and the system is shown to exhibit 16 qualitatively different phase portraits degeneracies of higher codimension than would be expected occur in this system giving rise to some highly non uniform behaviour both in phase and parameter space this behaviour seems to be generic in thermokinetic systems with more than one arrhenius nonlinearity

the averaged control system of fast oscillating control systems for control systems that either have a fast explicit periodic dependence on time and bounded controls or have periodic solutions and small controls we define an average control system that takes into account all possible variations of the control and prove that its solutions approximate all solutions of the oscillating system as the frequency of the oscillations tends to infinity the dimension of its velocity set is characterized geometrically when it is maximum the average system defines a finsler metric which is not twice differentiable in general for minimum time control this average system allows one to give a rigorous proof that averaging the hamiltonian given by the maximum principle is a valid approximation

new high order sufficient conditions for configuration tracking in this paper we propose new conditions guaranteeing that the trajectories of a mechanical control system can track any curve on the configuration manifold we focus on systems that can be represented as forced affine connection control systems and we generalize the sufficient conditions for tracking known in the literature the new results are proved by a combination of averaging procedures by highly oscillating controls with the notion of kinematic reduction

averaging theorems for highly oscillatory differential equations and iterated lie brackets using averaging techniques and developing proper algebraic formalisms we study the limiting process of ordinary differential equations with highly oscillatory right hand sides we give sufficient conditions generalizing earlier work by kurzweil and jarnik for a sequence u j u 1 j ldots u m j subseteq l 1 0 t bbb r m to be such that for every choice of smooth vector fields f k k 1 ldots m on a smooth manifold the trajectories of dot x sum k 1 mu k j t f k x converge to the trajectories of an extended system dot x sum k 1 r v k t f k x where the new directions f m 1 ldots f r are lie brackets of f 1 ldots f m

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

on the curves that may be approached by trajectories of a smooth control affine system in this paper we give a characterization of the set of curves that may be approached by trajectories of a smooth control affine nonlinear system in the topology of uniform convergence this characterization is in terms of the drift vector field and the distribution spanned by the lie algebra generated by the control vector fields the characterization is valid on open sets where this distribution has constant rank these results also characterize the support of diffusion processes with smooth coefficients

analysis of chemical kinetic systems over the entire parameter space iii a wet combustion system a simple generalization of semenov thermal explosion theory is studied here one exothermic reaction taking place at the same time as a second one involving water as a reactant evaporation and condensation are also considered the resulting two ordinary differential equations containing three parameters are treated by recently developed methods outlined in the first paper of this series this model is a simple representation of the combustion of wet cellulosic materials such as bagasse cotton and sawdust the theory predicts an amazing variety of behaviour arising in association with the degenerate singularities that occur these include degeneracies up to and including the quartic fold h3 2 bifurcation and degenerate double zero eigenvalue points

extremum seeking control with an adaptive dither signal we present an output feedback law that steers a fully actuated control affine system with general drift vector field to a minimum of the output function the proposed control law is based on a suitable approximation of descent directions by lie brackets in the high frequency limit a novelty of our approach is an adaptive choice of the frequency parameter in this way the problem of determining a sufficiently large frequency parameter becomes obsolete the adaptive choice of the frequency parameter also prevents finite escape times in the presence of a drift the proposed control law does not only lead to convergence into a neighborhood of the set of minima but leads to exact convergence for the case of an output function with a global minimum and no other stationary points we prove global convergence

pre control form of the generalized campbell baker hausdorff dynkin formula for affine nonholonomic systems abstract in this paper a pre control form of the generalized campbell baker hausdorff dynkin gcbhd formula is presented this form is dedicated to nonholonomic affine systems often encountered in robotics the simplest possible expression for the pre control gcbhd is obtained as control dependent functions pre multiply elements of the ph hall basis algorithmic aspects of deriving the formula are highlighted an application of the pre control form of the gcbhd formula in motion planning of nonholonomic systems is also reported

abnormal minimizers this paper constructs the first example of a singular abnormal minimizer for the lagrange problem with linear velocity constraints and quadratic definite lagrangian or equivalently for an optimal control system of linear controls with k controls n states and a running cost function that is quadratic positive definite in the controls in the example k 2 n 3 and the system is completely controllable the example is stable if both the control law and cost are perturbed the singular minimizer persists its importance is due in part to the fact that it is a counterexample to a theorem that has appeared several times in the differential geometry literature there the problem is called the problem of finding minimizing sub riemannian geodesics and it has been claimed that all minimizers are normal pontryagin extremals the mathematical theory of optimal processes wiley interscience new york 1962 if the number of states equals the number of controls then the problem is that of finding riemannian geodesics the main difficulty is proving minimality to do this the length cost of the abnormal is compared with all competing normal extremals a detailed asymptotic analysis of the differential equations governing the normals shows that they are all longer

