convergent finite difference solvers for viscosity solutions of the elliptic monge ampere equation in dimensions two and higher the elliptic monge ampere equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail novel solution methods are required for stability and convergence to the weak viscosity solution in this article we build a wide stencil finite difference discretization for the monge ampere equation the scheme is monotone so the barles souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation solutions of the scheme are found using a damped newton s method we prove convergence of newton s method and provide a systematic method to determine a starting point for the newton iteration computational results are presented in two and three dimensions which demonstrates the speed and accuracy of the method on a number of exact solutions which range in regularity from smooth to nondifferentiable

on the exponential stability of discrete time systems with applications in observer design exponential stability of discrete time systems stability domains and state bounds are obtained with the use of vector norm lyapunov functions for linear systems necessary and sufficient conditions are given for asymptotic stability and exponential bounds are derived for nonlinear systems sufficient conditions for exponential stability also yield a stability domain and exponential bounds applications are demonstrated in observer design for uncertain discrete time nonlinear systems

reduced explicit constrained linear quadratic regulators in this note it is studied how structural properties of certain linear systems can be exploited to derive reduced dimension multiparametric quadratic programs that lead to explicit piecewise linear feedback solutions to the state and input constrained linear quadratic regulation problem the reduced dimensionality typically results in suboptimal controllers of lower complexity with associated computational advantages in the online implementation at the heart of the methods are state space projections based on the singular value decomposition

set membership approximation theory for fast implementation of model predictive control laws in this paper the use of set membership sm methodologies is investigated in the approximation of model predictive control mpc laws for linear systems such approximated mpc laws are derived from a finite number n of exact control moves computed off line properties in terms of guaranteed approximation error closed loop stability and performance are derived assuming only the continuity of the exact predictive control law these results are achieved by means of two main contributions at first it will be shown that if the approximating function enjoys two key properties i e fulfillment of input constraints and explicit evaluation of a bound on the approximation error which converges to zero as n increases then it is possible to guarantee the boundedness of the controlled state trajectories inside a compact set their convergency to an arbitrary small neighborhood of the origin and satisfaction of state constraints moreover the guaranteed performance degradation in terms of maximum state trajectory distance can be explicitly computed and reduced to an arbitrary small value by increasing n then two sm approximations are investigated both enjoying the above key properties the first one minimizes the guaranteed approximation error but its on line computational time increases with n the second one has higher approximation error but lower on line computational time which is constant with n when the off line computed moves are suitably chosen the presented approximation techniques can be systematically employed to obtain an efficient mpc implementation for fast processes the effectiveness of the proposed techniques is tested on two numerical examples

pointwise rates of convergence for the oliker prussner method for the monge ampere equation we study the oliker prussner method exploiting its geometric nature we derive discrete stability and continuous dependence estimates in the max norm by using a discrete alexandroff estimate and the brunn minkowski inequality we show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain nodes still conform to the domain boundary this gives a suitable notion of operator consistency which combined with stability leads to pointwise rates of convergence for classical and non classical solutions of the monge ampere equation

design sensitivity analysis of coupled thermoviscoelastic systems abstract both the direct differentiation and adjoint variable methods for design sensitivity analysis of transient dynamic arbitrarily nonlinear thermoviscoelastic coupled systems are presented in this paper the approach is based on the thermodynamic description of simple materials due to coleman large strains as well as arbitrary material nonlinearities are accounted for in the derivations the domain parametrization or reference volume concept is used allowing a uniform treatment of both the shape and sizing design sensitivity analysis analytical examples demonstrating the use of the derived equations are given this approach provides a solid basis for discretization of the developed equations for subsequent use in numerical computations

approximate explicit constrained linear model predictive control via orthogonal search tree solutions to constrained linear model predictive control problems can be precomputed off line in an explicit form as a piecewise linear state feedback on a polyhedral partition of the state space avoiding real time optimization we suggest an algorithm that can determine an approximate explicit piecewise linear state feedback by imposing an orthogonal search tree structure on the partition this leads to a real time computational complexity that is logarithmic in the number of regions in the partition and the algorithm yields guarantees on the suboptimality asymptotic stability and constraint fulfillment

