a computational algorithm for a class of non smooth optimal control problems arising in aquaculture operations this paper introduces a computational approach for solving non linear optimal control problems in which the objective function is a discontinuous function of the state we illustrate this approach using a dynamic model of shrimp farming in which shrimp are harvested at several intermediate times during the production cycle the problem is to choose the optimal harvesting times and corresponding optimal harvesting fractions the percentage of shrimp stock extracted to maximize total revenue the main difficulty with this problem is that the selling price of shrimp is modelled as a piecewise constant function of the average shrimp weight and thus the revenue function is discontinuous by performing a time scaling transformation and introducing a set of auxiliary binary variables we convert the shrimp harvesting problem into an equivalent optimization problem that has a smooth objective function we then use an exact penalty method to solve this equivalent problem we conclude the paper with a numerical example

multipliers paramultipliers and weak strong uniqueness for the navier stokes equations abstract in this article we describe spaces p such that if u is a weak in the sense of leray j leray sur le mouvement d un fluide visqueux remplissant l espace acta math 63 1934 193 248 solution of the navier stokes system for some initial data u 0 and if u belongs to p then u is unique in the class of weak solutions we say then that weak strong uniqueness holds it turns out that the proof of such results relies on the boundedness of a trilinear functional f l 2 h l 2 h p r where belong to 0 1 in order to find optimal conditions for the boundedness of f we are led to describing spaces of multipliers and of paramultipliers that is functions which map by classical pointwise product or by paraproduct a given sobolev spaces in another given sobolev space the study of these spaces enables us to give conditions for weak strong uniqueness which generalise all previously known results from the famous serrin criterion j serrin the initial value problem for the navier stokes equations in r e langer ed nonlinear problems univ of wisconsin press 1963 pp 69 98 to the recent conditions formulated by lemarie rieusset p g lemarie rieusset recent developments in the navier stokes problem chapman and hall 2003

a diffusive single species model with periodic coefficients and its optimal harvesting policy in this paper we consider a logistic model with spatially nonhomogeneous diffusion under the email protected email protected email protected r x t u1 uk x t e x t u x t email protected 0 u x 0 email protected x email protected email protected email protected n 0 email protected 0 email protected email protected email protected where coefficients r k e are smooth t periodic functions this model describes the growth of the single species with the neumann boundary condition and initial value condition we investigate the global stability of a periodic solution and optimal harvesting policy furthermore we also consider a generalized single species model and its harvesting problem the results gained in this article extend the works in references hailong li logistic model for single species with spatial diffusion and its optimal harvesting policy j biomath 14 3 1999 293 300 in chinese ling bai ke wang gilpin ayala model with spatial diffusion and its optimal harvesting policy appl math comput 171 2005 531 546

navier stokes flow in the weighted hardy space with applications to time decay problem abstract the asymptotic expansions of the navier stokes flow in r n and the rates of decay are studied with aid of weighted hardy spaces fujigaki and miyakawa 12 miyakawa 28 proved the nth order asymptotic expansion of the navier stokes flow if initial data decays like 1 x n 1 and if nth moment of initial data is finite in the present paper it is clarified that the moment condition for initial data is essential in order to obtain higher order asymptotic expansion of the flow and to consider the rapid time decay problem the second author 39 established the weighted estimates of the strong solutions in the weighted hardy spaces with small initial data which belongs to l n and a weighed hardy space firstly the refinement of the previous work 39 is achieved with alternative proof then the existence time of the solution in the weighted hardy spaces is characterized without any hardy norm as a result in two dimensional case the smallness condition on initial data is completely removed as an application the rapid time decay of the flow is investigated with aid of asymptotic expansions and of the symmetry conditions introduced by brandolese 3

l3 solutions of the liquid crystals system abstract in this article we study the simplified system of the original ericksen leslie equations for the flow of liquid crystals it is shown that the l 3 solutions of the cauchy problem for the three dimensional incompressible liquid crystal system are smooth

extension criterion via two components of vorticity on strong solutions to the 3 d navier stokes equations we shall show that only two components of vorticity play an essential role to determine possibility of extension of the time interval for the local strong solution to the navier stokes equations then we shall apply our extension theorem to regularity criterion on weak solutions due to serrin and beirao da veiga chae choe proved the same criterion as beirao da veiga only by means of the two components of vorticity we deal with the critical case which they excluded our criterion may be regarded as the generalization of the result of beal kato majda from l to bmo

