directional stability radius a stability analysis tool for uncertain polynomial systems coefficients of characteristic polynomials for stable parametrically uncertain systems are allowed to perturb to some extent for stability stability radius is a useful tool to assess the allowance of the stability for the systems to enhance its usefulness we modify stability radius so that it takes into account of given restricted perturbations which we call directional stability radius for an application we show shifted hurwitz stability conditions and a stability analysis method for interval polynomial systems using the directional stability radii

staticization its dynamic program and solution propagation abstract stationary action formulations of dynamical systems are considered use of stationary action formulations allows one to generate fundamental solutions for classes of two point boundary value problems tpbvps one solves for stationary points of the payoff as a function of inputs rather than minimization maximization a task which is significantly different from that in optimal control problems both a dynamic programming principle dpp and a hamilton jacobi partial differential equation hj pde are obtained for a class of problems subsuming the stationary action formulation although convexity or concavity of the payoff may be lost as one propagates forward stationary points continue to exist and one must be able to use the dpp and or hj pde to solve forward to such time horizons in linear quadratic models this leads to a requirement for propagation of solutions of differential riccati equations past finite escape times such propagation is also required in nonlinear n body problem formulations where the potential is represented via semiconvex duality the dynamic programming tools developed here are applicable

static duality and a stationary action application abstract conservative dynamical systems propagate as stationary points of the action functional using this representation it has previously been demonstrated that one may obtain fundamental solutions for two point boundary value problems for some classes of conservative systems via solution of an associated dynamic program further such a fundamental solution may be represented as a set of solutions of differential riccati equations dres where the solutions may need to be propagated past escape times notions of static duality and stat quad duality are developed where the relationship between the two is loosely analogous to that between convex and semiconvex duality static duality is useful for smooth functionals where one may not be guaranteed of convexity or concavity some simple properties of this duality are examined particularly commutativity application to stationary action is considered which leads to propagation of dres past escape times via propagation of stat quad dual dres

the limits of sinai s simple random walk in random environment we study the sample path asymptotics of a class of recurrent diffusion processes with random potentials including examples of sinai s simple random walk in random environment and brox s diffusion process with brownian potential the main results consist of several integral criteria which completely characterize all the possible levy classes therefore providing a very precise image of the almost sure asymptotic behaviors of these processes

moderate deviation principles for stochastic differential equations with jumps moderate deviation principles for stochastic differential equations driven by a poisson random measure prm in finite and infinite dimensions are obtained proofs are based on a variational representation for expected values of positive functionals of a prm

a class of stability regions for which a kharitonov like theorem holds families of complex polynomials whose coefficients lie within given intervals are discussed in particular the problem of determining if all polynomials in a family have the property that all of their roots lie within a given region is discussed towards this end a notion of a kharitonov region is defined roughly speaking a kharitonov region is a region in the complex plane with the following property given any suitable family of polynomials in order to determine if all polynomials in the family have all of their roots in the region it suffices to check only the vertex polynomials of the family the main result is a sufficient condition for a given region to be a kharitonov region

the principle of least action and fundamental solutions of mass spring and n body two point boundary value problems two point boundary value problems for conservative systems are studied in the context of the least action principle one obtains a fundamental solution whereby two point boundary value problems are converted to initial value problems via an idempotent convolution of the fundamental solution with a cost function related to the terminal data the classical mass spring problem is included as a simple example the n body problem under gravitation is also studied there the least action principle optimal control problem is converted to a differential game where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential solutions are obtained as indexed sets of solutions of riccati equations

on slowdown and speedup of transient random walks in random environment we consider one dimensional random walks in random environment which are transient to the right our main interest is in the study of the sub ballistic regime where at time n the particle is typically at a distance of order o n from the origin kappa in 0 1 we investigate the probabilities of moderate deviations from this behaviour specifically we are interested in quenched and annealed probabilities of slowdown at time n the particle is at a distance of order o n nu 0 from the origin nu 0 in 0 kappa and speedup at time n the particle is at a distance of order n nu 1 from the origin nu 1 in kappa 1 for the current location of the particle and for the hitting times also we study probabilities of backtracking at time n the particle is located around n thus making an unusual excursion to the left for the slowdown our results are valid in the ballistic case as well

a max plus based algorithm for a hamilton jacobi bellman equation of nonlinear filtering the hamilton jacobi bellman hjb equation associated with the robust hinfty filter as well as the mortensen filter is considered these filters employ a model where the disturbances have finite power the hjb equation for the filter information state is a first order equation with a term that is quadratic in the gradient yet the solution operator is linear in the max plus algebra this property is exploited by the development of a numerical algorithm where the effect of the solution operator on a set of basis functions is computed off line the precomputed solutions are stored as vectors of coefficients of the basis functions these coefficients are then used directly in the real time computations

moderate deviations for diffusions with brownian potentials we present precise moderate deviation probabilities in both quenched and annealed settings for a recurrent diffusion process with a brownian potential our method relies on fine tools in stochastic calculus including kotani s lemma and lamperti s representation for exponential functionals in particular our result for quenched moderate deviations is in agreement with a recent theorem of comets and popov probab theory related fields 126 2003 571 609 who studied the corresponding problem for sinai s random walk in random environment

solving two point boundary value problems for a wave equation via the principle of stationary action and optimal control a new approach to solving two point boundary value problems for a wave equation is developed this new approach exploits the principle of stationary action to reformulate and solve such problems in the framework of optimal control in particular an infinite dimensional optimal control problem is posed so that the wave equation dynamics and temporal boundary data of interest are captured via the characteristics of the associated hamiltonian and choice of terminal payoff respectively in order to solve this optimal control problem for any such terminal payoff and hence solve any two point boundary value problem corresponding to the boundary data encapsulated by that terminal payoff a fundamental solution to the optimal control problem is constructed specifically the optimal control problem corresponding to any given terminal payoff can be solved via a max plus convolution of this fundamental solution with the specified terminal payoff crucially the fundamental solution is shown to be a quadratic fun

rates of convergence of diffusions with drifted brownian potentials we are interested in the asymptotic behaviour of a diffusion process with drifted brownian potential the model is a continuous time analogue to the random walk in random environment studied in the classical paper of kesten kozlov and spitzer we not only recover the convergence of the diffusion process which was previously established by kawazu and tanaka but also obtain all the possible convergence rates an interesting feature of our approach is that it shows a clear relationship between drifted brownian potentials and bessel processes

numerical integration of the differential matrix riccati equation two new bernoulli substitution methods for solving the riccati differential equation are tested numerically against direct integration of the riccati equation the chandrasekhar algorithm and the davison maki method on a large set of problems taken from the literature the first of these new methods was developed for the time invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n this method is fast and accurate when the extremal solutions of the associated algebraic riccati equation are well separated the second new method was developed as a means of eliminating the instabilities associated with the davison maki algorithm by using reinitialization at each time step the davison maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time invariant and time dependent problems from the results of our study we conclude that the modified davison maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the chandrasekhar algorithm is better

the most visited site of brownian motion and simple random walk let l t x be the local time at x for brownian motion and for each t let bar v t inf x underline underline 0 l t x vee l t x mathop sup limits y l t y the absolute value of the most visited site for brownian motion up to time t in this paper we prove that v t is transient and obtain upper and lower bounds for the rate of growth of v t the main tools used are the ray knight theorems and william s path decomposition of a diffusion an invariance principle is used to get analogous results for simple random walks we also obtain a law of the iterated logarithm for v t

a one dimensional diffusion process in a wiener medium we consider a one dimensional diffusion process in a random medium which is itself generated by a wiener process the asymptotic behaviour is investigated using the self similarity of the medium process the results are analogous to the lattice case

annealed tail estimates for a brownian motion in a drifted brownian potential we study brownian motion in a drifted brownian potential kawazu and tanaka j math soc japan 49 1997 189 211 exhibited two speed regimes for this process depending on the drift they supplemented these laws of large numbers by central limit theorems which were recently completed by hu shi and yor trans amer math soc 351 1999 3915 3934 using stochastic calculus we studied large deviations ann probab 29 2001 1173 1204 showing among other results that the rate function in the annealed setting that is after averaging over the potential has a flat piece in the ballistic regime in this paper we focus on this subexponential regime proving that the probability of deviating below the almost sure speed has a polynomial rate of decay and computing the exponent in this power law this provides the continuous time analogue of what dembo peres and zeitouni proved for the transient random walk in random environment comm math phys 181 1996 667 683 our method takes a completely different route making use of lamperti s representation together with an iteration scheme

a note on the stability of delta operator induced systems points out that stability of delta operator induced real polynomials is equivalent to a shifted hurwitz stability of their reciprocal counterparts this fact reveals a simple and clear link between the delta domain stability and hurwitz stability for real polynomials

limit law for transition probabilities and moderate deviations for sinai s random walk in random environment we consider a one dimensional random walk in random environment in the sinai s regime our main result is that logarithms of the transition probabilities after a suitable rescaling converge in distribution as time tends to infinity to some functional of the brownian motion we compute the law of this functional when the initial and final points agree also among other things we estimate the probability of being at time t at distance at least z from the initial position when z is larger than ln2t but still of logarithmic order in time

large deviations for a brownian motion in a drifted brownian potential we derive a large deviation principle both quenched and annealed for a one dimensional diffusion process in a drifted brownian environment providing the continuous time analogue of what comets gantert and zeitouni recently establish for the random walk in random environment a key ingredient kotani s lemma allows us to compute the corresponding rate functions the results are more explicit than in the discrete time setting

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

random walkers in one dimensional random environments exact renormalization group analysis sinai s model of diffusion in one dimension with random local bias is studied by a real space renormalization group which yields exact results at long times the effects of an additional small uniform bias force are also studied we obtain analytically the scaling form of the distribution of the position x t of a particle the probability of it not returning to the origin and the distributions of first passage times in an infinite sample as well as in the presence of a boundary and in a finite but large sample we compute the distribution of the meeting time of two particles in the same environment we also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward these quantities obey multifractal scaling characterized by generalized persistence exponents ensuremath theta g which we compute in the presence of a small bias the number of returns to the origin becomes finite characterized by a universal scaling function which we obtain the full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained as well as detailed analytical information about correlations between directions and times of successive jumps the two time distribution of the positions of a particle x t and x t ensuremath with tg t ensuremath is also computed exactly it is found to exhibit aging with several time regimes characterized by different behaviors in the unbiased case for t ensuremath t ensuremath ensuremath sim t ensuremath ensuremath alpha with ensuremath alpha g1 it exhibits a mathrm ln t mathrm ln t ensuremath scaling with a singularity at coinciding rescaled positions x t x t ensuremath this singularity is a novel feature and corresponds to particles that remain in a renormalized valley for closer times ensuremath alpha l1 the two time diffusion front exhibits a quasiequilibrium regime with a mathrm ln t ensuremath t ensuremath mathrm ln t ensuremath behavior which we compute the crossover to a t t ensuremath aging form in the presence of a small bias is also obtained analytically rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables are also characterized in detail connections with the green function of a one dimensional schr odinger problem and quantum spin chains are discussed