lasalle type theorem and its applications to infinite horizon optimal control of discrete time nonlinear stochastic systems based on discrete martingale theory the lasalle type theorem for general discrete time stochastic systems is obtained and the almost sure stability is in turn discussed as applications infinite horizon nonlinear optimal regulator is investigated and a dynamical programming equation called the hamilton jacobi bellman equation is also derived for discrete time nonlinear stochastic optimal control

brief paper robustness of exponential stability of a class of stochastic functional differential equations with infinite delay we regard the stochastic functional differential equation with infinite delay dx t f x t dt g x t dw t as the result of the effects of stochastic perturbation to the deterministic functional differential equation email protected t f x t where x t x t q c 0 r n is defined by x t q x email protected email protected 0 we assume that the deterministic system with infinite delay is exponentially stable in this paper we shall characterize how much the stochastic perturbation can bear such that the corresponding stochastic functional differential system still remains exponentially stable

renormalization and blow up for charge one equivariant critical wave maps we prove the existence of equivariant finite time blow up solutions for the wave map problem from 2 1 s 2 of the form u t r q lambda t r mathcal r t r where u is the polar angle on the sphere q r 2 arctan r is the ground state harmonic map t t 1 and mathcal r t r is a radiative error with local energy going to zero as t 0 the number nu frac 1 2 can be prescribed arbitrarily this is accomplished by first renormalizing the blow up profile followed by a perturbative analysis

existence uniqueness and continuation theorems for stochastic functional differential equations abstract the main aim of this paper is to develop some basic theories of stochastic functional differential equations sfdes firstly we establish stochastic versions of the well known picard local existence uniqueness theorem given by driver and continuation theorems given by hale and driver for functional differential equations fdes then we extend the global existence uniqueness theorems of wintner for ordinary differential equations odes driver for fdes and taniguchi for stochastic ordinary differential equations sodes to sfdes these show clearly the power of our new results

power flow management of microgrid networks using model predictive control in this paper we present a power flow management method for a network of cooperating microgrids within the context of a smart grid by formulating the problem in a model predictive control framework in order to reliably and economically provide the required power to the costumers the proposed method enables the network of microgrids to share the power generated from their renewable energy sources and minimize the power needed from the micro gas turbines to corroborate the viability of the proposed method we will illustrate simulation results on a model consisting of three microgrids in a network

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

on asymptotic convergence and boundedness of stochastic systems with time delay analyzing the long term behavior of a nonlinear dynamical system is an important but challenging task especially in the presence of noise and delays this paper investigates asymptotic convergence and boundedness properties of stochastic systems with time delay we are particularly interested in asymptotic convergence that may not be exponential as observed in many practical applications general criteria for checking both moment and almost sure asymptotic convergence and boundedness are established such criteria are mainly motivated by stochastic systems with time varying coefficients and multiple delays and or different orders of nonlinearities as shown by several examples the existing results cannot be applied to analyze such systems thus we generalize the existing theory by allowing the diffusion operator associated with a lyapunov function to satisfy a weaker assumption which involves time varying coefficients and several auxiliary functions to cope with possible multiple delays and or different orders of nonlinearities the results presented in this paper provide an effective tool for checking asymptotic convergence and boundedness properties of general stochastic nonlinear systems with time delay

stochastic versions of the lasalle theorem the main aim of this paper is to establish stochastic versions of the well known lasalle stability theorem from these stochastic versions follow many classical results on stochastic stability this shows clearly the power of our new results

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

existence and stability of a solution blowing up on a sphere for an l 2 supercritical nonlinear schrodinger equation we consider the quintic two dimensional focusing nonlinear schrodinger equation iut u u 4u which is l2 supercritical even though the existence of finite time blow up solutions in the energy space h1 is known very little is understood concerning the singularity formation numerics suggest the existence of a stable blow up dynamic corresponding to a self similar blowup at one point in space we prove the existence of a different type of dynamic and exhibit an open set among the h1 radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere this is the first result of an explicit description of a blow up dynamic in the l2 supercritical setting

the improved lasalle type theorems for stochastic functional differential equations abstract the main aim of this paper is to improve some results obtained by mao x mao the lasalle type theorems for stochastic functional differential equations nonlinear stud 7 2000 307 328 our new theorems give better results while conditions imposed are much weaker than in the paper mentioned above for example we need only the local lipschitz condition but neither the linear growth condition nor the bounded moment condition on the solutions to guarantee the existence and uniqueness of the global solution to the underlying stochastic functional differential equation sfde under the weaker conditions imposed in this paper we establish a generalised existence and uniqueness theorem which covers a wider class of nonlinear sfdes as demonstrated by the examples discussed in this paper moreover from our improved results follow some new criteria on the stochastic asymptotic stability for sfdes