optimal harvesting of diffusive models in a nonhomogeneous environment abstract we study the optimal harvesting strategy for populations whose dynamics is described by reaction diffusion equations the production function can be of logistic gilpin ayala or gompertz type the diffusion structure is discussed we suggest to consider u k where k is the carrying capacity of the environment rather than u and study optimal harvesting for models with this diffusion type maximum yield is investigated for both continuous and impulsive models for continuous harvesting the optimal policy is obtained for the impulsive equation some limit cases are considered the paper also outlines a variety of open problems

extinction criteria in stage structured population models with impulsive culling we propose stage structured population models for species whose adult members are subject to culling with a view to understanding the culling regimes that are likely to result in eradication of the species a purely time dependent model is proposed in which culling occurs at particular discrete times not necessarily equally spaced then a reaction diffusion model is proposed for a situation in which the adults can diffuse in this model the culling is continuous in time but occurs only at particular discrete points in space such a model might be appropriate for pheromone trapping of insects for both models conditions are obtained that are sufficient for species eradication

existence and positivity of solutions of a fourth order nonlinear pde describing interface fluctuations we study the partial differential equation which arose originally as a scaling limit in the study of interface fluctuations in a certain spin system in that application x lies in r but here we study primarily the periodic case x e s we establish existence uniqueness and regularity of solutions locally in time for positive initial data in h s and prove the existence of several families of lyapunov functions for the evolution from the latter we establish a sharp connection between existence globally in time and positivity preservation if o t is a maximal half open interval of existence for a positive solution of the equation with t 1 16r2 then t 03 and lim m w r exists and is constant we discuss also some explicit solutions and propose a generalization to higher dimensions 0 1994 john wiley

existence and stability of periodic quasisolutions in nonlinear parabolic systems with discrete delays the existence of periodic quasisolutions and the dynamics for a couple system of semilinear parabolic equations with discrete time delays and periodic coefficients are investigated using the method of upper and lower solutions it is shown that if the reaction function in the system possesses a mixed quasimonotone property and the periodic boundary value system has a pair of coupled t upper and lower solutions then the periodic boundary value problem has a pair of periodic quasisolutions and the sector between the quasisolutions is an attractor of the delayed periodic parabolic system under some additional conditions the periodic quasisolutions are exactly true periodic solutions of the periodic boundary value system finally three models arising from ecology are given to illustrate the obtained results

weighted estimate of the asymptotic profiles of the navier stokes flow in rn abstract we study the asymptotic profiles of the navier stokes flow and the weighted estimate of the spatial and temporal decay rate of the error term this paper extends the result of y fujigaki and t miyakawa in y fujigaki t miyakawa asymptotic profiles of nonstationary incompressible navier stokes flows in the whole space siam j math anal 33 3 2001 523 544 where they derived the asymptotic profiles of the navier stokes flow and estimated the temporal decay rate of the error term

global exponential stability of positive periodic solutions for a delayed nicholsonʼs blowflies model abstract this paper is concerned with a non autonomous delayed nicholson s blowflies model under proper conditions we employ a novel argument to establish a criterion on the global exponential stability of positive periodic solutions this answers an open problem proposed by berezansky et al 2010 2 we also provide numerical simulations to support the theoretical result

mild solutions to the time fractional navier stokes equations in r n abstract this paper addresses the existence and uniqueness of mild solutions to the navier stokes equations with time fractional differential operator of order 0 1 several interesting properties about the solution are also highlighted like regularity and decay rate in lebesgue spaces which will depend on the fractional exponent moreover it is shown that the l p exponent range which the solution belongs to is different from the range for the solution of the classical problem with 1

optimization problems for general simple population with n impulsive harvest in this paper n impulsive harvest problems of general simple population are discussed by models with dirac function the optimal impulsive harvest policies to protect the renewable resource better are obtained under conditions of fixed quantity per impulsive harvest then a concept of the sequence for optimal harvest moments for general simple population is presented which is beneficial to protect resource better and sustainable development finally we apply the conclusions to some special models

optimal harvesting of a stochastic delay tri trophic food chain model with levy jumps abstract in this paper the optimal harvesting of a stochastic delay tri trophic food chain model with levy jumps is considered we introduce two kinds of environmental perturbations in this model one is called white noise which is continuous and is described by a stochastic integral with respect to the standard brownian motion and the other one is jumping noise which is modeled by a levy process under some mild assumptions the critical values between extinction and persistent in the mean of each species are established the sufficient and necessary criteria for the existence of optimal harvesting policy are established and the optimal harvesting effort and the maximum of sustainable yield are also obtained we utilize the ergodic method to discuss the optimal harvesting problem the results show that white noises and levy noises significantly affect the optimal harvesting policy while time delays is harmless for the optimal harvesting strategy in some cases at last some numerical examples are introduced to show the validity of our results

nonholonomic motion planning steering using sinusoids methods for steering systems with nonholonomic c onstraints between arbitrary configurations are investigated suboptimal trajectories are derived for systems that are not in canonical form systems in which it takes more than one level of bracketing to achieve controllability are considered the trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level a class of systems that can be steered using sinusoids claimed systems is defined conditions under which a class of two input systems can be converted into this form are given

the initial boundary value problem for thin plate flow of incompressible non newtonian viscous fluids abstract in this paper the particular model that describes the thin plate flow of incompressible non newtonian viscous fluids is investigated this model which arises from polymer processing concerns the initial boundary problem of nonstationary thin plate flow of non newtonian viscous incompressible fluids by using monotone operator theory and schauder s fixed point theorem an existence and uniqueness theorem for a generalized solution is obtained and the regularity for this solution is tested

on the navier stokes equations in the half space with initial and boundary rough data in morrey spaces abstract we consider the initial boundary value problem for the navier stokes equations in the half space with data in morrey spaces existence of small global solutions is proved in spaces with the right homogeneity to allow self similar solutions moreover we analyze the long time behavior of the solutions and obtain a class of asymptotically self similar ones

global well posedness and long time decay of the 3d boussinesq equations abstract the incompressible 3d boussinesq equations is considered in this paper the global well posedness of mild solutions is proved by using fourier analysis and standard techniques the global solutions u will decay to zero in sense of the norm of the space 1 as time goes to infinity and the stability of global solutions is proved

on a larger class of stable solutions to the navier stokes equations in exterior domains where w w x w1 x wn x and x denote the unknown velocity vector and the unknown pressure of the fluid at point x respectively while f f x f i j x i j 1 n is the given tensor with div f n j 1 f 1 j xj nj 1 f n j xj denoting the external force in our previous paper 23 we established the theory of existence uniqueness and regularity for solutions to n s0 in the class w ln

long time decay to the lei lin solution of 3d navier stokes equations abstract in this paper we prove if u c 0 x 1 r 3 is a global solution of 3d navier stokes equations then u t x 1 decays to zero as time goes to infinity fourier analysis and standard techniques are used

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

global stability of large solutions to the 3d navier stokes equations we prove the stability of mildly decaying global strong solutions to the navier stokes equations in three space dimensions combined with previous results on the global existence of large solutions with various symmetries this gives the first global existence theorem for large solutions with approximately symmetric initial data the stability of unforced 2d flow under 3d perturbations is also obtained

gilpin ayala model with spatial diffusion and its optimal harvesting policy in this paper we consider spatially nonhomogeneous gilpin ayala diffusive equation suffered from exploitation with spatially variable harvesting effort e x email protected email protected r x u1 uk x q e x u t x 0 email protected u 0 x email protected x email protected email protected email protected n 0 email protected 0 email protected email protected email protected which describes the growth of the single species with neumann boundary condition and initial value condition we investigate the global stability of equilibrium solution and optimal harvesting policy it is found that in the case of ordinary differential equation our results generalize the spatially homogeneous equations discussed in c w clark mathematical bio economics the optimal management of renewable resources wiley new york 1976 c w clark mathematical bio economics the optimal management of renewable resources second ed wiley new york 1990 m fan k wang optimal harvesting policy for single population with periodic coefficients math biosci 152 1998 165 h li a class of single species models with periodic coefficients and their optimal harvesting policy j biomath 14 4 1999 293 our brief results also generalize the corresponding results in h li logistic model for single species with spatial diffusion and its optimal harvesting policy j biomath 14 3 1999 293 when q 1

optimal harvesting of a stochastic mutualism model with levy jumps in this letter a stochastic mutualism model with levy jumps and harvesting is considered under some simple assumptions sufficient and necessary criteria for the existence of optimal harvesting policy are established the optimal harvesting effort and the maximum of sustainable yield are also obtained the effects of random noises on the optimal harvesting of the model are discussed and some numerical simulations are introduced to illustrate the main results

regularity criterion for 3d mhd fluid passing through the porous medium in terms of gradient pressure abstract the aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in r 3 our results show that if p l with 2 3 3 1 then the weak solution is regular and unique if p l with 2 3 2 3 2 then the weak solution is regular and unique

dynamical systems that sort lists diagonalize matrices and solve linear programming problems abstract we establish a number of properties associated with the dynamical system h h h n where h and n are symmetric n by n matrices and a b ab ba the most important of these come from the fact that this equation is equivalent to a certain gradient flow on the space of orthogonal matrices we are especially interested in the role of this equation as an analog computer for example we show how to map the data associated with a linear programming problem into h 0 and n in such a way as to have h h h n evolve to a solution of the linear programming problem this result can be applied to find systems which solve a variety of genetic combinatorial optimization problems and it even provides an algorithm for diagonalizing symmetric matrices

on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain abstract we study the time decay of weighted norms of weak and strong solutions to the navier stokes equations in a 3d exterior domain moment estimates for weak solutions and weighted l q estimates for strong solutions are deduced both of which seem to be optimal the relation is discussed between the space time decay and the vanishing of the total net force exerted by the fluid to the body a class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish

approximating the ideal free distribution via reaction diffusion advection equations abstract we consider reaction diffusion advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness where fitness is defined as a logistic local population growth rate we show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics the modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution the analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability the models are quasilinear parabolic equations with nonlinear boundary conditions

explicit design of time varying stabilizing control laws for a class of controllable systems without drift abstract this paper gives a systematic way to design time variant feedback control laws for a class of controllable nonlinear systems this class contains a lot of systems which cannot be stabilized via a time invariant feedback control law the interest of this work lies in the design method since a general existence results is already available the techniques employed here are basic they mainly involve classical lyapunov analysis

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

optimal harvesting of a stochastic delay logistic model with levy jumps the optimal harvesting problem of a stochastic time delay logistic model with levy jumps is considered in this article we first show that the model has a unique global positive solution and discuss the uniform boundedness of its pth moment with harvesting then we prove that the system is globally attractive and asymptotically stable in distribution under our assumptions furthermore we obtain the existence of the optimal harvesting effort by the ergodic method and then we give the explicit expression of the optimal harvesting policy and maximum yield

effects of supplying alternative food in a predator prey model with harvesting in this paper we propose a tritrophic predator prey model with harvesting where the top predator population is partially supported with alternative food we report the consequences of providing alternative food to the top predator in a top predator harvested model the extinction criterion for top predator population local stability of equilibrium points and persistence conditions are discussed theoretically pontryagins maximum principle is used to characterize the optimal control of harvesting we have derived the condition of hopf bifurcation by varying harvesting effort the bifurcation diagrams of the system with respect to harvesting effort in presence of alternative food are given our analysis show that alternative food can prevent top predator extinction risk at higher harvesting effort and plays a vital role for biological conservation of species

biological population management based on a hamilton jacobi bellman equation with boundary blow up establishment of a long term biological population management policy requires balancing its cost and utility disutility we approach this issue from the viewpoint of stochastic control as an effici

impulsive harvesting and by catch mortality for the theta logistic model in this paper we investigate the population dynamics described by the theta logistic model with periodic impulsive harvesting and by catch mortality we examine the existence and stability of two positive periodic solutions by using qualitative methods and cobwebs then the sufficient conditions under which the unique positive periodic solution exists and is semi stable are established and qualifications for the solutions approach zero are also obtained further choosing the maximum sustainable yield as the management objective we investigate the optimal harvesting policy for the theta logistic model with periodic impulsive harvesting moreover the corresponding theta logistic difference equation is considered subject to the impulsive perturbation and the dynamics which is parallel to that for the differential equation is examined the main results extend and generalize the classical results for populations described by the autonomous logistic equation in renewable resources management

optimal impulsive harvesting policies for single species populations in this paper we address an optimal impulsive control problem with the aim of optimizing the harvesting rate for a non autonomous logistic model in addressing this problem we combine bang bang and singular controls for an impulsive optimization problem in which the singular controls are blocked at certain harvesting times in our approach we first obtain the singular controls by applying a pontryagin optimal control framework for impulsive systems second we investigate and derive a number of relationships characterizing the optimal harvesting rates and present an optimization principle the optimal path lies as close as possible to the singular path finally based on this optimization principle we obtain analytical expressions for the optimal harvesting policy for our problem and use numerical simulations to demonstrate the effectiveness of our method this study develops concepts and theory related to continuous control problems and applies them to impulsive control problems and extends the related work of xiao et al 2006 and braverman and mamdani 2008

persistence and extinction in spatial models with a carrying capacity driven diffusion and harvesting abstract for the reaction diffusion equation u t x t d u t x k t x r t x u t x g k t x u t x e t x u t x with the general type of growth diffusion stipulated by the carrying capacity k and harvesting existence positivity persistence extinction and stability of solutions are investigated in numerical simulations the results are compared to the model with a regular diffusion

optimal harvesting of a stochastic delay competitive lotka volterra model with levy jumps abstract this paper systematically investigates the optimal harvesting of a stochastic delay competitive lotka volterra model with levy jumps under some simple assumptions the sufficient conditions for extinction and stable in the time average of each species are established the stability in distribution of this model is proved under our assumptions finally the sufficient and necessary criteria for the existence of optimal harvesting policy are established and the explicit expression of the optimal harvesting effort and the maximum of sustainable yield are also obtained and some numerical simulations are introduced to demonstrate the theoretical results

global smooth solutions of mhd equations with large data abstract in this paper we establish the global existence of smooth solutions of the three dimensional mhd system for a class of large initial data both the initial velocity and magnetic field can be arbitrarily large in the critical norm

initial and boundary values for lαq lp solution of the navier stokes equations in the half space abstract in this paper we study the initial and boundary value problem of the navier stokes equations in the half space we prove the existence of weak solution u l q 0 l p r n 1 2 1 n p 2 q 0 n p with u l loc q 2 0 l loc p 2 r n for the solenoidal initial data h b p q 1 n p r n and the boundary data g l q 0 b p p 1 p r n 1 when h b p q 1 n p r n g l q 0 b p p 1 p r n 1 is small enough moreover the solution is unique in the class l q 0 t l p r n for any t if 0 and for some t if 0

existence and generalized gevrey regularity of solutions to the kuramoto sivashinsky equation in rn abstract motivated by the work of foias and temam c foias r temam gevrey class regularity for the solutions of the navier stokes equations j funct anal 87 1989 359 369 we prove the existence and gevrey regularity of local solutions to the kuramoto sivashinsky equation in r n with initial data in the space of distributions the control on the gevrey norm provides an explicit estimate of the analyticity radius in terms of the initial data in the particular case when n 1 our analysis allows for initial data that are less smooth than that considered by grujic and kukavica z grujic i kukavica space analyticity for the navier stokes and related equations with initial data in l p j funct anal 152 1998 447 466

commutator estimate and its application to regularity criteria of the dissipative quasi geostrophic equation abstract a new commutator estimate with respect to a nonlinear convection upper bounded by a single partial derivative component in hilbert spaces is obtained as an application regularity criteria on the supercritical quasi geostrophic equation are obtained provided that solution growth conditions are assumed to involve a single partial derivative component

forward self similar solutions of the fractional navier stokes equations abstract we study forward self similar solutions to the 3 d navier stokes equations with the fractional diffusion first we construct a global time forward self similar solutions to the fractional navier stokes equations with 5 6 1 for arbitrarily large self similar initial data by making use of the so called blow up argument moreover we prove that this solution is smooth in r 3 0 in particular when 1 we prove that the solution constructed by korobkov and tsai 2016 16 satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system which implies this solution has the same properties as a solution which was constructed in jia and sverak 2014 13

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

asymptotic decay of solutions to 3d mhd equations abstract in the very recent work wang and wang 2014 proved that the initial value problem for the three dimension incompressible mhd equations has a global solutions u b c 0 1 provided that the norms of the initial data are bounded exactly by the minimal value of the viscosity coefficients in this paper we prove that global solutions u b that was obtained in wang and wang 2014 is asymptotic to zero in sense of the norm of the space 1 as time goes to infinity moreover stability of global solutions is also discussed

note on some periodic optimal harvesting problems for age structured population dynamics we consider two optimal harvesting problems related to age structured population dynamics with logistic term and time periodic of period t control and vital rates we derive the first order necessary optimality conditions for both problems we prove that in the particular case of time independent vital rates and under a natural hypothesis on the logistic term the optimal control for the first problem is constant the second problem concerns the constant harvesting efforts which act on subintervals of 0 t the maximizing here is treated with respect to the subinterval where the control acts we find the directional derivative of the cost functional some final comments concerning the population dynamics with diffusion are given as well

global mild solutions of navier stokes equations we establish a global well posedness of mild solutions to the three dimensional incompressible navier stokes equations if the initial data are in the space x 1 defined by x 1 f e d r 3 3 1 f d and if the norms of the initial data in x 1 are bounded exactly by the viscosity coefficient

energy decay for a weak solution of the navier stokes equation with slowly varying external forces we consider the navier stokes system with slowly decaying external forces formula we show that the energy norm of a weak solution has non uniform decay u t 2 0 t under suitable conditions on the data f and a which make the energy of solution bounded in time also we show the exact rate of the decay uniform decay of the energy u t 2 c 1 t e for external forces with a given explicit decay rate

optimality for a diffusive predator prey system in a spatially heterogeneous environment incorporating a prey refuge abstract in this paper we consider the optimal harvesting controls for a diffusive predator prey system which is formulated by prey refuge and modified leslie gower functional response the locally and globally asymptotic stability of unique positive constant steady state is discussed under homogeneous neumann boundary condition the nonexistence and existence of nonconstant steady states are obtained by leray schauder degree theory also the optimal harvesting strategies to get maximum sustainable yield msy and maximum economic yield mey are correspondingly given

nonholonomic control systems on riemannian manifolds this paper gives a general formulation of the theory of nonholonomic control systems on a riemannian manifold modeled by second order differential equations and using the unique riemannian connection defined by the metric the main concern is to introduce a reduction scheme replacing some of the second order equations by first order equations the authors show how constants of motion together with the nonholonomic constraints may be combined to yield such a reduction the theory is applied to a particular class of nonholonomic control systems that may be thought of as modeling a generalized rolling ball this class reduces to the classical example of a ball rolling without slipping on a horizontal plane

sliding mode control in dynamic systems the concept of sliding modes in abstract dynamic systems described by a semigroup of state space transformations is introduced the sliding mode design procedure is used for designing finite observers sliding mode control for systems with delays and differential difference systems which is illustrated by sliding mode control of longitudinal oscillations

control and stabilization of nonholonomic dynamic systems a class of inherently nonlinear control problems has been identified the nonlinear features arising directly from physical assumptions about constraints on the motion of a mechanical system models are presented for mechanical systems with nonholonomic constraints represented both by differential algebraic equations and by reduced state equations control issues for this class of systems are studied and a number of fundamental results are derived although a single equilibrium solution cannot be asymptotically stabilized using continuous state feedback a general procedure for constructing a piecewise analytic state feedback which achieves the desired result is suggested

optimal harvesting policy of a stochastic food chain population model this note studies the optimal harvesting policy of a stochastic three species food chain model the almost sufficient and necessary conditions for the existence of optimal harvesting policy are established meanwhile the optimal harvesting effort and the maximum of expectation of sustainable yield are obtained

on optimal control problems of a class of impulsive switching systems with terminal states constraints abstract the global optimal control problem is proposed for a special class of hybrid dynamical systems i e impulsive switching systems then the necessary condition of the above problem the minimum principle is given ekeland s variational principle and the matrix cost functional structure expression are utilized in the process of the proof based on the main result a special linear hybrid impulsive and switching system hiss is illustrated and the optimal control algorithm is presented moreover the cases of pure impulsive systems and pure switched systems are included in this paper

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data

stabilization of nonholonomic systems using isospectral flows in this paper we derive and analyze a discontinuous stabilizing feedback for a lie algebraic generalization of a class of kinematic nonholonomic systems introduced by brockett new directions in applied mathematics p hilton and g young eds springer verlag new york 1982 pp 11 27 the algorithm involves discrete switching between isospectral and norm decreasing flows we include a rigorous analysis of the convergence

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

note on prodi serrin ladyzhenskaya type regularity criteria for the navier stokes equations in this article we prove new regularity criteria of the prodi serrin ladyzhenskaya type for the cauchy problem of the three dimensional incompressible navier stokes equations our results improve the classical lr 0 t ls regularity criteria for both velocity and pressure by factors of certain negative powers of the scaling invariant norms u l 3 and u h 1 2

large time behaviour of solutions to the 3d nse in xσ spaces abstract in this paper we study the incompressible navier stokes equations in l 2 r 3 x 1 r 3 in the global existence case we establish that if the solution u is in the space c r l 2 x 1 then for 3 2 the decay of u t x is at least of the order of t 2 3 4 fourier analysis and standard techniques are used

optimal control and geodesic flows in this paper we analyze and generalize from the point of view of the maximum principle a class of nonlinear optimal control problems originally introduced in brockett 1994 the optimal control problems discussed here lie on the adjoint orbits of compact lie groups or on grassmann manifolds and the structure of the equations that arise depends on the metric used the so called normal metric we consider some special cases of the general problem and their meaning from the point of view of riemannian geometry and hamiltonian mechanics

space time regularity of the koch tataru solutions to navier stokes equations abstract in koch tataru 2001 the authors have proved local and global well posedness of the navier stokes equations with small initial data u 0 b m o 1 and then the spatial analyticity of the koch tataru solution stated as t k 2 k u x t for any positive integer k has been presented by germain pavlovic staffilani 2007 where x t is the space defined by koch tataru 2001 see also definition 1 2 in this paper we shall present the space time regularity more precisely for any positive integer m k we have t m k 2 t m k u x t

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

time decay rates of the l3 norm for strong solutions to the navier stokes equations in r3 abstract let u c 0 l 3 r 3 be a strong solution of the cauchy problem for the 3d navier stokes equations with the initial value u 0 we prove that the time decay rates of u in the l 3 norm coincide with ones of the heat equation with the initial value u 0 our proofs use the theory about the existence of local strong solutions time decay rates of strong solutions when the initial value is small enough and uniqueness arguments

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter

well posedness for the navier stokes nernst planck poisson system in triebel lizorkin space and besov space with negative indices abstract this paper is concerned with the well posedness of the navier stokes nerst planck poisson system nsnpp let s p 2 n p we prove that the nsnpp has a unique local solution u v w e u t e v t e v t for u 0 v 0 w 0 in a subspace i e v u 1 v v 1 v v 1 of f 1 2 b p s p b p s p with u 0 0 we also prove that there exists a unique small global solution u v w e u e v e v for any small initial data u 0 v 0 w 0 f 1 2 b p s p b p s p with u 0 0

competitive cooperative models with various diffusion strategies the paper is concerned with different types of dispersal chosen by competing species we introduce a model with the diffusion type term a u p which includes some previously studied systems as special cases where a positive space dependent function p can be interpreted as a chosen dispersal strategy the well known result that if the first species chooses p proportional to the carrying capacity while the second does not then the first species will bring the second one to extinction is also valid for this type of dispersal however we focus on the case when the ideal free distribution is attained as a combination of the two strategies adopted by the two species then there is a globally stable coexistence equilibrium its uniqueness is justified if both species choose the same dispersal strategy non proportional to the carrying capacity then the influence of higher diffusion rates is negative while of higher intrinsic growth rates is positive for survival in a competition this extends the result of dockery et al 1998 for the regular diffusion to a more general type of dispersal