a finite difference method for the numerical solution of the sal nikov thermokinetic oscillator problem a finite difference method is developed for solving two coupled ordinary differential equations that model a sequence of chemical reactions the initial value problem is highly nonlinear and involves three parameters various types of theoretical solution of this problem the sal nikov thermokinetic oscillator problem may be found depending on these parameters this is because the stationary point is surrounded by up to two limit cycles the well known first order explicit euler method and an implicit finite difference method of the same order are used to compute the solution it is shown that this implicit method may in fact be used explicitly and extensive numerical experiments are made to confirm the superior stability properties of the alternative method

heat kernel analysis on semi infinite lie groups this paper studies brownian motion and heat kernel measure on a class of infinite dimensional lie groups we prove a cameron martin type quasi invariance theorem for the heat kernel measure and give estimates on the lp norms of the radon nikodym derivatives we also prove that a logarithmic sobolev inequality holds in this setting

phragmen lindelof alternative results for the shallow water equations for transient compressible viscous flow abstract in this paper the spatial behavior of the viscous shallow water equations in a two dimensional channel domain is studied a differential inequality for suitable energy associated with the solutions of the viscous shallow water equations in a semi infinite channel is derived from which we show the phragmen lindelof alternative results the main tool used is the energy method our results can be viewed as a version of saint venant s principle to the transient compressible viscous flow

lie and chronologico algebraic tools for studying stability of time varying systems abstract we will study stability and asymptotic stability for time varying systems described by odes of the form x f e 1 t x where f t x is 1 periodic with respect to t and e 0 is a small parameter since the discovery of stabilizing effect of vibration in the reverse pendulum example there have been a lot of study regarding stability of such systems and design of fast oscillating stabilizing feedback laws in this paper we suggest an approach which is kind of high order averaging procedure based on lie algebraic formalism and the formalism of chronological calculus this latter is a method of asymptotic analysis for flows generated by time variant ode we apply the approach to study stability issues for linear and nonlinear systems in particular we derive conditions of stability for the second and third order linear differential equations with periodic fast oscillating coefficients we study output feedback stabilization of bilinear systems and consider high order averaging procedure for nonlinear systems under homogeneity assumptions at the end we study the problem of stabilization of nonholonomic control linear systems by means of time varying feedbacks

robust point stabilization of underactuated mechanical systems via the extended chained form this paper addresses point stabilization for the extended chained form ecf a control system that may be used to model a number of mechanical underactuated systems a control law is proposed based on well known hybrid open loop feedback techniques which exponentially stabilizes the origin of a dynamic extension of the ecf and ensures a degree of robustness to additive disturbance terms that may represent for instance model uncertainties numerical simulations are included to illustrate the performance of the presented stabilizers

practical stability and stabilization presents a practical stability result for dynamical systems depending on a small parameter this result is applied to a practical stability analysis of fast time varying systems studied in averaging theory and of highly oscillatory systems studied by sussmann and liu 1999 furthermore the problem of practically stabilizing control affine systems with drift is discussed

on modified equations for discretizations of odes the theory of modified equations mes for discretizations of odes is reconsidered obstructions to convergence of series expansions of mes are pinpointed and alternative approaches are presented which provide more accurate descriptions of numerical approximations through mes we emphasize how structural assumptions on the ode can be used to improve estimates then we give arguments for a slightly alternative approach based on time dependent mes which avoids the asymptotic nature traditionally associated with mes some applications of the theory are also provided

a method for the complete qualitative analysis of two coupled ordinary differential equations dependent on three parameters in this paper recent advances in bifurcation theory are specialized to systems describable by two coupled ordinary differential equations odes containing at most three independent parameters for such systems by plotting in the relevant parameter plane the locus of successively degenerate singular points a complete description of all the qualitatively distinct behaviour of the system can be obtained the description is in terms of phase portraits and bifurcation diagrams even though much use is made of existing results obtained via local analyses the results of this technique cover the entire parameter space furthermore because the information is built up in successive stages the question of whether the parameters universally unfold a given degeneracy does not arise this can mean a major saving in effort particularly for degenerate hopf points finally if as is often the case the parameters appear in the system in a simple way the procedure can be applied analytically because the variables which will appear nonlinearly can be used to parametrize the relevant loci

from time ordered products to magnus expansion recent work by lam on the decomposition of time ordered products appearing in the time evolution operator in terms of sums of products of nested commutators is endowed with a recursive algorithm a sufficient condition for absolute convergence is subsequently obtained connection with magnus expansion is established