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

efficient model predictive control for nonlinear systems via function approximation techniques this note deals with the problem of fast implementation of nonlinear model predictive control using approximated control laws at first accuracy properties of a generic approximated controller are introduced together with their influence on closed loop stability and performance then exploiting such results it is shown how set membership sm function approximation theory can be systematically employed to improve the accuracy performance of the resulting controller given by the sum of with a sm approximating function satisfies the above mentioned properties even if they are not met by alone a nonlinear oscillator example shows the effectiveness of the proposed methodology

adjoint sensitivity analysis for nonlinear dynamic thermoelastic systems an adjoint approach is presented to formulate explicit sensitivities for a general functional associated with a nonlinear dynamic thermoelastic system uncoupled thermoelastic response and small deformation behavior are assumed the formulation uses the lagrange multiplier method to define an adjoint system the convolution operator to incorporate transient effects and domain parameterization to describe shape variations variations of the functional are expressed in explicit form with respect to perturbations of the design fields structural shape material properties applied loads prescribed boundary conditions and initial conditions the functional is defined in terms of the design fields and the implicit response fields displacement temperature strain temperature gradient stress heat flux vector reaction force and reaction surface flux in an example problem the finite element method is used to evaluate the real and adjoint responses and the shape sensitivities for a tank design problem the tank is modeled as a nonlinear quasistatic uncoupled thermoelastic system

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

suboptimal receding horizon estimation via noise blocking abstract for discrete time linear systems we propose a suboptimal approach to constrained estimation so that the associated computation burden is reduced this is achieved by enforcing a move blocking mb structure in the estimated process noise sequence pns we show that full information estimation fie and receding horizon estimation rhe with mb are both stable in the sense of an observer the techniques in proving stability are inspired by those that have been proposed for standard rhe to be specific stability results are mainly achieved by i carefully embellishing the general assumptions for standard rhe to accommodate the mb requirement ii exploiting the principle of optimality as well as convexity of the quadratic programs qps associated with fie and rhe iii relying on the fact that the kalman filter is the best linear estimator in the least squares sense a crucial requirement in achieving stability for mb rhe is that the segment structure ss of the pns of mb fie for the optimization steps within the receding horizon i e steps between t n and t 1 has to be enforced in the mb rhe optimization as a result the mb rhe strategy becomes a dynamic estimator with a periodically varying computational complexity the theoretical results have been illustrated with examples

global exponential asymptotic stability in nonlinear discrete dynamical systems abstract for the nonlinear discrete dynamical system x k 1 tx k on bounded closed and convex set d r n we present several sufficient and necessary conditions under which the unique equilibrium point is globally exponentially asymptotically stable the infimum of exponential bounds of convergent trajectories is also derived

moving horizon control of linear systems with input saturation and plant uncertainty part 1 robustness we present a moving horizon feedback system based on constrained optimal control algorithms for linear plants with input saturation the system is a non conventional sampled data system its sampling periods vary from sampling instant to sampling instant and the control during the sampling time is not constant but determined by the solution of an open loop optimal control problem this is a two part paper in this part we show that the proposed moving horizon control system is robustly stable whether the state of the plant is measurable or not in the second part we show that the proposed moving horizon control system is capable of following a class of reference inputs and suppressing a class of disturbances experimental results show that the behaviour of the moving horizon control system is superior to that resulting from some alternative control laws

vibration analysis of linear coupled thermoviscoelastic thin plates by a variational approach abstract according to the integral type constitutive relation of linear coupled thermoviscoelasticity a mathematical model of thin plates is set up by the introduction of structural functions and thermal functions in the sense of the kirchhoff s hypothesis the corresponding integral type variational formulations are presented by means of modern convolution bilinear forms as well as classical cartesian bilinear forms the ritz method in the spatial domain and the differentiating method in the temporal domain are used to approximate the mathematical model in a system of rectangular cartesian coordinates by properties of inequality and parabola the structure of dynamic solution to vibration of a thermoviscoelastic thin plate under a harmonic thermal load is studied in the space splayed by material parameter and loading parameter the influences of thermal excitation frequency mechanical relaxation time and thermal relaxation time on amplitude and phase difference of steady state vibration of a square plate are investigated by amplitude frequency analysis and phase frequency analysis double peak resonance vibration of thermoviscoelastic plates exists for given parameters

fast algorithm for a constrained infinite horizon lq problem this paper presents an efficient algorithmic solution to the infinite horizon linear quadratic optimal control problem for a discrete time siso plant subject to bound constraints on a scalar variable the solution to the corresponding quadratic programming problem is based on the active set method and on dynamic programming it is shown that the optimal solution can be updated after inclusion or removal of an active constraint by a simple procedure requiring in the order of kn operations n being the system order and k the time at which the constraint is included or removed

