a stochastic version of center manifold theory random dynamical systems arise naturally if the influence of white or real noise on the parameters of a nonlinear determinstic dynamical system is studied in this situation lyapunov exponents attached to the linearized flow replace the real parts of the eigenvalues and describe the stability behavior of the linear system if at least one of them vanishes then it is possible to prove the existence of a stochastic analogue of the deterministic center manifold the asymptotic behavior of the entire system can then be derived from the lower dimensional system restricted to this stochastic center manifold a dynamical characterization of the stochastic center manifold is given and approximation results are proved

a variational approach to the polynomial optimization of multivariabie control with coloured measurement noise polynomial optimization has been applied to multivariabie control in the presence of stochastic inputs including coloured measurement noise the method has been derived using the calculus of variations which is thought to be a simpler formulation than completing squares both discrete time and continuous time systems are considered

entropy formula for random transformations we exhibit random strange attractors with random sinai bowen ruelle measures for the composition of independent random diffeomorphisms

random traveling wave and bifurcations of asymptotic behaviors in the stochastic kpp equation driven by dual noises abstract in this paper we study the influences of dually environmental noises on the traveling wave which develops from the deterministic kpp equation we prove that if the strengths of noises satisfy some condition the solution of the stochastic kpp equation with heaviside initial condition develops a random traveling wave whose wave speed is deterministic and depends on the strengths of noises if the strengths of noises satisfy some other conditions the solution tends to zero as time tends to infinity therefore there exist bifurcations of asymptotic behaviors of solution induced by the strengths of dual noises

an impact of noise on invariant manifolds in nonlinear dynamical systems invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems however these nonlinear systems are often subject to random fluctuations or noises it is thus desirable to quantify the impact of noises on the invariant manifolds when the noise intensity is small this impact is estimated via asymptotic analysis in the context of liapunov perron formulation namely the random invariant manifold is represented as a perturbation of the deterministic invariant manifold with a well defined bound for the deviation

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

optimal multivariable lqg control using a single diophantine equation the general solution of the optimal lqg problem using the polynomial equation approach involves the solution of a couple of two sided matrix polynomial equations the conditions under which a single one sided equation which is related to the original two the implied diophantine equation yields the unique optimal controller are derived in this paper the result has considerable relevance to aspects such as computation and self tuning control applications

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

polynomial optimization of stochastic continuous time control for unstable plants the matrix polynomial method of optimization of linear multivariable continuous time control for unstable plants with stationary stochastic inputs is derived in a constructive manner new sufficient conditions are included but certain extra conditions imposed previously for continuous time optimization are shown to be unnecessary kucera s second diophantine equation is derived directly from the requirement that the system be stable as in discrete time the second equation need not be used in the optimization when a certain coprimeness exists

optimal multivariable regulation with disturbance measurement feedforward abstract the problem of optimal stochastic regulation based on output feedback and disturbance measurement feedforward for multivariable systems is solved using the polynomial equation approach two independent couples of polynomial matrix equations are derived which yield the feedback and the feedforward parts of the regulator

polynomial optimization of stochastic discrete time control for unstable plants the matrix polynomial method of optimization of linear multivariable discrete time control for unstable plants with stationary stochastic inputs is derived in a constructive manner new sufficient conditions are obtained and kucera s second diophantine equation is found directly from the requirement that the system be stable although the second equation must be satisfied it is proved that it need not be used in the optimization when a certain coprimeness exists

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

wong zakai approximations and center manifolds of stochastic differential equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of the stochastic differential equation driven by a l dimensional brownian motion we prove that the solutions of wong zakai approximations converge in the mean square to the solutions of the stratonovich stochastic differential equation we also show that for a simple multiplicative noise the center manifold of the wong zakai approximations converges to the center manifold of the stratonovich stochastic differential equation

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

on the spatial asymptotic behavior of stochastic flows in euclidean space we study asymptotic growth rates of stochastic flows on r d and their derivatives with respect to the spatial parameter under lipschitz conditions on the local characteristics of the generating semimartingales in a first step these conditions are seen to imply moment inequalities for the flow of the form e sup 0t x 0t y p 1 0 t t in a second step we deduce the growth rates from an integrated version of these moment inequalities using the continuity lemma of garsia rodemich and rumsey we provide two examples to show that our results are sharp

generalized polynomial optimization of stochastic feedback control a purely algebraic polynomial approach is used to optimize the discrete time feedback controller for a multivariable plant subject to stochastic reference and disturbance and coloured output measurement noise a quadratic cost criterion with dynamic weightings is used that includes sensitivity and complementary sensitivity as well as tracking error and control input the method involves the solution of only one diophantine equation

smooth convergence of random center manifolds for spdes in varying phase spaces abstract in this paper we study the limiting behavior of center manifolds for a class of singularly perturbed stochastic partial differential equations in terms of the phase spaces we first prove the existence and smoothness of random center manifolds for these equations then we establish the smooth convergence of these random center manifolds as the phase spaces approach to their singular limit

fractional order theory of thermoelasticity in this work a new theory of thermoelasticity is derived using the methodology of fractional calculus the theories of coupled thermoelasticity and of generalized thermoelasticity with one relaxation time follow as limit cases a uniqueness theorem for this model is proved a variational principle and a reciprocity theorem are derived

synchronization by noise we provide sufficient conditions for synchronization by noise i e under these conditions we prove that weak random attractors for random dynamical systems consist of single random points in the case of sde with additive noise these conditions are also essentially necessary in addition we provide sufficient conditions for the existence of a minimal weak point random attractor consisting of a single random point as a result synchronization by noise is proven for a large class of sde with additive noise in particular we prove that the random attractor for an sde with drift given by a multidimensional double well potential and additive noise consists of a single random point all examples treated in tearne probab theory relat fields 141 1 2 1 18 2008 are also included

technical communique descriptor wiener state estimators based on the modern time series analysis method a new time domain wiener filtering approach is presented asymptotically stable wiener state estimators are presented for discrete linear stochastic descriptor systems they can be implemented via the autoregressive moving average arma recursive filters they can handle the optimal state filtering smoothing and prediction problems in a unified framework and can simply be obtained based on the arma innovation model the solution of the diophantine equations and riccati equations is avoided so that the computational burden is reduced a simulation example shows the effectiveness of the new approach

spatial estimates for stochastic flows in euclidean space we study the behavior for large x of kunita type stochastic flows t x on r d driven by continuous spatial semimartingales for this class of flows we prove new spatial estimates for large x under very mild regularity conditions on the driving semimartingale random field it is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering stochastic functional and partial differential equations some examples and counterexamples are given

polynomial approach to wiener filtering a rigorous derivation is given of the polynomial method for finding wiener filters in both discrete time and continuous time in each case the method involves one spectral factorization and the solution of one diophantine equation the theory is applicable in multivariable situations but simple scalar examples are included here to compare the polynomial approach with the basic wiener method

a variational approach to the polynomial optimization of multivariable control with coloured noise at the controller output a method is derived for the design of an optimal continuous time controller for a multivariable plant in the presence of stochastic inputs which in this case includes coloured noise at the controller output the design processes are polynomial and the derivation is based on the calculus of variations an example illustrates the design process

the wong zakai approximations of invariant manifolds and foliations for stochastic evolution equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of a class of stochastic evolution equations with a multiplicative white noise we prove that the solutions of wong zakai approximations almost surely converge to the solutions of the stratonovich stochastic evolution equation we also show that the invariant manifolds and stable foliations of the wong zakai approximations converge to the invariant manifolds and stable foliations of the stratonovich stochastic evolution equation respectively

lqg multivariable controllers minimum variance interpretation for use in self tuning systems it is first demonstrated that every lqg optimal control problem can be rewritten as a minimum output variance control problem for a related system model this system may be defined in either minimum or non minimum phase forms the optimal controller derived from the equivalent minimum variance problem is identical to the lqg optimal controller during the solution of the equivalent problem an equation is derived which is similar to the implicit relationships used in direct self tuning control schemes this enables the problems in constructing a multivariable implicit lqg self tuning controller using polynomial matrix methods to be explored the inherent difficulties become clear but a possible strategy is evident finally the results are specialized to the scalar case and a very simple implicit lqg self tuning algorithm is described

perfect cocycles through stochastic differential equations we prove that if is a random dynamical system cocycle for whicht t x is a semimartingale then it is generated by a stochastic differential equation driven by a vector field valued semimartingale with stationary increment helix and conversely this relation is succinctly expressed as semimartingale cocycle exp semimartingale helix to implement it we lift stochastic calculus from the traditional one sided time to two sided timet and make this consistent with ergodic theory we also prove a general theorem on the perfection of a crude cocycle thus solving a problem which was open for more than ten years