tail estimates for markovian rough paths we work in the context of markovian rough paths associated to a class of uniformly subelliptic dirichlet forms 26 and prove a better than exponential tail estimate for the accumulated local p variation functional which has been introduced and studied in 17 we comment on the significance of these estimates to a range of currently studied problems including the recent results of ni hao 32 and chevyrev and lyons 18

phragmen lindelof type results for harmonic functions with nonlinear boundary conditions this paper is concerned with investigating the asymptotic behavior of harmonic functions defined on a three dimensional semi infinite cylinder where homogeneous nonlinear boundary conditions are imposed on the lateral surface of the cylinder such problems arise in the theory of steady state heat conduction the classical phragmen lindelof theorem states that harmonic functions which vanish on the lateral surface of the cylinder must either grow exponentially or decay exponentially with distance from the finite end of the cylinder here we show that the results are significantly different when the homogeneous dirichlet boundary condition is replaced by the nonlinear heatloss or heat gain type boundary condition we show that polynomial growth or decay or exponential growth or decay may occur depending on the form of the nonlinearity explicit estimates for the growth or decay rates are obtained

stabilization by artificial wiener processes in this technical note we restage the positive effects of gaussian white noises for continuous time driftless input affine nonholonomic systems compared with previous studies we found that the positive effects are hidden in the shadows of nonuniqueness of stochastic integrals over multi dimensional wiener processes furthermore we found that the positive effects are artificially restaged by numerical calculations with use of euler scheme for ordinary differential equations our results demonstrate that the positive effects act as hidden independent control inputs for nonholonomic systems this implies that our results provide a constructive procedure for designing control inputs such that the origins become p th moment exponentially or finite time stable

homogeneous state feedback stabilization of homogenous systems we show that for any asymptotically controllable homogeneous system in euclidean space not necessarily lipschitz at the origin there exists a homogeneous control lyapunov function and a homogeneous possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system if the system satisfies the usual local lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate if the system satisfies a global lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate the control lyapunov function and the feedback are based on the lyapunov exponents of a suitable auxiliary system and admit a numerical approximation

an asymptotic analysis of the sal nikov thermokinetic oscillator the sal nikov thermokinetic oscillator is studied in the limiting case where the dimensionless heat capacity tends to zero this is equivalent to the no fuel consumption approximation in classical thermal explosion theory and is equally revealing in that many exact results can be obtained by simple algebraic methods regions in parameter space are found where although the system is asymptotically stable a large single excursion occurs before the steady state is approached these regions border the region of oscillations which in the limiting case are of the relaxation type all the interesting behaviour requires rt e frac 1 4 an obvious parallel with thermal explosion theory the unstable limit cycles that occur in the sal nikov oscillator disappear in this limiting case however the requirements for an unstable limit cycle to exist in the relaxation limit are discussed the homoclinic bifurcation in the limiting case is also examined and it is shown that this bifurcation can in theory be calculated exactly in addition an extension to the sal nikov oscillator scheme in a closed system to include fuel consumption is studied both numerically and in a limiting case it is shown that the full scheme exhibits finite trains of almost periodic behaviour before monotonically approaching equilibrium

stabilization of bilinear sparse matrix control systems using periodic inputs abstract in this brief paper using results from the theory of averaging of bilinear systems we provide a graph theoretic characterization for the existence of periodic control inputs that stabilize a sparse matrix system to the origin in particular we introduce a class of extensions to the directed graph corresponding to a given sparse matrix system which when contains a stable sparse matrix system implies that the original system is stabilizable using periodic inputs when this condition holds we provide a systematic procedure for designing such controllers our technical approach combines ideas from the theory of bilinear control systems and averaging theory with graph theoretic results

the inverse problem for rough controlled differential equations we provide a necessary and sufficient condition for a rough control driving a differential equation to be reconstructable to some order from observing the resulting controlled evolution physical examples and applications in stochastic filtering and statistics demonstrate the practical relevance of our result