on the numerical solution of the equation frac partial 2 z partial x 2 frac partial 2 z partial y 2 left frac partial 2 z partial x partial y right 2 f and its discretizations i the equation indicated in the title is the simplest representative of the class of nonlinear equations of monge ampere type equations with such nonlinearities arise in dynamic meteorology geometric optics elasticity and differential geometry in some special cases heuristic procedures for numerical solution are available but in order for them to be successful a good initial guess is required for a bounded convex domain nonnegativef and dirichlet data we consider a special discretization of the equation based on its geometric interpretation for the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence no special information about an initial guess is required and to initiate the iterates a routine step is made the method is self correcting and is structurally suitable for a parallel computer the computer program modules and several examples are presented in two appendices

approximate model predictive control laws for constrained nonlinear discrete time systems analysis and offline design the objective of this work consists in the offline approximation of possibly discontinuous model predictive control laws for nonlinear discrete time systems while enforcing hard constraints on state and input variables obtaining an offline approximation of the receding horizon control law may lead to a very significant reduction of the online computational burden with respect to algorithms based on iterated optimization thus allowing the application to fast dynamics plants the proposed approximation scheme allows to cope with discontinuous control laws such as those arising from constrained nonlinear finite horizon optimal control problems a detailed stability analysis of the closed loop system driven by the approximated state feedback controller shows that the devised technique guarantees the input to state practical stability with respect to the non fading approximation induced errors two examples are provided to show the effectiveness of the method when the approximator is chosen either as a d

nonlinear model predictive control based on collective neurodynamic optimization in general nonlinear model predictive control nmpc entails solving a sequential global optimization problem with a nonconvex cost function or constraints this paper presents a novel collective neurodynamic optimization approach to nmpc without linearization utilizing a group of recurrent neural networks rnns the proposed collective neurodynamic optimization approach searches for optimal solutions to global optimization problems by emulating brainstorming each rnn is guaranteed to converge to a candidate solution by performing constrained local search by exchanging information and iteratively improving the starting and restarting points of each rnn using the information of local and global best known solutions in a framework of particle swarm optimization the group of rnns is able to reach global optimal solutions to global optimization problems the essence of the proposed collective neurodynamic optimization approach lies in the integration of capabilities of global search and precise local search the simulation results of many cases are discussed to substantiate the effectiveness and the characteristics of the proposed approach

on constrained infinite time linear quadratic optimal control abstract this work presents a solution to the infinite time linear quadratic optimal control itlqoc problem with state and control constraints it is shown that a single finite dimensional convex program of known size can yield this solution properties of the resulting value function with respect to initial conditions are also established and are shoen to be useful in determining the aforementioned problem size an example illustrating the method is finally presented

explicit sub optimal linear quadratic regulation with state and input constraints optimal feedback solutions to the infinite horizon lqr problem with state and input constraints based on receding horizon real time quadratic programming are well known in this paper we develop an explicit solution to the same problem eliminating the need for real time optimization it is shown that the resulting feedback controller is piecewise linear this explicit functional structure is exploited for efficient real time implementation a suboptimal strategy based on a suboptimal choice of a finite horizon and imposing additional limitations on the allowed switching between active constraint sets on the horizon is suggested in order to address the computer memory and processing capacity requirements of the explicit solution

real time heuristic search abstract we apply the two player game assumptions of limited search horizon and commitment to moves in constant time to single agent heuristic search problems we present a variation of minimax lookahead search and an analog to alpha beta pruning that significantly improves the efficiency of the algorithm paradoxically the search horizon reachable with this algorithm increases with increasing branching factor in addition we present a new algorithm called real time a for interleaving planning and execution we prove that the algorithm makes locally optimal decisions and is guaranteed to find a solution we also present a learning version of this algorithm that improves its performance over successive problem solving trials by learning more accurate heuristic values and prove that the learned values converge to their exact values along every optimal path these algorithms effectively solve significantly larger problems than have previously been solvable using heuristic evaluation functions

quantitative characterizations of exponential convergence property of nonlinear discrete dynamical systems in this technical note we consider the exponential convergence property of the trajectories of nonlinear discrete dynamical systems with exponential stability based on an invariant quantity derived from lyapunov functions of the systems the essential and quantitative relationship between exponential convergence property of trajectories and the lyapunov functions is revealed

robust dynamic programming for min max model predictive control of constrained uncertain systems we address min max model predictive control mpc for uncertain discrete time systems by a robust dynamic programming approach and develop an algorithm that is suitable for linearly constrained polytopic systems with piecewise affine cost functions the method uses polyhedral representations of the cost to go functions and feasible sets and performs multiparametric programming by a duality based approach in each recursion step we show how to apply the method to robust mpc and give conditions guaranteeing closed loop stability finally we apply the method to a tutorial example a parking car with uncertain mass

simultaneous material load shape variations of thermoelastic structures a continuous approach is adopted in order to find the total variation of a general performance criterion the present analysis may find important physical applications in the simultaneous shape optimization and control of large space structures in time by applied thermal and or mechanical loads the adjoint variable method and the material derivative concept are used to find the sensitivity expressions

ultra fast stabilizing model predictive control via canonical piecewise affine approximations this paper investigates the use of canonical piecewise affine pwa functions for approximation and fast implementation of linear mpc controllers the control law is approximated in an optimal way over a regular simplicial partition of a given set of states of interest the stability properties of the resulting closed loop system are analyzed by constructing a suitable pwa lyapunov function the main advantage of the proposed approach to the implementation of mpc controllers is that the resulting stabilizing approximate mpc controller can be implemented on chip with sampling times in the order of tens of nanoseconds

approximate explicit receding horizon control of constrained nonlinear systems an algorithm for the construction of an explicit piecewise linear state feedback approximation to nonlinear constrained receding horizon control is given it allows such controllers to be implemented via an efficient binary tree search avoiding real time optimization this is of significant benefit in applications that requires low real time computational complexity or low software complexity the method has a priori guarantee of asymptotic stability with region of attraction being a close inner approximation to the stabilizable set this is achieved by ensuring that the approximation error does not exceed the stability margin

a patchy approximation of explicit model predictive control the explicit solution of multi parametric optimisation problems mpop has been used to construct an off line solution to relatively small and medium sized constrained control problems the control design principles are based on receding horizon optimisation and generally use linear prediction models for the system dynamics in this context it can be shown that the optimal control law is a piecewise linear pwl state feedback defined over polytopic cells of the state space however as the complexity of the related optimisation problems increases the memory footprint and implementation of such explicit optimal solution may be burdensome for the available hardware principally due to the high number of polytopic cells in the state space partition in this article we provide a solution to this problem by proposing a patchy pwl feedback control law which intend to approximate the optimal control law the construction is based on the linear interpolation of the exact solution at the vertices of a feasibl

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

dynamic programming for constrained optimal control of discrete time linear hybrid systems in this paper we study the solution to optimal control problems for constrained discrete time linear hybrid systems based on quadratic or linear performance criteria the aim of the paper is twofold first we give basic theoretical results on the structure of the optimal state feedback solution and of the value function second we describe how the state feedback optimal control law can be constructed by combining multiparametric programming and dynamic programming

adaptive fuzzy control with guaranteed convergence of optimal approximation error with no a priori knowledge of plant boundary functions a novel direct adaptive fuzzy controller afc for a class of single input single output siso uncertain affine nonlinear systems is developed in this paper based on the theory of fuzzy logic systems flss with variable universes of discourse uds sufficient conditions that guarantee that the optimal fuzzy approximation error fae is locally convergent are given by the use of the output tracking error and its derivatives as input variables and by the selection of suitable adjusting parameters a variable ud fls with an optimal fae local convergence is constructed and its parameter adaptive law is derived by virtue of the lyapunov stability theorem under the assumption that the optimal fae is bounded it is proved that the closed loop system is asymptotically stable in the sense that all variables are uniformly ultimately bounded and that the tracking errors converge to zero the proposed approach eliminates the influence of the fae on the tracking errors by means of the inherent mechanism of the variable ud fls thus it has the potential to achieve high control performance without additional compensation under only a few fuzzy rules simulation studies demonstrate the superiority of the proposed afc in terms of the settling time tracking accuracy smoothness of the control input and robustness against external disturbances and parameter variations

set membership identification of nonlinear systems in the paper the problem of identifying nonlinear dynamic systems described in nonlinear regression form is considered using finite and noise corrupted measurements most methods in the literature are based on the estimation of a model within a finitely parametrized model class describing the functional form of involved nonlinearities a key problem in these methods is the proper choice of the model class typically realized by a search from the simplest to more complex ones linear bilinear polynomial neural networks etc in this paper an alternative approach based on a set membership framework is presented not requiring assumptions on the functional form of the regression function describing the relations between measured input and output but assuming only some information on its regularity given by bounds on its gradient in this way the problem of considering approximate functional forms is circumvented moreover noise is assumed to be bounded in contrast with statistical methods which rely on assumptions such as stationarity ergodicity uncorrelation type of distribution etc whose validity may be difficult to test reliably and is lost in presence of approximate modeling in this paper necessary and sufficient conditions are given for the validation of the considered assumptions an optimal interval estimate of the regression function is obtained providing its uncertainty range for any assigned regressor values the set estimate allows to derive an optimal identification algorithm giving estimates with minimal guaranteed l p error on the assigned domain of the regressors the properties of the optimal estimate are investigated and its worst case l p identification error is evaluated the presented approach is tested and compared with other nonlinear methods on the identification of a water heater a mechanical system with input saturation and a vehicle with controlled suspensions

a modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations this paper develops a fully discrete modified characteristic finite element method for a coupled system consisting of the fully nonlinear monge ampere equation and a transport equation the system is the eulerian formulation in the dual space for b j hoskins semigeostrophic flow equations which are widely used in meteorology to model frontogenesis to overcome the difficulty caused by the strong nonlinearity we first formulate at the differential level a vanishing moment approximation of the semigeostrophic flow equations a methodology recently proposed by the authors x feng and m neilan j sci comput 38 2009 pp 74 98 which involves approximating the fully nonlinear monge ampere equation by a family of fourth order quasilinear equations we then construct a fully discrete modified characteristic finite element method for the regularized problem it is shown that under certain mesh constraint the proposed numerical method converges with an optimal order rate of convergence in particular the obtained error bounds show explicit dependence on the regularization parameter varepsilon numerical tests are also presented to validate the theoretical results and to gauge the efficiency of the proposed fully discrete modified characteristic finite element method

brief paper an integrated perturbation analysis and sequential quadratic programming approach for model predictive control computationally efficient algorithms are critical in making model predictive control mpc applicable to broader classes of systems with fast dynamics and limited computational resources in this paper we propose an integrated formulation of perturbation analysis and sequential quadratic programming inpa sqp to address the constrained optimal control problems the proposed algorithm combines the complementary features of perturbation analysis and sqp in a single unified framework thereby leading to improved computational efficiency and convergence property a numerical example is reported to illustrate the proposed method and its computational effectiveness

variational principle for shape design sensitivity analysis in this paper the adjoint method of design sensitivity analysis is stated as a general variational principle that is applicable to a wide range of problems the principle gives a very simple and straightforward method of obtaining the design sensitivity expression for a functional dependent on the state fields the sensitivity expression involves certain adjoint fields and explicit design variations of the functional and the governing equations for the primary state fields the principle is stated in terms of an augmented functional that is defined by adding to the response functional whose sensitivity is desired the equilibrium equation of the primary problem then explicit design variation of the augmented functional gives the total design variation of the response functional stationarity of the augmented functional with respect to the state fields defines the adjoint problem for the response functional the principle is proved and applied to several classes of linear and nonlinear problems such as field problems structural problems and dynamic response problems it is shown that the design sensitivity expressions derived in the literature using long procedures are obtained quite routinely by use of the principle

sensitivity analysis for coupled thermoelastic systems abstract this paper presents an adjoint approach derived from the reciprocal theorem for the sensitivity analysis of linear dynamic thermoelastic systems the variation of a general response functional is expressed in explicit form with respect to variations of the design fields which consist of the material properties applied loads prescribed boundary initial conditions and the structural shape the functional is dependent on these design quantities as well as the following implicity defined response fields displacement temperature stress strain heat flux temperature gradient reaction forces and reaction surface flux the formulation incorporates the reciprocal relation between variations of the real system design and response fields and an adjoint state here convolution is employed in lieu of time mappings used in other transient adjoint sensitivity derivations specializations of the formulation to uncoupled combined quasi static uncoupled and steady state thermoelasticity are also presented the finite element method is used to demonstrate the application of the formulation to a problem in automobile engine design

the monge ampere equation various forms and numerical solution we present three novel forms of the monge ampere equation which is used e g in image processing and in reconstruction of mass transportation in the primordial universe the central role in this paper is played by our fourier integral form for which we establish positivity and sharp bound properties of the kernels this is the basis for the development of a new method for solving numerically the space periodic monge ampere problem in an odd dimensional space convergence is illustrated for a test problem of cosmological type in which a gaussian distribution of matter is assumed in each localised object and the right hand side of the monge ampere equation is a sum of such distributions

technical communique evaluation of piecewise affine control via binary search tree we present an algorithm for generating a binary search tree that allows efficient computation of piecewise affine pwa functions defined on a polyhedral partition this is useful for pwa control approaches such as explicit model predictive control as it allows the controller to be implemented online with small computational effort the computation time is logarithmic in the number of regions in the pwa partition