construction of solutions with exactly k blow up points for the schrodinger equation with critical nonlinearity we consider the nonlinear schrodinger equation i partial u mathord left vphantom i partial u partial t right kern nulldelimiterspace partial t delta u left u right 4 mathord left vphantom 4 n right kern nulldelimiterspace n uandu left 0 right varphi left right 1 whereu 0 t n for any given pointsx1 x2 x k in n we construct a solution of eq 1 u t which blows up in a finite timet at exactlyx1 x2 x k in addition we describe the precise behavior of the solutionu t whent t at the blow up points x1 x2 x k and in n x1 x2 x k

further results on global adaptive stabilisation for a class of uncertain stochastic nonlinear systems this paper considers the global adaptive stabilisation for a class of uncertain stochastic nonlinear systems comparing with the existing relevant literature the systems under investigation allow more serious unknowns uncertainties more importantly by remarkably reducing the dimension of dynamic compensator for system unknowns from n to 2 and considerably diminishing the overparametrisation in the existing literature this paper proposes an improved design scheme for adaptive state feedback controllers it is turned out that with the designed controller in loop all the closed loop system states are globally bounded and the original system states converge to the origin both in the sense of probability one a simulation example is also given to demonstrate the effectiveness of the theoretical results

blowup behaviour for the nonlinear klein gordon equation we analyze the blowup behaviour of solutions to the focusing nonlinear klein gordon equation in spatial dimensions d ge 2 we obtain upper bounds on the blowup rate both globally in space and in light cones the results are sharp in the conformal and sub conformal cases the argument relies on lyapunov functionals derived from the dilation identity we also prove that the critical sobolev norm diverges near the blowup time

output feedback control using small gain conditions for stochastic nonlinear systems with siiss inverse dynamics this article further discusses output feedback control for stochastic nonlinear systems with stochastic integral input to state stability siiss inverse dynamics based on the stochastic lasalle theorem and small gain type conditions on siiss an output feedback controller using the backstepping method is constructively designed to guarantee that all the closed loop signals are bounded almost surely and the stochastic closed loop system is globally asymptotically stable in probability

the blow up dynamic and upper bound on the blow up rate for critical nonlinear schrodinger equation we consider the critical nonlinear schr odinger equation iut u u 4 n u with initial condition u 0 x u0 in dimension n 1 for u0 h1 local existence in the time of solutions on an interval 0 t is known and there exist finite time blow up solutions that is u0 such that limt t 1

blow up of the critical norm for some radial l2 super critical nonlinear schrodinger equations we consider the nonlinear schr odinger equation iu t delta u u p 1 u in dimension n geq 3 in the l 2 super critical range 1 frac 4 n p frac n 2 n 2 the corresponding scaling invariant space is dot h s c with 0 s c 1 and this covers the physically relevant case n p 3 the existence of finite time blow up solutions is known let p c frac n 2 p 1 so that dot h s c subset l p c let u t in dot h s c cap dot h 1 be a radially symmetric blow up solution which blows up at 0 t infty we prove that the scaling invariant l p c norm also blows up with a lower bound u t l p c geq log t t c n p rm as t to t

a martingale approach for testing diffusion models based on infinitesimal operator abstract i develop an omnibus specification test for diffusion models based on the infinitesimal operator the infinitesimal operator based identification of the diffusion process is equivalent to a martingale hypothesis for the processes obtained by a transformation of the original diffusion model my test procedure is then constructed by checking the martingale hypothesis via a multivariate generalized spectral derivative based approach that delivers a n 0 1 asymptotical null distribution for the test statistic the infinitesimal operator of the diffusion process is a closed form function of drift and diffusion terms consequently my test procedure covers both univariate and multivariate diffusion models in a unified framework and is particularly convenient for the multivariate case moreover different transformed martingale processes contain separate information about the drift and diffusion specifications this motivates me to propose a separate inferential test procedure to explore the sources of rejection when a parametric form is rejected simulation studies show that the proposed tests have reasonable size and excellent power performance an empirical application of my test procedure using eurodollar interest rates finds that most popular short rate models are rejected and the drift misspecification plays an important role in such rejections

