on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

holder continuity of quasiminimizers with nonstandard growth abstract we show the holder continuity of quasiminimizers of the energy functionals f x u u d x with nonstandard growth under the general structure conditions z p x b x y r x g x f x y z z p x b x y r x g x the result is illustrated by showing that weak solutions to a class of a b harmonic equations div a x u u b x u u are quasiminimizers of the variational integral of the above type and thus are holder continuous our results extend work by chiado piat and coscia 1997 fan and zhao 2000 and giaquinta and giusti 1984

cooperative distributed model predictive control abstract in this paper we propose a cooperative distributed linear model predictive control strategy applicable to any finite number of subsystems satisfying a stabilizability condition the control strategy has the following features hard input constraints are satisfied terminating the iteration of the distributed controllers prior to convergence retains closed loop stability in the limit of iterating to convergence the control feedback is plantwide pareto optimal and equivalent to the centralized control solution no coordination layer is employed we provide guidance in how to partition the subsystems within the plant we first establish exponential stability of suboptimal model predictive control and show that the proposed cooperative control strategy is in this class we also establish that under perturbation from a stable state estimator the origin remains exponentially stable for plants with sparsely coupled input constraints we provide an extension in which the decision variable space of each suboptimization is augmented to achieve pareto optimality we conclude with a simple example showing the performance advantage of cooperative control compared to noncooperative and decentralized control strategies

global l estimates and holder continuity of weak solutions to elliptic equations with the general nonstandard growth conditions abstract we study the elliptic equations in divergence form with the general nonstandard growth conditions involving lebesgue measurable functions on in this paper we prove global l estimates and holder continuity of weak solutions for these equations our results improve the one in giusti 2003 even for the case when p x p and q x q are constants

local boundedness of general minimizers with nonstandard growth abstract we prove the local boundedness of local minimizers of f x u u d x where f is subject to the general structural conditions z p x b x y r x g x f x y z z p x b x y r x g x

distributed satisficing mpc with guarantee of stability this technical note develops conditions for the closed loop stability of a new class of cooperative distributed model predictive control called satisficing whereby the relative importance of each controller changes dynamically depending on the prevailing conditions instead of being adjusted a priori in the satisficing approach the less satisfied controller will gain more importance the main properties of the proposed approach are illustrated with an unstable dynamic system composed of three subsystems

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

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

ordinary differential equations in the oscillation theory of partial half linear differential equation abstract in the paper we study the damped half linear partial differential equation div a x u p 2 u b x u p 2 u c x u p 2 u 0 using radialization method we derive general oscillation results which allow to deduce new oscillation criteria for this equation from oscillation criteria for ordinary differential equations using careful radialization we improve several known oscillation criteria

conditions under which suboptimal nonlinear mpc is inherently robust abstract we address the inherent robustness properties of nonlinear systems controlled by suboptimal model predictive control mpc i e when a suboptimal solution of the generally nonconvex optimization problem rather than an element of the optimal solution set is used for the control the suboptimal control law is then a set valued map and consequently the closed loop system is described by a difference inclusion under mild assumptions on the system and cost functions we establish nominal exponential stability of the equilibrium and with a continuity assumption on the feasible input set we prove robust exponential stability with respect to small but otherwise arbitrary additive process disturbances and state measurement estimation errors these results are obtained by showing that the suboptimal cost is a continuous exponential lyapunov function for an appropriately augmented closed loop system written as a difference inclusion and that recursive feasibility is implied by such nominal exponential cost decay these novel robustness properties for suboptimal mpc are inherited also by optimal nonlinear mpc we conclude the paper by showing that in the absence of state constraints we can replace the terminal constraint with an appropriate terminal cost and the robustness properties are established on a set that approaches the nominal feasibility set for small disturbances the somewhat surprising and satisfying conclusion of this study is that suboptimal mpc has the same inherent robustness properties as optimal mpc

harnack s inequality for quasiminimizers with nonstandard growth conditions we prove harnack inequalities for quasiminimizers of the variable exponent dirichlet energy integral by employing the de giorgi method

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

global boundedness and holder continuity of quasiminimizers with the general nonstandard growth conditions abstract in this paper we study the integral functionals with the general nonstandard growth we prove the boundedness and holder continuity of quasiminimizers of these functionals our results for quasiminimizers improve variable exponent case and generalize constant exponent case studied in direct methods in the calculus of variations 2003 by giusti

sobolev embedding theorems for spaces wk p x ω this paper gives a sobolev type embedding theorem for the generalized lebesgue sobolev space wk p x where is an open domain in rn n 2 with cone property and p x is a lipschitz continuous function defined on satisfying 1 p p nk the main result can be stated as follows for any measurable function q x x formula with p x q x p x np x n kp x there exists a continuous embedding from wk p x to lq x