analytical expression of explicit mpc solution via lattice piecewise affine function an analytical expression of the explicit solution to linear model predictive control mpc is proposed by the introduction of a lattice piecewise affine pwa function a systematic procedure is developed for building a lattice pwa representation from a continuous explicit mpc solution obtained by a multi parametric program a simple method is presented to remove the redundant parameters in the lattice expression of mpc control laws the effectiveness of this approach is supported by the study of three benchmark mpc problems the proposed analytical expression provides a very efficient and practically viable method for implementing the explicit mpc solutions regarding its online calculation and memory space requirements

stability of model predictive control with mixed constraints we derive stability conditions for model predictive control mpc with hard constraints on the inputs and soft constraints on the outputs for an infinitely long output horizon we show that with state feedback mpc is globally asymptotically stabilizing if and only if all the eigenvalues of the open loop system are in the closed unit disk with output feedback we show that the results hold if all the eigenvalues are strictly inside the unit circle the online optimization problem defining mpc can be posed as a finite dimensional quadratic program even though the output constraints are specified over an infinite horizon

discretization of functionals involving the monge ampere operator gradient flows in the wasserstein space have become a powerful tool in the analysis of diffusion equations following the seminal work of jordan kinderlehrer and otto jko the numerical applications of this formulation have been limited by the difficulty to compute the wasserstein distance in dimension geqslant 2 one step of the jko scheme is equivalent to a variational problem on the space of convex functions which involves the monge ampere operator convexity constraints are notably difficult to handle numerically but in our setting the internal energy plays the role of a barrier for these constraints this enables us to introduce a consistent discretization which inherits convexity properties of the continuous variational problem we show the effectiveness of our approach on nonlinear diffusion and crowd motion models

brief paper using hash tables to manage the time storage complexity in a point location problem application to explicit model predictive control the online computational burden of linear model predictive control mpc can be moved offline by using multi parametric programming so called explicit mpc the solution to the explicit mpc problem is a piecewise affine pwa state feedback function defined over a polyhedral subdivision of the set of feasible states the online evaluation of such a control law needs to determine the polyhedral region in which the current state lies this procedure is called point location its computational complexity is challenging and determines the minimum possible sampling time of the system a new flexible algorithm is proposed which enables the designer to trade off between time and storage complexities utilizing the concept of hash tables and the associated hash functions the proposed method solves an aggregated point location problem that overcomes prohibitive complexity growth with the number of polyhedral regions while the storage processing trade off can be optimized via scaling parameters the flexibility and power of this approach is supported by several numerical examples

brief an algorithm for multi parametric quadratic programming and explicit mpc solutions explicit solutions to constrained linear model predictive control problems can be obtained by solving multi parametric quadratic programs mp qp where the parameters are the components of the state vector we study the properties of the polyhedral partition of the state space induced by the multi parametric piecewise affine solution and propose a new mp qp solver compared to existing algorithms our approach adopts a different exploration strategy for subdividing the parameter space avoiding unnecessary partitioning and qp problem solving with a significant improvement of efficiency

a numerical method for the elliptic monge ampere equation with transport boundary conditions the problem of optimal mass transport arises in numerous applications including image registration mesh generation reflector design and astrophysics one approach to solving this problem is via the monge ampere equation while recent years have seen much work in the development of numerical methods for solving this equation very little has been done on the implementation of the transport boundary condition in this paper we propose a method for solving the transport problem by iteratively solving a monge ampere equation with neumann boundary conditions to enable mappings between variable densities we extend an earlier discretization of the equation to allow for right hand sides that depend on gradients of the solution b d froese and a m oberman siam j numer anal 49 2011 pp 1692 1714 this discretization provably converges to the viscosity solution the resulting system is solved efficiently with newton s method we provide several challenging computational examples that demonstra

nsm constrained approximation of lipschitz functions from data the problem of approximating a function from a set of its values in the presence of constraints is considered solving this problem is useful in system identification state estimation and control the typical approach to this problem consists in 1 deriving an approximation without considering the constraints and 2 using the constraints to clip the approximation however using this approach may give poor results and in any case no optimality properties can be guaranteed in this paper a nonlinear set membership method is developed allowing optimal constrained approximation the effectiveness of the proposed method is shown through its application to the problem of model predictive control approximation

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

the boundary method for semi discrete optimal transport partitions and wasserstein distance computation abstract we introduce a new technique which we call the boundary method for solving semi discrete optimal transport problems with a wide range of cost functions the boundary method reduces the effective dimension of the problem thus improving complexity for cost functions equal to a p norm with p 1 we provide mathematical justification convergence analysis and algorithmic development our testing supports the boundary method with these p norms as well as other more general cost functions

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

brief paper set membership approximation of discontinuous nonlinear model predictive control laws in this paper the use of set membership sm methods is investigated in order to derive off line an approximation of a discontinuous nonlinear model predictive control nmpc law the approximating function can then be evaluated on line instead of solving the nonlinear program embedded in the nmpc scheme this way a significant decrease of computational times may be obtained thus allowing the application of nmpc also to systems with fast dynamics it is shown that the knowledge of the discontinuities is needed to achieve an approximated controller with arbitrarily small approximation error by exploiting such a knowledge sm techniques already developed for the case of continuous nmpc laws are generalized in order to approximate discontinuous ones the stability of the origin of the closed loop system with the approximated control law is analyzed and a numerical example is employed to illustrate the features of the proposed approach

on characterizations of exponential stability of nonlinear discrete dynamical systems on bounded regions in this paper we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space through introducing two new concepts the lip constant of a nonlinear operator and strongly equivalent metrics of a norm we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics or if and only if the lip constant of the nonlinear operator is less than one in the latter case we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the lip constant based on the obtained results we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the lip constant and strongly equivalent metrics the obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems

heuristically enhanced feedback control of constrained discrete time linear systems recent advances in computer technology have spurred new interest in the use of feedback controllers based upon on line minimization for the control of constrained linear systems still the use of computers in the feedback loop has been hampered by the fact that the amount of time available for computation in most sampled data systems is not enough to achieve a complete solution using conventional algorithms several ad hoc techniques have been proposed but their applicability is restricted by the lack of supporting theory in this paper we present a theoretical framework to analyze the stability of the closed loop system resulting from the use of on line optimization in the feedback loop using these results we show that a suboptimal algorithm based upon the use of heuristic search techniques yields asymptotically stable systems provided that enough computation power is available to solve at each sampling interval an optimization problem considerably simpler than the original the controller presented in this paper is valuable for situations where the customary approaches of using pontryagin s minimum principle or storing a family of extremal curves are not applicable due to limitations in the computational resources available

numerical solution of the optimal transportation problem using the monge ampere equation a numerical method for the solution of the elliptic monge ampere partial differential equation with boundary conditions corresponding to the optimal transportation ot problem is presented a local representation of the ot boundary conditions is combined with a finite difference scheme for the monge ampere equation email protected s method is implemented leading to a fast solver comparable to solving the laplace equation on the same grid several times theoretical justification for the method is given by a convergence proof in the companion paper 4 solutions are computed with densities supported on non convex and disconnected domains computational examples demonstrate robust performance on singular solutions and fast computational times

designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport we study here the problem of determining a system of two refractive interfaces transforming a plane wavefront of a given shape and radiation intensity into a coherent output plane wavefront with prescribed output position shape and intensity such interfaces can be refracting surfaces of two different lenses or of one lens in geometrical optics approximation the analytic formulation of this problem in both cases requires construction of maps with controlled jacobian though this jacobian can be expressed as a second order partial differential equation of monge ampere type for a scalar function defining one of the refracting surfaces its analysis is not straightforward in this paper we use a geometric approach for reformulating the problem in certain associated measures and defining weak solutions existence and uniqueness of weak solutions in lipschitz classes for both cases are established by variational methods our results show in particular that two types of interfaces exist in each case for the same data one of these types always consists of two interfaces one of which is concave or convex and the second convex or concave while the interfaces of the second type may be neither convex nor concave the availability of a design with convex concave lenses is particularly important for fabrication the truly geometric nature of this problem permits its statement and investigation in mathbb r n 1 n geqq 1

variational method for design sensitivity analysis in nonlinear structural mechanics this paper describes a unified variational theory of design sensitivity analysis of linear and nonlinear structures for shape nonshape and material selection problems the concepts of reference volume and adjoint structure are used to develop the unified viewpoint a general formula for design sensitivity analysis is derived several analytical linear and nonlinear examples are used to interpret various terms of the formula and demonstrate its use such analytical examples give insight into the application of the general formula to more complex problems that must be treated numerically adjoint state fields are also interpreted as sensitivities for the functional related to the primary structure

c 0 penalty methods for the fully nonlinear monge ampere equation in this paper we develop and analyze c 0 penalty methods for the fully nonlinear monge ampere equation det d 2 u f in two dimensions the key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric stable and consistent with the continuous linearization we are then able to show the well posedness of the penalty method as well as quasi optimal error estimates using the banach fixed point theorem as our main tool numerical experiments are presented which support the theoretical results

fast finite difference solvers for singular solutions of the elliptic monge ampere equation the elliptic monge ampere equation is a fully nonlinear partial differential equation which originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail in this article we build a finite difference solver for the monge ampere equation which converges even for singular solutions regularity results are used to select a priori between a stable provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain this allows singular solutions to be computed using a stable method and regular solutions to be computed more accurately the resulting nonlinear equations are then solved by newton s method computational results in two and three dimensions validate the claims of accuracy and solution speed a computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities

piecewise linear lq control for systems with input constraints abstract a piecewise linear control plc law based on lq theory is derived ths law raises an lq gain as the controlled error converges towards the origin it can be computed off line and it guarantees that input bounds are never exceeded without causing input saturation a succession of positively invariant sets are defined of diminishing size and associated with each is the corresponding highest lq gain possible in the presence of the input bounds the formulation gives rise to an iteration function whose solution is a fixed point fp the fp defines the optimal gain and the invariant set which comprise ellipsoids fp existence uniqueness and convergence properties are proven we explain the algorithms for the generation and subsequent implementation of the plc law the notion of a switching function is introduced consideration is given to the computational aspects of the fp the plc scheme is demonstrated for the linearized dynamics of the simple pendulum and the puma 560 robot a discrete time plc law can also be derived using the discrete time algebraic riccati equation

finite element methods for fully nonlinear second order pdes based on a discrete hessian with applications to the monge ampere equation the purpose of this paper is twofold first we modify a method due to lakkis and pryer where the notion of a discrete hessian is introduced to compute fully nonlinear second order pdes the discrete hessian used in our approach is entirely local making the resulting linear system within the newton iteration much easier to solve the second contribution of this paper is to analyze both lakkis and pryer s method and its modification in parallel applied to the two dimensional monge ampere equation in both cases we show the well posedness of the methods as well as derive optimal error estimates numerical experiments are presented which i back up the theoretical findings and ii indicate that the methods are able to capture weak viscosity solutions

optimal control of sampled data piecewise affine systems this paper discusses the optimal continuous time control problem of a class of piecewise affine pwa systems where the switching action of the discrete state is determined at each sampling time according to a condition on the continuous state such a system is called here the sampled data pwa sd pwa system first important remarks on the control design of this system via the continuous or discrete time pwa model are pointed out which motivate us to use the sd pwa model next based on the good properties of the proposed model an optimal continuous time controller of the sd pwa systems is proposed

global exponential stability of impulsive delay difference equation abstract in this paper an impulsive delay difference equation is considered by establishing an impulsive delay difference inequality and using the properties of cone and eigenspace of the spectral radius of nonnegative matrices some new sufficient conditions for global exponential stability of the impulsive delay difference equation are obtained the results extend and improve the earlier publications an example is given to demonstrate the effectiveness of the theory

a numerical method for variational problems with convexity constraints we consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex this problem is at the interface between convex analysis convex optimization variational problems and partial differential equation techniques the approach is to approximate the nonpolyhedral cone of convex functions by a polyhedral cone which can be represented by linear inequalities this approach leads to an optimization problem with linear constraints which can be computed efficiently hundreds of times faster than existing methods

design sensitivity analysis in dynamic thermoviscoelasticity with implicit integration design sensitivity equations for coupled thermoviscoelastic systems are discretized via the finite element method the approach is developed for structural systems by using the total lagrangian approach and the reference domain concept the discretization is based on implicit integration schemes the implementation of finite element and design sensitivity analysis utilizes the christensen naghdi free energy function and the direct differentiation approach which is most suitable for the systems under consideration a relationship between the discretized sensitivity equations and the equations for the original analysis is shown use of the same discretization for analysis and sensitivity analysis is emphasized partial derivatives of the right hand side required for sensitivity calculations are implemented via the central difference method which provides greater flexibility without sacrificing accuracy examples of analysis and design sensitivity analysis for a thermoviscoelastic non linear truss and a plate with a hole are given the calculated sensitivity results are verified by comparison with overall finite difference calculations

quantitative studies on asymptotic growth behaviors of trajectories of nonlinear discrete dynamical systems this technical note studies quantitatively asymptotic growth behaviors of trajectories agbt of nonlinear autonomous discrete dynamical system that has unbounded domain non lipschitz continuous nonlinear operator and stable or unstable equilibrium point we explain how trajectory motion speed is quantitatively determined in the system and study exact computation and sharp estimation of the smallest exponential bound of trajectories we characterize exponential stability and asymptotic stability of the system from a new point of view and provide a simple condition to distinguish them from each other these results extend existing results that were obtained in some special cases of the system and are helpful for quantitative analysis and understanding of agbt of the system