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

profiles and quantization of the blow up mass for critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window for a suitable class of initial data in the energy space h1 we prove that the solution splits in two parts the first part corresponds to the singular part and accumulates a quantized amount of l2 mass at the blow up point the second part corresponds to the regular part and has a strong l2 limit at blow up time

dirichlet forms and semilinear elliptic equations with measure data we propose a probabilistic definition of solutions of semilinear elliptic equations with possibly nonlocal operators associated with regular dirichlet forms and with measure data using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right hand side of the equation is monotone and satisfies mild integrability assumption and the measure is smooth we also study regularity of solutions under the assumption that the measure is smooth and has finite total variation some applications of our general results are given

on collapsing ring blow up solutions to the mass supercritical nls we consider the nonlinear schr odinger equation i pa tu delta u u u p 1 0 in dimension n 2 and in the mass super critical and energy subcritical range 1 4n p min n 2n 2 5 for initial data u0 h1 with radial symmetry we prove a universal upper bound on the blow up speed we then prove that this bound is sharp and attained on a family of collapsing ring blow up solutions first formally predicted by gavish fibich and wang

on the structure of general mean variance hedging strategies we provide a new characterization of mean variance hedging strategies in a general semimartingale market the key point is the introduction of a new probability measure p which turns the dynamic asset allocation problem into a myopic one the minimal martingale measure relative to p coincides with the variance optimal martingale measure relative to the original probability measure p

worst case formulations of model predictive control for systems with bounded parameters two different predictive control formulations are developed based on minimization of the worst case quadratic cost for systems with bounded parameters the two formulations differ on the assumptions made about the future inputs in optimizing the current input one assumes open loop control while the other considers closed loop control their closed loop properties such as asymptotic stability are examined we then focus on a moving average model with an integrator and derive computationally simpler suboptimal algorithms

gradient estimates for sdes driven by multiplicative levy noise abstract gradient estimates are derived for the first time for the semigroup associated to a class of stochastic differential equations driven by multiplicative levy noise in particular the estimates are sharp for stable type noises to derive these estimates a new derivative formula of bismut elworthy li s type is established for the semigroup by using the malliavin calculus and a finite jump approximation argument

on input to state stability of stochastic retarded systems with markovian switching this note develops a razumikhin type theorem on pth moment input to state stability of hybrid stochastic retarded systems also known as stochastic retarded systems with markovian switching which is an improvement of an existing result an application to hybrid stochastic delay systems verifies the effectiveness of the improved result

convergence rate of numerical solutions to sfdes with jumps abstract in this paper we are interested in numerical solutions of stochastic functional differential equations with jumps under a global lipschitz condition we show that the p th moment convergence of euler maruyama numerical solutions to stochastic functional differential equations with jumps has order 1 p for any p 2 this is significantly different from the case of stochastic functional differential equations without jumps where the order is 1 2 for any p 2 it is therefore best to use the mean square convergence for stochastic functional differential equations with jumps moreover under a local lipschitz condition we reveal that the order of mean square convergence is close to 1 2 provided that local lipschitz constants valid on balls of radius j do not grow faster than log j

stability of the log log bound for blow up solutions to the critical non linear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation open image in new window with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for initial data with negative energy finite time blow up with a universal sharp upper bound on the blow up rate corresponding to the so called log log law has been proved in 10 11 we focus in this paper onto the positive energy case where at least two blow up speeds are known to possibly occur we establish the stability in energy space h1 of the log log upper bound exhibited in the negative energy case and a sharp lower bound on blow up rate in the other regime which corresponds to known explicit blow up solutions

global regulation for a class of stochastic high order nonlinear systems with multiple unknown control directions this paper concerns the global regulation for a class of stochastic high order nonlinear systems which admit more general system form than the existing works for details except for the serious sy

stability analysis of stochastic functional differential equations with infinite delay and its application to recurrent neural networks in this paper we investigate the stochastic functional differential equations with infinite delay some sufficient conditions are derived to ensure the pth moment exponential stability and pth moment global asymptotic stability of stochastic functional differential equations with infinite delay by using razumikhin method and lyapunov functions based on the obtained results we further study the pth moment exponential stability of stochastic recurrent neural networks with unbounded distributed delays the result extends and improves the earlier publications two examples are given to illustrate the applicability of the obtained results

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

brief paper small gain control method for stochastic nonlinear systems with stochastic iiss inverse dynamics this paper investigates the small gain type conditions on stochastic iiss siiss systems and makes full use of these conditions in the design and analysis of the controller the contributions are as follows 1 a new proof of the stochastic lasalle theorem is provided 2 the small gain type conditions on siiss are developed and their relationship is discussed 3 based on the stochastic lasalle theorem and siiss small gain type conditions the adaptive controllers are designed to guarantee that all of the closed loop signals are bounded almost surely and the stochastic closed loop systems are globally asymptotically stable in probability

nondispersive radial solutions to energy supercritical non linear wave equations with applications in this paper we establish optimal pointwise decay estimates for non dispersive compact radial solutions to non linear wave equations in 3 dimensions in the energy supercritical range as an application we show for the full energy supercritical range in the defocusing case that if the scale invariant sobolev norm of a radial solution remains bounded in its maximal interval of existence then the solution must exist for all times and scatter

attraction stability and robustness for stochastic functional differential equations with infinite delay this paper establishes the stochastic lasalle theorem to locate limit sets for stochastic functional differential equations with infinite delay from which some criteria on attraction boundedness stability and robustness are obtained to illustrate the applications of our results clearly this paper considers a scalar stochastic integro differential equation with infinite delay as an example

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

stability in distribution of stochastic functional differential equations abstract in this paper we investigate the stability in distribution for a class of stochastic functional differential equations sfdes which include stochastic differential delay equations sddes although stability in distribution has been studied by several authors recently there is so far no stability in distribution criterion on sfdes where the terms involved the delay components are highly nonlinear not bounded by linear functions in this paper we will establish the sufficient criteria on the stability in distribution for a class of highly nonlinear sfdes two examples will be given to illustrate our new results we also explain the reason why the existing stability in distribution criteria are not applicable by these two examples

standing ring blow up solutions to the n dimensional quintic nonlinear schrodinger equation we consider the quintic nonlinear schr odinger equation in dimension n 3 this problem is energy critical in dimension n 3 and energy super critical for n 4 we prove the existence of a radially symmetric blow up mechanism with l2 concentration along the unit sphere of rn this singularity formation is moreover stable by smooth and radially symmetric perturbation of the initial data this result extends the result obtained for n 2 in 29 and is the first result of description of a singularity formation in the energy supercritical class for nls type problems our main tool is the proof of the propagation of regularity outside the blow up sphere in the presence a so called log log type singularity

new criteria on asymptotic behavior of neutral stochastic functional differential equations this paper investigates the asymptotic behavior of neutral stochastic functional differential equations nsfdes under both the local lipschitz condition and the one dependent on the diffusion operator and on a coercivity term which is more general than the classical growth condition some sufficient conditions for stability with general decay rate and boundedness of nsfdes are derived via the lyapunov analysis method and some stochastic analysis techniques our results not only cover a wide class of highly nonlinear nsfdes but they can also deal with general stability issues including the polynomial stability and the exponential stability finally an illustrative example is provided to show the effectiveness of our theoretical results

weak tail conditions for local martingales the following conditions are necessary and jointly sufficient for an arbitrary c adl ag local martingale to be a uniformly integrable martingale a the weak tail of the supremum of its modulus is zero b its jumps at the first exit times from compact intervals converge to zero in l 1 on the events that those times are finite and c its almost sure limit is an integrable random variable

on universality of blow up profile for l 2 critical nonlinear schrodinger equation we consider finite time blow up solutions to the critical nonlinear schrodinger equation iut u u 4 nu with initial condition u0 h1 existence of such solutions is known but the complete blow up dynamic is not understood so far for a specific set of initial data finite time blow up with a universal sharp upper bound on the blow up rate has been proved in 22 23

stochastic functional differential equations with infinite delay abstract the stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay the main aim of this paper is to close this gap we establish criteria of pth moment t bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay and we also illustrate the result with an example

stochastic functional differential equations with infinite delay existence and uniqueness of solutions solution maps markov properties and ergodicity abstract this work is devoted to stochastic functional differential equations sfdes with infinite delay first existence and uniqueness of the solutions of such equations are examined because the solutions of the delay equations are not markov a viable alternative for studying further asymptotic properties is to use solution maps or segment processes by examining solution maps this work investigates the markov properties as well as the strong markov properties also obtained are adaptivity and continuity mean square boundedness and convergence of solution maps from different initial data this paper then examines the ergodicity of underlying processes and establishes existence of the invariant measure for sfdes with infinite delay under suitable conditions

razumikhin type theorem for stochastic functional differential systems via vector lyapunov function abstract this paper is concerned with input to state stability of sfdss by using stochastic analysis techniques razumikhin techniques and vector lyapunov function method vector razumikhin type theorem has been established on input to state stability for sfdss novel sufficient criteria on the pth moment exponential input to state stability are obtained by the established vector razumikhin type theorem when input is zero an improved criterion on exponential stability is obtained two examples are provided to demonstrate validity of the obtained results