a necessary condition for feedback stabilization abstract we give a new necessary condition for the existence of a static or dynamic continuous asymptotically stabilizing feedback control for the system x dot f x u we apply our condition to give an example of a system in r 3 which is small time locally controllable satisfies brockett s second and third condition and cannot be locally asymptotically stabilized by employing a static or dynamic continuous feedback law

necessary conditions of asymptotic stability for unilateral dynamical systems in this paper we develop a mathematical tool that can be used to state necessary conditions of asymptotic stability of isolated stationary solutions of a class of unilateral dynamical systems more precisely nonlinear evolution variational inequalities are considered instability criteria are also given applications can be found in mechanics or electrical circuits

mean field approximation and a small parameter in turbulence theory numerical and physical experiments on two dimensional 2d turbulence show that the differences of transverse components of velocity field are well described by gaussian statistics and kolmogorov scaling exponents in this case the dissipation fluctuations are irrelevant in the limit of small viscosity in general one can assume the existence of a critical space dimensionality d d c at which the energy flux and all odd order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant at dl d c the flow can be described by the mean field theory leading to the observed gaussian statistics and kolmogorov scaling of transverse velocity differences it is shown that in the vicinity of d d c the ratio of the relaxation and translation characteristic times decreases to zero thus giving rise to a small parameter of the theory the expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d d c the resulting equation describes experimental data on two dimensional turbulence and demonstrates the onset of intermittency as d ensuremath d c g0 and r stackrel ensuremath rightarrow l 0 in three dimensional flows in close agreement with experimental data in addition some exact relations between correlation functions of velocity differences are derived it is also predicted that the single point probability density function of transverse velocity components in developing as well as in the large scale stabilized two dimensional turbulence is a gaussian

necessary conditions for stability and attractivity of continuous systems considered in this paper are control systems of the form f x u for such systems a number of related necessary conditions for various forms of stability and attractivity are presented the paper starts by showing that brockett s necessary condition for stabilizability via smooth feedback still persists if is continuous and the class of allowable u increased to include continuous feedbacks using similar ideas to those used to prove the continuous brockett result again only assuming continuity of f and u necessary conditions are then derived for global attractivity and for ultimate boundedness

on brockett s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback the necessity of brockett s condition for stabilizability of nonlinear systems by smooth feedback is shown by an argument based on properties of a degree for set valued maps to persist when the class of controls is enlarged to include discontinuous feedback

topological obstructions to submanifold stabilization we consider the problem of local asymptotic feedback stabilization via a continuously differentiable feedback law of a control system x f x u defined in euclidean space rn with f being continuously differentiable to a compact connected oriented m dimensional submanifold m of rn with codimension strictly larger than one we obtain necessary conditions on the topology of m for such a stabilizing feedback law to exist this extends the work done in where only the codimension one case was treated we also briefly discuss the case where the control is only assumed continuous

optimal control of mobile sources for heat conductivity processes abstractwe consider a problem of optimal control of the mobile sources for heat conductivity processes described by parabolic equation and systems of ordinary differential equations sufficient conditions for frechet differentiability of the performance criterion and an expression for its gradient were determined the necessary optimality condition was established in the form of the integral principles of maximum the theoretical conclusions are illustrated by the solution of a numerical example

locality and stability of the cascades of two dimensional turbulence we investigate and clarify the notion of locality as it pertains to the cascades of two dimensional turbulence the mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions first introduced by l vov and procaccia as a point of departure we use a revised version of the system of hypotheses that was proposed by frisch for three dimensional turbulence we show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non perturbative statistical locality we also investigate the stability conditions for both cascades we have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade for the enstrophy cascade statistical stability requires large scale dissipation and a vanishing downscale energy dissipation a careful discussion of the subtle notion of locality is given at the end of the paper

topological properties of asymptotically stable sets topological properties of the domain of attraction for dynamical systems are investigated the main purpose of this paper is to prove that a compact asymptotically stable attractor of a dynamical system defined on a locally compact metric space is a deformation retract of its domain of attraction in a weak sense that is made precise under additional local assumptions the attractor can be shown to be a retract a deformation retract or a strong deformation retract the well known result that the domain of attraction of an asymptotically stable equilibrium is contractible follows as a corollary

pointwise control of burgers equation a numerical approach abstract we describe a methodology for numerically solving pointwise control problems for the viscous burger s equation in one space dimension the solution methods are based on a combination of finite difference and finite element techniques for the time and space discretizations with iterative algorithms such as quasi newton methods and gmres numerical experiments validate the methods

using machine learning to predict catastrophes in dynamical systems nonlinear dynamical systems which include models of the earth s climate financial markets and complex ecosystems often undergo abrupt transitions that lead to radically different behavior the ability to predict such qualitative and potentially disruptive changes is an important problem with far reaching implications even with robust mathematical models predicting such critical transitions prior to their occurrence is extremely difficult in this work we propose a machine learning method to study the parameter space of a complex system where the dynamics is coarsely characterized using topological invariants we show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner we are able to predict with high accuracy the locations of critical transitions in parameter space

a priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control we consider finite element approximations of parabolic control problems with pointwise control the state equation exhibits low regularity due to the control imposed pointwisely this introduces some difficulties for both theoretical and numerical analysis to discretize the optimal control problem we use variational discretization together with piecewise linear and continuous finite elements for the space discretization of the state and we use the backward euler scheme for time discretization we prove a priori error estimates for the control state and adjoint state numerical experiments are provided which support the theoretical results

two dimensional turbulence in the inverse cascade range numerical and physical experiments on forced two dimensional navier stokes equations show that transverse velocity differences are described by normal kolmogorov scaling proportional r 2n 3 and obey gaussian statistics since nontrivial scaling is a sign of the strong nonlinearity of the problem these two results seem to contradict each other a theory explaining these observations is presented in this paper the derived self consistent expression for the pressure gradient contributions leads to the conclusion that small scale transverse velocity differences are governed by a linear langevin like equation stirred by a nonlocal universal solution dependent gaussian random force this explains the experimentally observed gaussian statistics of transverse velocity differences and their kolmogorov scaling the solution for the pdf of longitudinal velocity differences is based on the numerical smallness of the energy flux in two dimensional turbulence the theory makes a few quantitative predictions that can be tested experimentally

oscillations in multi stable monotone systems with slowly varying feedback the study of dynamics of gene regulatory networks is of increasing interest in systems biology a useful approach to the study of these complex systems is to view them as decomposed into feedback loops around open loop monotone systems key features of the dynamics of the original system are then deduced from the input output characteristics of the open loop system and the sign of the feedback this paper extends these results showing how to use the same framework of input output systems in order to prove existence of oscillations if the slowly varying strength of the feedback depends on the state of the system

evidence for the double cascade scenario in two dimensional turbulence statistical features of homogeneous isotropic two dimensional turbulence is discussed on the basis of a set of direct numerical simulations up to the unprecedented resolution 32768 2 by forcing the system at intermediate scales narrow but clear inertial ranges develop both for the inverse and for direct cascades where the two kolmogorov laws for structure functions are simultaneously observed the inverse cascade spectrum is found to be consistent with kolmogorov kraichnan prediction and is robust with respect the presence of an enstrophy flux the direct cascade is found to be more sensible to finite size effects the exponent of the spectrum has a correction with respect theoretical prediction which vanishes by increasing the resolution

global dynamics of neural nets with infinite gain abstract we consider a model of neural and gene networks where the nonlinearities in the system of differential equations are discontinuous and piecewise constant we develop a framework for study of such systems as a first step we associate to the system a graph g on a hypercube and show how the collection of strongly connected components of g relates to the dynamics of the flow on the set of rays through the origin in the second step we discuss the relationship between the invariant sets of the ray flow and the invariant sets of the original flow we provide a sufficient condition for a one to one correspondence between these sets finally we study the class of binary networks within this framework under certain conditions we can determine the structure of an invariant set corresponding to the lowest strongly connected component of the hypercube graph

optimal pointwise control of semilinear parabolic equations we study optimal control problems for semilinear parabolic equations with pointwise controls in a bounded domain of r when the nonlinear term in the state equation is of the form y y we prove the existence of solutions for such equations when 1 n n 2 we next study a control problem with a terminal observation we prove existence of optimal controls and a pontryagin principle for these problems