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

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

continuous dependence with respect to the input of trajectories of control affine systems we study the continuous dependence on the input of trajectories of control affine systems belonging to the class c0 m of all systems sigma of the form sigma dot x f 0 x sum i 1 m u i t f i x where f0 fm are continuous vector fields on some open subset of mathbb r n and the control functions belong to l 1 0 t mathbb r m we give a simple necessary and sufficient condition for a control sequence u j j 1 infty to cal t 0 converge to a control u infty i e to be such that for every system sigma in c0 the trajectories generated by the uj converge as j rightarrow infty to the trajectories generated by u infty we also characterize cal t k convergence the concept of control convergence that arises when we use instead of c0 m the class ck m of systems sigma where the fi are of class ck for k ge 1 in the scalar input case and we explain how the analogous characterization for the multi input case fails to be true unless one restricts oneself to the class c k comm m of systems for which the vector fields f0 fm commute as a preliminary we define a topology of trajectory convergence or t convergence on the set of all time varying vector fields omega times i ni x t mapsto f x t in mathbb r n where omega is an open subset of mathbb r n and i is an interval and we study some of its properties this enables us to make the definition of cal t k convergence precise for sequences and more generally for nets by saying that a net u alpha alpha in a in l 1 0 t mathbb r m cal t k converges to a limit u infty if for every system sigma in c0 m the time varying vector fields x t mapsto f 0 x sum i 1 m u i alpha t f i x cal t k converge to x t mapsto f 0 x sum i 1 m u i infty t f i x

asymptotic expansion of stochastic flows we study the asymptotic expansion in small time of the solution of a stochastic differential equation we obtain a universal and explicit formula in terms of lie brackets and iterated stochastic stratonovich integrals this formula contains the results of doss 6 sussmann 15 fliess and normand cyrot 7 krener and lobry 10 yamato 17 and kunita 11 in the nilpotent case and extends to general diffusions the representation given by ben arous 3 for invariant diffusions on a lie group the main tool is an asymptotic expansion for deterministic ordinary differential equations given by strichartz 14

stochastic expansions and hopf algebras we study solutions to nonlinear stochastic differential systems driven by a multi dimensional wiener process a useful algorithm for strongly simulating such stochastic systems is the castell gaines method which is based on the exponential lie series when the diffusion vector fields commute it has been proved that at low orders this method is more accurate in the mean square error than corresponding stochastic taylor methods however it has also been shown that when the diffusion vector fields do not commute this is not true for strong order one methods here we prove that when there is no drift and the diffusion vector fields do not commute the exponential lie series is usurped by the sinh log series in other words the mean square error associated with a numerical method based on the sinh log series is always smaller than the corresponding stochastic taylor error in fact to all orders our proof uses the underlying hopf algebra structure of these series and a two alphabet associative algebra of shuffle and concatenation operations we illustrate the benefits of the proposed series in numerical studies

thermal stabilization of chemical reactors ii bifurcation analysis of the endex cstr the steady states of the dynamical endex cstr problem are analysed in terms of degenerate bifurcation singularity theory steady state analysis is used in two ways 1 as a device to assist in understanding and predicting complex dynamical behaviour and 2 as a practical design and operational tool by applying the concept of quasi static parameter variation in the bifurcation analysis consideration is given to the effects of thermal mismatching kinetic mismatching and variations in the thermal exchange mass flux and heat loss rate parameters on the structure of the parameter space the qualitative equivalence of the bifurcation structure of the kinetically matched perfectly coupled adiabatic endex scheme and the single reaction adiabatic cstr is demonstrated with reference to the role of the reaction enthalpy effects numerical analysis shows that either kinetic mismatching of the reactions or imperfect heat exchange may introduce hopf bifurcations into the adiabatic system this is a result that is both philosophically and practically important because it shows that limit cycles are not restricted to non adiabatic thermokinetic systems the coefficients of thermal exchange mass flux and heat loss are found to induce distortions of the surface of saddle node bifurcations the limit point shell through codimension 2 bifurcations the steady state and oscillatory state degeneracies are discussed with reference to the design and operational implications for a working system

the origins and limits of thermal steady state multiplicity in the continuous stirred tank reactor in this study of multiplicity in the exothermal continuous stirred tank reactor cstr problem the formalism of singularity theory is extended to bifurcation surfaces and some outstanding questions regarding the structure of the parameter space are answered the multiplicity boundaries of the adiabatic cstr model are shown to form a three dimensional surface an irreducible object which is fundamental to our understanding of this model and its more complex gradientless perturbations extension of the analysis to the diabatic model reveals the need for finite coupling between feed and coolant temperatures in order to achieve a a smooth embedding of the diabatic into the adiabatic parameter space this resolves a paradox previously unreported in the singularity theory treatment of the cstr the existence of which leads to a subtle problem of overdetermination at the theoretical level and prevents the smooth application of singularity theory to reactor design at the applied level it is shown that the highest order singularity the winged cusp is present only for zero or finitely coupled feed and coolant temperatures otherwise it is trapped at infinity the results of this analysis are shown to have particular relevance to practical chemical reactor design and will be applied in forthcoming studies of a novel method of thermal stabilization

efficient strong integrators for linear stochastic systems we present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of wiener processes these schemes are based on the neumann stochastic taylor and magnus expansions first we consider the case when the governing linear diffusion vector fields commute with each other but not with the linear drift vector field we prove that numerical methods based on the magnus expansion are more accurate in the mean square sense than corresponding stochastic taylor integration schemes second we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations consequently for general nonlinear stochastic differential equations with noncommuting vector fields we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order third we consider the consequences in two numerical studies one of which is an application arising in stochastic linear quadratic optimal control

a global steering method for nonholonomic systems abstract in this paper we present an iterative steering algorithm for nonholonomic systems also called driftless control affine systems and we prove its global convergence under the sole assumption that the lie algebraic rank condition larc holds true everywhere that algorithm is an extension of the one introduced in jean et al 2005 21 for regular systems the first novelty here consists in the explicit algebraic construction starting from the original control system of a lifted control system which is regular the second contribution of the paper is an exact motion planning method for nilpotent systems which makes use of sinusoidal control laws and which is a generalization of the algorithm described in murray and sastry 1993 29 for chained form systems

a combinatorial approach to the generalized baker campbell hausdorff dynkin formula abstract the aim of this paper is to obtain an explicit expression instead of using a recursive method for the nth term coefficient of the generalized baker campbell hausdorff dynkin gbchd formula the gbchd formula has been applied to control theory specially to nonholonomic motion planning

lie algebras associated with the exponential solutions of nonautonomous linear differential equations a new free lie algebra structure generated by a continuous set of operators is introduced the friedrichs criterion which is known for a finite countable number of generators is extended to the continuous case based on friedrichs criterion we obtain an explicit expression for the coefficients of the magnus expansion which is central for one to get approximately the exponential solutions of nonautonomous linear differential equations

recursion formulas for the lie integral abstract an explicit definition is given of a non linear integral which is termed the lie integral this integral is defined for lie algebra valued functions in terms of the representations of the algebra various properties of the integral are investigated and applications are given to differential equations applications to differential geometry will be given elsewhere

spatial decay bounds in time dependent pipe flow of an incompressible viscous fluid in this paper the authors investigate the flow of an incompressible viscous fluid in a semi infinite cylindrical pipe if the net entrance flow is nonzero then the fluid velocity will not tend to zero as the distance from the entrance end tends to infinity when the fluid adheres at the cylinder wall and the fluid is initially at rest assuming that the entrance velocity data are small enough and that the fluid flow converges to a laminar flow as the distance down the pipe tends to infinity it is shown that the convergence in energy measure is at least exponential

algebraic structures and stochastic differential equations driven by levy processes we construct an efficient integrator for stochastic differential systems driven by lvy processes an efficient integrator is a strong approximation that is more accurate than the corresponding stoc

brownian chen series and atiyah singer theorem the purpose of this work is to give a new and short proof of the atiyah singer local index theorem for the dirac operator on the spin bundle this proof is obtained by using heat semigroups approximations based on the truncation of brownian chen series

analysis and design of oscillatory control systems this paper presents analysis and design results for control systems subject to oscillatory inputs i e inputs of large amplitude and high frequency the key analysis results are a series expansion characterizing the averaged system and various lie algebraic conditions that guarantee the series can be summed various example systems provide insight into the results with regards to design we recover and extend a variety of point stabilization and trajectory tracking results using oscillatory controls we present novel developments on stabilization of systems with positive trace and on tracking for second order underactuated systems

a note on the notion of geometric rough paths we use simple sub riemannian techniques to prove that every weak geometric p rough path a geometric p rough path in the sense of 20 is the limit in sup norm of a sequence of canonically lifted smooth paths uniformly bounded in p variation thus clarifying the two different definitions of a geometric p rough path our proofs are sufficiently general to include the case of holder and modulus type regularity this allows us to extend a few classical results on holder spaces and p variation spaces to the non commutative setting necessary for the theory of rough paths as an application we give a precise description of the support of enhanced fractional brownian motion and prove a conjecture by ledoux et al

the exponential lie series for continuous semimartingales we consider stochastic differential systems driven by continuous semimartingales and governed by non commuting vector fields we prove that the logarithm of the flowmap is an exponential lie series this relies on a natural change of basis to vector fields for the associated quadratic covariation processes analogous to stratonovich corrections the flowmap can then be expanded as a series in compositional powers of vector fields and the logaritm of the flowmap can thus be expanded in the lie algebra of vector fields further we give a direct self contained proof of the corresponding chen strichartz formula which provides an explicit formula for the lie series coefficients such exponential lie series are important in the development of strong lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves