monotone methods for a finite difference system of reaction diffusion equation with time delay abstract the aim of this paper is to present some monotone iterative schemes for computing the solution of a system of nonlinear difference equations which arise from a class of nonlinear reaction diffusion equations with time delays the iterative schemes lead to computational algorithms as well as existence uniqueness and upper and lower bounds of the solution an application to a diffusive logistic equation with time delay is given

global attractor of a coupled finite difference reaction diffusion system with delays abstract in the study of asymptotic behavior of solutions for reaction diffusion systems an important concern is to determine whether and when the system has a global attractor which attracts all positive time dependent solutions the aim of this paper is to investigate the global attraction problem for a finite difference system which is a discrete approximation of a coupled system of two reaction diffusion equations with time delays sufficient conditions are obtained to ensure the existence and global attraction of a positive solution of the corresponding steady state system applications are given to three types of lotka volterra reaction diffusion models where time delays may appear in the opposing species

gaussian measures of dilatations of convex symmetric sets we prove that the inequality 1 ta t 1 a holds for any centered gaussian measure on a separable banach space f any convex closed symmetric set a f and t 1 where x 1 x x 2 1 2 x x exp y 2 2 dy as an application the best constants in comparison of moments of gaussian vectors are calculated

saturating constructions for normed spaces ii we prove several results of the following type given finite dimensional normed space v there exists another space x with log dim x o log dim v and such that every subspace or quotient of x whose dimension is not too small contains a further subspace isometric to v this sheds new light on the structure of such large subspaces or quotients resp large sections or projections of convex bodies and allows us to solve several problems stated in the 1980s by v milman

adaptive sampled data tracking for input constrained exothermic chemical reaction models we consider digital input constrained adaptive and non adaptive output feedback control for a class of nonlinear systems which arise as models for controlled exothermic chemical reactors our objective is set point control of the temperature of the reaction with pre specifiedasymptotic tracking accuracy set by the designer our approach is basedon tracking controllers but in a context of piecewise constant sampled data output feedbacks and possibly adapted sampling periods the approach does not require any knowledge of the systems parameters does not invoke an internal model is simple in its design copes with noise corruptedoutput measurements andrequires only a feasibility assumption in terms of the reference temperature andthe input constraints

approximation design of optimal controllers for nonlinear systems with sinusoidal disturbances abstract this paper considers an infinite time optimal damping control problem for a class of nonlinear systems with sinusoidal disturbances a successive approximation approach saa is applied to design feedforward and feedback optimal controllers by using the saa the original optimal control problem is transformed into a sequence of nonhomogeneous linear two point boundary value tpbv problems the existence and uniqueness of the optimal control law are proved the optimal control law is derived from a riccati equation matrix equations and an adjoint vector sequence which consists of accurate linear feedforward and feedback terms and a nonlinear compensation term and the nonlinear compensation term is the limit of the adjoint vector sequence by using a finite term of the adjoint vector sequence we can get an approximate optimal control law a numerical example shows that the algorithm is effective and robust with respect to sinusoidal disturbances

gaussian processes and almost spherical sections of convex bodies on presente une preuve simple avec des estimations pointues du theoreme de dvoretzky sur l existence des sections presque spheriques ayant une grande dimension dans des corps convexes arbitraires dans r n

on the optimal control of nonlinear systems abstract the lie algebra of tensors on a hilbert space is used to obtain optimal controls for a class of nonlinear systems

existence and multiplicity of positive solutions for classes of singular elliptic pdes we consider the boundary value problem u g u u in u 0on where rn is a bounded domain is a nonnegative function in l such that 0 on some subset of of positive measure and g 0 r is continuous we establish the existence of three positive solutions when g 0 0 positone the graph of s 1g s is roughly s shaped and 0 we also prove that there exists at least one positive solution when g 0 0 and 0 1 we employ the method of sub super solutions to prove our results

successive approximation procedure of optimal tracking control for nonlinear similar composite systems abstract the optimal tracking control otc problem for a class of affine nonlinear composite systems with similar structure is considered by using a modeling technique the nonlinear similar composite system is first transformed into some quasi decoupled subsystems then the high order strongly coupled nonlinear two point boundary value tpbv problem is transformed into a sequence of linear decoupled tpbv problems through a successive approximation procedure the obtained otc law consists of an accurate linear term and a nonlinear compensation term which is the limit of the adjoint vector sequence a suboptimal tracking control law is obtained by truncating a finite iterative result of the adjoint vector sequence as its nonlinear compensation term

bounds for projection constants and 1 summing norms it is shown that projection constants k xn of dimensional normed spaces xn satisfy l xn 0 is a numer ical constant similarly the 1 summing norms of the identity of xn can be estimated by ttx xn fn c fn these estimates are the best possi ble for prime n translation invariant dimensional spaces xn such that k xn sfh 2l jn and itx xn 3v d xn lf fn 1 introduction and results the main results of the paper complete the investigation of an upper bound for the projection constants of arbitrary finite dimensional banach spaces which had started with the kadec snobar kas estimate k xn yfn for any n dimensional space xn this estimate was further studied in several papers koll ko kol lew2 in particular it has recently been shown by lewis lew2 that k xf yfh n 5 our main theorem gives an essential strengthening stated in the abstract upper bounds for projection constants are intimately connected to lower bounds for the 1 summing norm of the identity operator for spaces with enough symmetries these bounds follow from each other by a simple formal argument in general it turns out that similar but somewhat simpler calcula tions than for the projection constants lead to the lower bound for nx xn both upper and lower bounds are the best possible up to terms of higher order the approach used by konig in ko based on some number theoretic results gives the relevant constructions the same approach leads to a con struction of a finite dimensional space with large gordon lewis constant this provides the first deterministic example of such a space our notation essentially follows tj for completeness let us recall the main

monotone iterative methods for finite difference system of reaction diffusion equations this paper presents an existence comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary value problems the basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system using upper and lower solutions as two distinct initial iterations two monotone sequences from a suitable linear system are constructed it is shown that these two sequences converge monotonically from above and below respectively to a unique solution of the nonlinear discrete equations this formulation leads to a well posed problem for the nonlinear discrete system applications are given to several models arising from physical chemical and biological systems numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences

on numerical asymptotic behavior of finite element solutions for parabolic equations in this paper we investigate the numerical asymptotic behavior of the finite element solutions for linear parabolic equations under some appropriate conditions we also give some results of numerical experiments in the two dimensional problems to indicate the effectiveness of our results

a separation theorem for nonlinear systems in this paper we have obtained a nonlinear separation result for controlled stochastic systems the result is based on a sequential technique introduced by the second author which has been applied with significant success for nonlinear deterministic systems

global optimal feedback control for general nonlinear systems with nonquadratic performance criteria abstract optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria that permit state and control dependent time varying weighting parameters is solved classically using a sequence of linear quadratic and time varying problems the proposed method introduces an approximating sequence of riccati equations asre to explicitly construct nonlinear time varying optimal state feedback controllers for such nonlinear systems under very mild conditions of local lipschitz continuity the sequences converge globally to nonlinear optimal stabilizing feedback controls the computational simplicity and effectiveness of the asre algorithm is an appealing alternative to the tedious and laborious task of solving the hamilton jacobi bellman partial differential equation so the optimality of the asre control is studied by considering the original nonlinear nonquadratic optimization problem and the corresponding necessary conditions for optimality derived from pontryagin s maximum principle global optimal stabilizing state feedback control laws are then constructed this is compared with the optimality of the asre control by considering a nonlinear fighter aircraft control system which is nonaffine in the control numerical simulations are used to illustrate the application of the asre methodology which demonstrate its superior performance and optimality

optimal stochastic regulators with state dependent weights abstract we introduce two optimal regulators for linear stochastic systems the first is of a linear state feedback form and it generalises the linear quadratic regulator by introducing state dependent weights in the cost functional the second is a certain risk sensitive version of the first and it is of a nonlinear state feedback form both regulators are applied to the optimal investment problem

asymptotic behavior of solutions for finite difference equations of reaction diffusion abstract the method of upper lower solutions and the associated existence comparison theorem for continuous reaction diffusion equations are extended to a finite difference system which is a discrete version of the continuous problem of either parabolic type or elliptic type under appropriate boundary condition it is shown by using the existence comparison theorem for the discrete system that for a certain class of initial functions the corresponding time dependent solution converges monotonically to a steady state solution this monotone convergence leads to the stability property of the steady state solution as well as an estimate of the stability region in the framework of the finite difference system numerical results are given for an enzyme substrate reaction model where multiple steady state solutions exist for a certain range of physical parameters in this situation the time dependent solution converges either to the maximal solution or to the minimal solution depending on the corresponding initial function

sliding mode control for a class of non affine nonlinear systems abstract the objective of this paper is to present a sliding mode control smc technique for a class of nonlinear and control non affine systems a rotor supported by a pair of magnetic bearings is introduced as a nonlinear system which is designed as a sequence of linear time varying ltv systems and iterated for each time the sliding surface is designed for each iterated ltv system by using the time response of the previous iteration the surface parameters are selected so that the poles of reduced order ltv system remain on the left hand side of the complex plane it is shown that the response of ltv approximations converges to the nonlinear system s response which is validated by using the non affine nonlinear dynamical equations of rotor active magnetic bearing system

numerical solutions of reaction diffusion equations with nonlocal boundary conditions abstract the purpose of this paper is to present some iterative methods for numerical solutions of a class of nonlinear reaction diffusion equations with nonlocal boundary conditions using the finite difference method and the method of upper and lower solutions we present some monotone iterative schemes for both the time dependent and the steady state finite difference systems each monotone iterative scheme gives a computational algorithm for numerical solutions and an existence comparison theorem for the corresponding finite difference system the existence comparison theorems are used to investigate the asymptotic behavior of the discrete time dependent solution in relation to the discrete maximal and minimal solutions of the steady state problem numerical results are given to a model problem where the solution of the continuous problem is explicitly known and its values at the mesh points are used to compare with the numerical solutions obtained by the monotone iterative schemes

on a class of suboptimal controls for infinite dimensional bilinear systems the suboptimal control of a bilinear system is considered with respect to a quadratic cost criterion the feedback control is in the space of formal power series on a hilbert space

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

finite difference reaction diffusion systems with coupled boundary conditions and time delays abstract this paper is concerned with finite difference solutions of a coupled system of reaction diffusion equations with nonlinear boundary conditions and time delays the system is coupled through the reaction functions as well as the boundary conditions and the time delays may appear in both the reaction functions and the boundary functions the reaction diffusion system is discretized by the finite difference method and the investigation is devoted to the finite difference equations for both the time dependent problem and its corresponding steady state problem this investigation includes the existence and uniqueness of a finite difference solution for nonquasimonotone functions monotone convergence of the time dependent solution to a maximal or a minimal steady state solution for quasimonotone functions and local and global attractors of the time dependent system including the convergence of the time dependent solution to a unique steady state solution also discussed are some computational algorithms for numerical solutions of the steady state problem when the reaction function and the boundary function are quasimonotone all the results for the coupled reaction diffusion equations are directly applicable to systems of parabolic ordinary equations and to reaction diffusion systems without time delays

nonlinear model based predictive control of control nonaffine systems this paper proposes a new nonlinear model based predictive control nlmpc algorithm designed to handle control nonaffine systems nonlinear in the manipulated variable which is based on a reinterpretation of the prediction equation as a taylor series expansion they key feature of this algorithm lies in the use of a process output prediction that accounts for changes in process dynamics as a function of the operating point as well as of the magnitude of the process input change from this algorithm a suboptimal nlmpc algorithm is derived to improve computational efficiency for real time applications the performance improvement of the new algorithm is illustrated by comparing it with a standard nonlinear model based predictive control algorithm using three examples the first two examples are simulations of simple static nonlinear systems and the third example is a simulation of a semibatch acrylonitrile butadiene nbr emulsion polymerization process

finite horizon control constrained nonlinear optimal control using single network adaptive critics to synthesize fixed final time control constrained optimal controllers for discrete time nonlinear control affine systems a single neural network nn based controller called the finite horizon single network adaptive critic is developed in this paper inputs to the nn are the current system states and the time to go and the network outputs are the costates that are used to compute optimal feedback control control constraints are handled through a nonquadratic cost function convergence proofs of 1 the reinforcement learning based training method to the optimal solution 2 the training error and 3 the network weights are provided the resulting controller is shown to solve the associated time varying hamilton jacobi bellman equation and provide the fixed final time optimal solution performance of the new synthesis technique is demonstrated through different examples including an attitude control problem wherein a rigid spacecraft performs a finite time attitude maneuver subject to control bounds the new formulation has great potential for implementation since it consists of only one nn with single set of weights and it provides comprehensive feedback solutions online though it is trained offline

numerical solutions of a three competition lotka volterra system abstract this paper is concerned with finite difference solutions of a lotka volterra reaction diffusion system with three competing species the reaction diffusion system is discretized by the finite difference method and the investigation is devoted to the finite difference system for the time dependent solution and its asymptotic behavior in relation to the corresponding steady state problem three monotone iterative schemes for the computation of the time dependent solution are presented and the sequences of iterations are shown to converge monotonically to a unique positive solution also discussed is the asymptotic behavior of the time dependent solution in relation to various steady state solutions a simple condition on the competing rate constants is obtained which ensures that for every nontrivial nonnegative initial function the corresponding time dependent solution converges either to a unique positive steady state solution or to one of the semitrivial steady state solutions the above results lead to the coexistence and permanence of the competing system as well as computational algorithms for numerical solutions some numerical results from these computational algorithms are given all the conclusions for the reaction diffusion equations are directly applicable to the finite difference solution of the corresponding ordinary differential system

lagrangian manifolds and asymptotically optimal stabilizing feedback control abstract approximations to nonlinear optimal control based on solving a riccati equation which varies with the state have been put forward in the literature it is known that such algorithms are asymptotically optimal given large scale asymptotic stability this paper presents an analysis for estimating the size of the region on which large scale asymptotic stability holds this analysis is based on a geometrical construction of a viscosity type lyapunov function from a stable lagrangian manifold this produces a less conservative estimate than existing approaches in the literature by considering regions of state space over which the stable manifold is multi sheeted rather than just single sheeted

semilinear elliptic boundary value problems in chemical reactor theory abstract this paper is devoted to the study of semilinear elliptic boundary value problems arising in chemical reactor theory which obey the simple arrhenius rate law and newtonian cooling we prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless heat evolution rate