connected simple systems and the conley index of isolated invariant sets the object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows simple explicit formulae are derived for the maps which play a central role in the theory the presentation is self contained

turbulence without pressure we develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible we find explicit forms of certain probability distributions demonstrate that the breakdown of galilean invariance is responsible for intermittency and establish the operator product expansion we also indicate how the effects of pressure can be turned on perturbatively

a priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time in this paper we consider an optimal control problem which is governed by a linear parabolic equation and is subject to state constraints pointwise in time optimal order error estimates are developed for a space time finite element discretization of this problem numerical examples confirm the theoretical results as a by product of our analysis we derive a new regularity result for the optimal control

local asymptotic feedback stabilization to a submanifold topological conditions abstract we consider the problem of local asymptotic feedback stabilization of a control system x f x u defined in r n to a compact connected oriented embedded codimension one submanifold p of r n using a continuous feedback law this generalizes the problem of local asymptotic feedback stabilization to a point which has been previously considered in the control theory literature it is natural to expect the topology of p to play a role in deciding whether or not local asymptotic stabilization to p of the system x f x u is feasible via continuous feedback and our aim in this paper is precisely to outline necessary conditions on the topology of p for stabilization via continuous feedback to be achievable

measure valued directional sparsity for parabolic optimal control problems a directional sparsity framework allowing for measure valued controls in the spatial direction is proposed for parabolic optimal control problems it allows for controls which are localized in space where the spatial support is independent of time well posedness of the optimal control problems is established and the optimality system is derived it is used to establish structural properties of the minimizer an a priori error analysis for finite element discretization is obtained and numerical results illustrate the effects of the proposed cost functional and the convergence results

zeta functions periodic trajectories and the conley index abstract we use the conley index to produce a simple sufficient condition stated in terms of the betti numbers of the cohomology conley index for the existence of periodic orbits for flows which admit a cross section

conley index condition for asymptotic stability in this paper we use conley index theory to develop necessary conditions for stability of equilibrium and periodic solutions of nonlinear continuous time systems the conley index is a topological generalization of the morse theory which has been developed to analyze dynamical systems using topological methods in particular the conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set the conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set including system stability properties efficient numerical algorithms using homology theory have been developed in the literature to compute the conley index and can be used to deduce the stability properties of nonlinear dynamical systems

closure of two dimensional turbulence the role of pressure gradients inverse energy cascade regime of two dimensional turbulence is investigated by means of high resolution numerical simulations numerical computations of conditional averages of transverse pressure gradient increments are found to be compatible with a recently proposed self consistent gaussian model an analogous low order closure model for the longitudinal pressure gradient is proposed and its validity is numerically examined in this case numerical evidence for the presence of higher order terms in the closure is found the fundamental role of conditional statistics between longitudinal and transverse components is highlighted

optimal a priori error estimates of parabolic optimal control problems with pointwise control in this paper we consider a parabolic optimal control problem with a pointwise dirac type control in space but variable in time in two space dimensions to approximate the problem we use the standard continuous piecewise linear approximation in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we show almost optimal h 2 k convergence rate for the control in l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou a priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control tech report 2011 07 hamburger beitrage zur angewandten mathematik hamburg germany 2011

the conley index for fast slow systems ii multidimensional slow variable we develop a qualitative theory for fast slow systems with a one dimensional slow variable using conley index theory for singularity perturbed systems conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0 these orbits persist when the derivative of the slow variable is small and nonzero

three dimensional forced burgers turbulence supplemented with a continuity equation we investigate turbulent limit of the forced burgers equation supplemented with a continuity equation in three dimensions the scaling exponent of the conditional two point correlation function of density i e r x1 r x2 udu ux12x2u 2a 3 is calculated self consistently in the nonuniversal region from which we obtain a 353 also we derive an equation governing the evolution of the probability density function pdf of longitudinal velocity increments in length scale from which a possible mechanism for the dependence of the inertial pdf to one point urms is developed

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional