control of singularly perturbed hybrid stochastic systems we study a class of optimal stochastic control problems involving two different time scales the fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process under a fundamental ergodicity property for a class of infinitesimal control systems associated with the fast mode we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system both the finite and infinite discounted horizon cases are considered we show how an approximate optimal control law can be constructed from the solution of the limit control problem in the particular case where the infinitesimal control systems possess the so called turnpike property i e characterized by the existence of global attractors the limit control problem can be given an interpretation related to a decomposition approach

on existence of limit occupational measures set of a controlled stochastic differential equation we establish that under certain conditions the set of occupational measures as well as the set of mathematical expectations of occupational measures generated by the admissible controls and the corresponding solutions of a controlled stochastic differential equation csde converge with the time horizon tending to infinity to a set called limit occupational measures set loms and we show that this limit set coincides with the set of stationary marginal distributions of the csde we also demonstrate the applicability of our results for averaging of singularly perturbed csde

multiparameter singular perturbations of linear systems with multiple time scales a joint multitime scale multiparameter singular perturbation is formulated and resolved in the context of linear time varying systems an interesting feature of the solution procedure is a hierarchial scheme of aggregating and arranging of the groups of small parameters according to their order the scheme provides a suitable framework for establishing qualitative properties of multitime scale systems

tracking fast trajectories along a slow dynamics a singular perturbations approach controlled coupled slow and fast motions are examined in a singular perturbations setting the objective is to minimize a cost functional that takes into account both the fast motion supposing say tracking a fast target and the slow dynamics a method is offered to cope with the possibility that the fast flow has nonstationary limits invariant measures of the fast motion are then the controlled objects on the infinitesimal scale optimal amalgamation of them on the slow scale induces the variational limit whose solutions are near optimal solutions of the perturbed system

on discrete algebraic riccati equations a rank characterization of solutions abstract we give a rank characterization of the solution set of discrete algebraic riccati equations dares involving real matrices our characterization involves the use of invariant subspaces of the system matrix we observe that a dare can be considered as a specific type of a non symmetric continuous time algebraic riccati equation we show how to extract the low rank solutions from the full rank solution of a homogeneous dare we also study the eigen structure of various feedback closed loop matrices associated with solutions of a dare we show as a consequence of our observations that a solution x of a dare for controllable systems and certain specific types of uncontrollable systems satisfies the inequality x m i n x x m a x we characterize conditions under which the solution set of dare is bounded unbounded we use the method of invariant subspaces to identify some invariant sets of a difference riccati equation

the empirical distribution of the fourier coefficients of a sequence of independent identically distributed long tailed random variables suppose x1 x2 are independent identically distributed random variables and suppose n 1 x1 xn converges in distribution to a symmetric stable law of index 2 for s 1 n set text y ns n 1 alpha sum limits j 1 n x j text cos 2 pi js n text let n be the empirical distribution of yns s 1 n then n converges in distribution but not in probability

cramer type moderate deviation for the maximum of the periodogram with application to simultaneous tests in gene expression time series where 1 j q and q n 1 2 the periodogram is a fundamental tool in spectral analysis and is often used to detect periodic patterns in various real applications such as the analysis of gene expression data and the study of earthquake theoretical properties of the periodogram have been extensively studied an chen and hannan 1983 obtained the logarithm law for the maximum of the periodogram davis and mikosch 1999 mikosch resnick and samorodnitsky 2000 lin and liu 2009a obtained the asymptotic distribution for the maximum of the periodogram under the i i d and linear process cases the heavy tailed case and nonlinear time series case respectively fay and soulier 2001 proved the central limit theorem for functionals of the periodogram shao and wu 2007 obtained asymptotic distributions for the periodogram and the empirical distribution function of the periodogram for a wide class of nonlinear processes when xn are independent and identically distributed i i d random variables with var x1 2 and e x1 2 0 davis and mikosch 1999 show that

a distributional version of functional equations and their stabilities abstract generalizing the results in j math anal appl 286 2003 177 186 j math anal appl 295 2004 107 114 arch math to appear j math anal appl 299 2004 578 586 that consider the hyers ulam stability problems of several functional equations in the spaces of the schwartz tempered distributions and the fourier hyperfunctions we consider the stability problems of the functional equations in the space of distributions

on a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems a representation of the limit occupational measures set of a control system in terms of the vector function defining the system s dynamics is established applications in averaging of singularly perturbed control systems are demonstrated

singular perturbation of a finite horizon problem with state space constraints we study the singular perturbation of optimal control problems for nonlinear systems with constraints on the fast state variables and a cost functional either of bolza type or involving the exit time of the system from a given domain under a controllability assumption on the fast variables we show that these variables become controls in the limit problem our method consists of passing to the limit in the associated hamilton jacobi bellman hjb equations by means of some tools in the theory of viscosity solutions

multiscale singularly perturbed control systems limit occupational measures sets and averaging an averaging technique for nonlinear multiscale singularly perturbed control systems is developed issues concerning the existence and structure of limit occupational measures sets generated by such systems are discussed general results are illustrated with special cases

tracking periodic signals with the overtaking criterion the overtaking criterion is suggested to evaluate the tracking of periodic signals on an infinite horizon existence uniqueness and a feedback form are established the asymptotic dynamics of the solution and its relation to the asymptotic performance of the finite time optimizers are examined

singularly perturbed ordinary differential equations with dynamic limits coupled slow and fast motions generated by ordinary differential equations are examined the qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after invariant measures of the parametrised fast flow are employed to describe the limit behaviour rather than algebraic equations which are used in the standard reduced order approach in the case of a unique invariant measure for each parameter the limit of the slow motion is governed by a chattering type equation without the uniqueness the limit of the slow motion solves a differential inclusion the fast flow in turn converges in a statistical sense to the direct integral respectively the set valued direct integral of the invariant measures

convergence to convex compact sets in infinite dimensions we provide a simple test for the convergence of a sequence of subsets in an infinite dimensional space to a compact convex one the test reduces roughly the convergence problem to finite dimensions applications are portrayed demonstrating how convexification phenomena established in finite dimensions are easily generalized to infinite dimensions

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

on the construction of asymptoticallly optimal controls for singularly perturbed systems we construct asymptotically optimal controls for singularly perturbed nonlinear systems where both the slow and the fast motions are controlled for this purpose we give necessary optimality conditions for an averaged system for the slow motion

averaging method for one sided lipschitz differential inclusions with generalized solutions we investigate the convergence of the solution set for singularly perturbed differential inclusions using an averaging technique the limits of the fast solutions are considered as generalized solutions i e radon probability measures of the degenerate system exploiting in addition the one sided lipschitz condition we prove the existence of a limit of the solution set in an appropriate topology

stability of exponential equations in schwartz distributions abstract we reformulate the superstability of exponential equation and cosine functional equation j a baker the stability of cosine equation proc amer math soc 80 1980 411 416 in some spaces of generalized functions such as the schwartz distributions sato hyperfunctions and gelfand generalized functions which completes the previous results of partial generalizations of the stability problems j chung a distributional version of functional equations and their stabilities nonlinear anal 62 2005 1037 1051 j chung s y chung d kim the stability of cauchy equations in the space of schwartz distributions j math anal appl 295 2004 107 114

linear programming approach to deterministic long run average problems of optimal control we establish that deterministic long run average problems of optimal control are asymptotically equivalent to infinite dimensional linear programming problems lpps and we establish that these lpps can be approximated by finite dimensional lpps the solutions of which can be used for construction of the optimal controls general results are illustrated with numerical examples

singular perturbations in ergodic control of diffusions ergodic control of a nondegenerate diffusion with two time scales is studied in the limiting case as the time scale separation increases to infinity it is shown that the limit problem is another ergodic control problem for the slow time scale component alone with its dynamics averaged over the controlled invariant probability measures for the fast component these measures in turn can be treated as the effective control variable

the maximum of the periodogram of a non gaussian sequence it is a well known fact that the periodogram ordinates of an lid mean zero gaussian sequence at the fourier frequencies constitute an lid exponential vector hence the maximum of these periodogram ordinates has a limiting gumbel distribution we show for a non gaussian lid mean zero finite variance sequence that this statement remains valid we also prove that the point process constructed from the periodogram ordinates converges to a poisson process this implies the joint weak convergence of the upper order statistics of the periodogram ordinates these results are in agreement with the empirically observed phenomenon that various functionals of the periodogram ordinates of an lid finite variance sequence have very much the same asymptotic behavior as the same functionals applied to an lid exponential sample

a generalized negative imaginary lemma and riccati based static state feedback negative imaginary synthesis abstract in this paper we present a generalized negative imaginary lemma based on a generalized negative imaginary system definition then an algebraic riccati equation method is given to determine if a system is negative imaginary also a state feedback control procedure is presented that stabilizes an uncertain system and leads to the satisfaction of the negative imaginary property the controller synthesis procedure is based on the proposed negative imaginary lemma using this procedure the closed loop system can be guaranteed to be robustly stable against any strict negative imaginary uncertainty such as in the case of unmodeled spill over dynamics in a lightly damped flexible structure a numerical example is presented to illustrate the usefulness of the results

viscosity solutions methods for singular perturbations in deterministic and stochastic control viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order degenerate parabolic hamilton jacobi bellman equations this characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions the results apply to cases where the usual order reduction method does not give the correct limit and to systems with fast state variables depending nonlinearly on the control some connections with ergodic control and periodic homogenization are discussed

characterization of the optimal trajectories for the averaged dynamics associated to singularly perturbed control systems abstract the aim of this paper is to study singularly perturbed control systems firstly we provide linearized formulation version for the calculus of the value function associated with the averaged dynamics secondly we obtain necessary and sufficient conditions in order to identify the optimal trajectory of the averaged system

the maximum of the periodogram for a heavy tailed sequence we consider the maximum of the periodogram based on an infinite variance heavy tailed sequence for alpha 1 we show that the maxima constitute a weakly convergent sequence and find its limiting distribution for 1 alpha 2 we show that the sequence of the maxima is not tight and find a normalization that makes it tight

stability of functional equations in the spaces of distributions and hyperfunctions making use of the fundamental solution of the heat equation we prove the stability theorems of quadratic functional equation and d alembert equation in the spaces of schwartz distributions and sato hyperfunctions

the stability of cauchy equations in the space of schwartz distributions abstract we reformulate the stability theorem of d h hyers in the schwartz tempered distributions and prove that every additive tempered distribution can be approximated by a linear function we also consider the superstability of the cauchy equation f x y f x f y

contiguity of the whittle measure for a gaussian time series for a stationary time series whittle constructed a likelihood for the spectral density based on the approximate independence of the discrete fourier transforms of the data at certain frequencies whittle s likelihood has been widely used in the literature for con structing estimators in this paper we show that for a gaussian time series the whittle measure is mutually contiguous with the actual distribution of the data as a consequence most asymptotic properties of estimators and test statistics derived under the whittle measure can be carried over to the actual distribution

discontinuous control problems with state constraints linear formulations and dynamic programming principles abstract this paper aims at studying a class of discontinuous deterministic control problems under state constraints using a linear programming approach as for classical control problems gaitsgory and quincampoix 2009 16 goreac and serea 2011 19 the primal linear problem is stated on some appropriate space of probability measures naturally the support of these measures is contained in the set of constraints this linearized value function and its dual can alternatively be seen as the limit of standard penalized problems second we provide a semigroup property for this set of probability measures leading to dynamic programming principles for control problems under state constraints an abstract principle is provided for general bounded costs linearized versions are obtained under further semi continuity assumptions

a frequency domain bootstrap for ratio statistics in time series analysis the asymptotic properties of the bootstrap in the frequency domain based on studentized periodogram ordinates are studied it is proved that this bootstrap approximation is valid for ratio statistics such as autocorrelations by using edgeworth expansions it is shown that the bootstrap approximation even outperforms the normal approximation the results carry over to whittle estimates in a simulation study the behavior of the bootstrap is studied for empirical correlations and whittle estimates

suboptimal control of singularly perturbed systems and periodic optimization a traditional approach to singularly perturbed optimal control problems is based on an approximation of these problems by reduced problems which are obtained via the formal replacement of the fast variables by the states of equilibrium of the fast subsystems considered with frozen slow variables and controls it is shown that such an approximation is valid if and only if certain families of periodic optimization problems admit steady state solutions it is also shown how the solutions of these problems can be used to construct suboptimal controls for singularly perturbed problems when approximation by reduced problems is not possible

duality in linear programming problems related to deterministic long run average problems of optimal control it has been established recently that under mild conditions deterministic long run average problems of optimal control are asymptotically equivalent to infinite dimensional linear programming problems lpps and that these lpps can be approximated by finite dimensional lpps in this paper we introduce the corresponding infinite and finite dimensional dual problems and study duality relationships we also investigate the possibility of using solutions of finite dimensional lpps and their duals for numerical construction of the optimal controls in periodic optimization problems the construction is illustrated with a numerical example

averaging of three time scale singularly perturbed control systems abstract we investigate the asymptotic properties of singularly perturbed control systems with three time scales we apply the averaging method to construct a limiting system for the slowest motion in the form of a differential inclusion sufficient conditions for the uniform convergence of the slowest trajectories are given

limits of singularly perturbed control problems with statistical dynamics of fast motions we describe the limit behavior of admissible trajectories in a singularly perturbed control system as the small parameter tends to zero a general case is considered where in the limit the fast motion may infinitely rapidly oscillate in time invariant measures of the parameterized fast flow are employed to describe the limit behavior and construct the limit control problem the notion of relatively slow controls is introduced approximating properties of the limit problem within the families of relatively slow controls are verified the results are illustrated by examples

suboptimization of singularly perturbed control systems a technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems the technique is applicable in particular in the case when the optima

the problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures abstract we obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems in order to describe the value functions we employ an infinite dimensional dual formulation instead of using the characterization via hamilton jacobi partial differential equations in this paper we consider control problems with both infinite and finite horizons the reflection is given by the normal cone to a proximal retract set

on the approximation of the boundary layers for the controllability problem of nonlinear singularly perturbed systems abstract a new systematic approach to the construction of approximate solutions to a class of nonlinear singularly perturbed feedback control systems using the boundary layer functions especially with regard to the possible occurrence of the boundary layers is proposed for example problems with feedback control such as the steady states of the thermostats where the controllers add or remove heat depending upon the temperature registered in another place of the heated bar can be interpreted with a second order ordinary differential equation subject to a nonlocal three point boundary condition the o accurate approximation of behavior of these nonlinear systems in terms of the exponentially small boundary layer functions is given at the end of this paper we formulate the unsolved controllability problem for nonlinear systems

on bootstrapping kernel spectral estimates an approach to bootstrapping kernel spectral density estimates is described which is based on resampling from the periodogram of the original data we show that it is asymptotically valid under suitable conditions and we illustrate its performance for a medium sized time series sample with a small simulation study

limit occupational measures set for a control system and averaging of singularly perturbed control systems abstract we discuss conditions for the existence of the limit occupational measures set for a control system we approximate slow components of the trajectories of a singularly perturbed control system by the solutions of a differential inclusion the differential inclusion is obtained via averaging the slow subsystem over measures from the limit occupational measures set constructed for the associated system describing the fast motions with frozen slow ones

optimal programs on infinite horizon 2 we consider the limit behavior as n to infty of the expression sum i 0 n 1 v x i x i 1 in a compact metric space k where v is a real valued function defined on k times k we study the structure of v weakly optimal programs and establish a turnpike theorem for a generic continuous function v we also prove the existence of an almost periodic v optimal program for a generic continuous function v

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods