how to detect integrability in cellular automata ultra discrete equations are generalized cellular automata in the sense that the dependent and independent variables take only integer values we present a new method for identifying integrable ultra discrete equations which is the equivalent of the singularity confinement property for difference equations and the painleve property for differential equations using this criterion we find integrable ultra discrete equations which include the ultra discrete painleve equations

linearizable mappings and the low growth criterion we examine a family of discrete second order systems which are integrable through reduction to a linear system these systems were previously identified using the singularity confinement criterion here we analyse them using the more stringent criterion of nonexponential growth of the degrees of the iterates we show that the linearizable mappings are characterized by a very special degree growth the ones linearizable by reduction to projective systems exhibit zero growth i e they behave like linear systems while the remaining ones derivatives of riccati and gambier mapping lead to linear growth this feature may well serve as a detector of integrability through linearization

sufficient second order optimality conditions for semilinear control problems with pointwise state constraints second order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state in contrast to former publications on this subject the cone of critical directions is the smallest possible in the sense that the second order sufficient conditions are the closest to the associated necessary ones the theory is developed for elliptic distributed controls in domains up to dimension three moreover problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one respectively

optimal control problem for an ecosystem with two competing preys and one predator abstract this work is devoted to the optimal control problem for an ecosystem composed of one predator and two competing preys and our goal is to maximize the total density of the three populations we first investigate the existence and uniqueness of the positive strong solution for the controlled system and find an optimal solution under given initial conditions then we establish the first order necessary optimality condition and point out that the optimal control has a bang bang form moreover the second order necessary and sufficient optimality conditions are established

characterization of the solution to the optimal h model reduction problem abstract in this paper the h norm approximation of a given stable proper rational transfer function by a lower order stable proper rational transfer function is studied the solution method is based on the authors observation that the h model reduction problem can be converted into a hankel norm model reduction problem a simple characterization of the solution to the optimal h model reduction problem is developed

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

second order optimality conditions for semilinear elliptic control problems with finitely many state constraints this paper deals with necessary and sufficient optimality conditions for control problems governed by semilinear elliptic partial differential equations with finitely many equality and inequality state constraints some recent results on this topic for optimal control problems based upon results for abstract optimization problems are compared with some new results using methods adapted to the control problems meanwhile the lagrangian formulation is followed to provide the optimality conditions in the first case the lagrangian and hamiltonian functions are used in the second statement finally we prove the equivalence of both formulations

discrete dynamical systems associated with root systems of indefinite type a geometric charactrization of the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but which exhibits chaotic behavior is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be a realization of a cremona isometry on the picard group of the surface it is also shown that the group of cremona isometries is isomorphic to an extended weyl group of indefinite type a method to construct the mappings associated with some root systems of indefinite type is also presented

l optimization and hankel approximation an important l infty optimization problem arising in feedback system design is shown to be equivalent to an optimal hankel norm approximation problem this enables the problem to be solved using state space methods based on balanced realizations

bbb l 2 sufficient conditions for end constrained optimal control problems with inputs in a polyhedron an hbox bbb l 2 local optimality sufficiency theorem is proved for a class of structured infinite dimensional nonconvex programs with constraints of the form u in omega and h u 0 where omega is a set of lebesgue measurable essentially bounded vector valued functions u cdot 0 1 rightarrow hbox bbb r m with range in a polyhedron u and h is a smooth map of the space of essentially bounded functions u cdot into bbb r k the sufficiency theorem is based on formal counterparts of the finite dimensional karush kuhn tucker sufficient conditions in a cartesian product of polyhedra a strengthened variant of pontryagin s necessary condition and structure and continuity conditions on the first and second differentials of the objective function and equality constraint functions the new sufficient conditions are directly applicable to nonconvex continuous time bolza optimal control problems with control quadratic hamiltonians unqualified affine inequality constraints on vector valued control inputs and equality constraints on the terminal state vector or equivalent isoperimetric constraints on integrals of functions depending on the state and control variables

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

optimal control problems with partially polyhedric constraints this paper discusses a class of state constrained optimal control problems for which it is possible to formulate second order necessary or sufficient conditions for local optimality or quadratic growth that do not involve all curvature terms for the constraints this kind of result is classical in the case of polyhedric control constraints our theory of optimization problems with partially polyhedric constraints allows to extend these results to the case when the control constraints are polyhedric in the presence of state constraints satisfying some specific hypotheses the analysis is based on the assumption that some strict semilinearized qualification condition is satisfied we apply the theory to some optimal control problems of elliptic equations with state and control constraints

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

a legendre galerkin spectral method for optimal control problems governed by stokes equations in this paper we study the legendre galerkin spectral approximation of distributed optimal control problems governed by stokes equations we show that the discretized control problems satisfy the well known babuska brezzi conditions by choosing an appropriate pair of discretization spaces for the velocity and the pressure constructing suitable base functions of the discretization spaces leads to sparse coefficient matrices we first derive a priori error estimates in both h 1 and l 2 norms for the legendre galerkin approximation of the unconstrained control problems then both a priori and a posteriori error estimates are obtained for control problems with the constraints of an integral type thanks to the higher regularity of the optimal control finally some illustrative numerical examples are presented to demonstrate the error estimates

second order analysis for strong solutions in the optimal control of parabolic equations in this paper we provide a second order analysis for strong solutions in the optimal control of parabolic equations we consider the case of box constraints on the control and final integral constraints on the state in contrast to sufficient conditions assuring quadratic growth in the weak sense i e when the cost increases at least quadratically for admissible controls uniformly near to the nominal one see e g 16 26 our main result provides a sufficient condition for quadratic growth of the cost for admissible controls whose associated states are uniformly near to the state of the nominal one as a consequence of our results for qualified problems with a strictly convex and quadratic hamiltonian we prove that both notions of quadratic growth coincide

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

an integrability test for differential difference systems the authors present a novel integrability criterion that is adapted to differential difference systems it is based on the recently introduced notion of singularity confinement and uses elements from the singularity analysis approach which has proved to be a most successful integrability criterion for continuous systems a study of the toda lattice illustrates the applicability of the method

second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations this paper deals with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition the case of boundary control is studied pointwise constraints on the control and certain equality and set constraints on the state are considered second order sufficient conditions for local optimality of controls are established

second order analysis for bang bang control problems of pdes in this paper we derive some sufficient second order optimality conditions for control problems of partial differential equations pdes when the cost functional does not involve the usual quadratic term for the control or higher nonlinearities for it though not always in this situation the optimal control is typically bang bang two different control problems are studied the second differs from the first in the presence of the l 1 norm of the control this term leads to optimal controls that are sparse and usually take only three different values we call them bang bang bang controls though the proofs are detailed in the case of a semilinear elliptic state equation the approach can be extended to parabolic control problems some hints are provided in the last section to extend the results

optimal control of navier stokes equations by oseen approximation a non standard sequential quadratic programming type iterational process based on oseen s approximation is proposed and analyzed to solve an optimal control problem for the steady state navier stokes equations further numerical approximation by a finite element method and sample computational experiments are presented too

optimal hankel model reduction for nonminimal systems a basis free descriptor system representation is shown to facilitate the computation of all minimum degree and optimal kth order all pass extensions and hankel norm approximants the descriptor representation has the same simple form for both the optimal and suboptimal cases the method makes hankel model reduction practical for nonminimal and nearly nonminimal systems by eliminating the ill conditioned calculation of a minimal balanced realization a simple numerically sound method based on singular value decomposition enables the results to be expressed in state space form

augmented lagrangian sqp methods for nonlinear optimalcontrol problems of tracking type an augmented lagrangian method with second order update is developed and its relationship to the sequential quadratic programming method is described the rate of convergence proof depends on a second order sufficient optimality condition which is shown to be satisfied for a class of nonlinear optimal control problems of tracking type numerical examples are included which demonstrate the globalizing effect of the augmented lagrangian method

first and second order optimality conditions for a class of optimal control problems with quasilinear elliptic equations a class of optimal control problems for quasilinear elliptic equations is considered where the coefficients of the elliptic differential operator depend on the state function first and second order optimality conditions are discussed for an associated control constrained optimal control problem main emphasis is laid on second order sufficient optimality conditions to this aim the regularity of the solutions to the state equation and its linearization is studied in detail and the pontryagin maximum principle is derived one of the main difficulties is the nonmonotone character of the state equation

algebraic entropy and the space of initial values for discrete dynamical systems a method to calculate the algebraic entropy of a mapping which can be lifted to an isomorphism of a suitable rational surface the space of initial values is presented it is shown that the degree of the nth iterate of such a mapping is given by its action on the picard group of the space of initial values it is also shown by construction that the degree of the nth iterate of every painleve equation in sakai s list is o n2 and therefore its algebraic entropy is zero

optimal control strategies for a new ecosystem governed by reaction diffusion equations abstract an optimal control problem for a new ecosystem described by reaction diffusion equations is studied in this contribution different from the classical prey predator law the phenomenon that the growth of predator population gradually reduces due to the after effects of prey population is ubiquitous in the real world such as allelopathy phenomenon in planktonic ecosystems to model this relationship between predator and prey populations together with diffusion behaviors a novel model with general holling type functional response is presented after that the existence and uniqueness of the global positive strong solution to the ecosystem is obtained by the method of semigroups of operators on this basis an optimal control problem of the ecosystem is considered the existence of optimal control strategy is proved by using minimal sequence then the first order necessary optimality condition is obtained and the corresponding optimal control is found to be of on off form furthermore the second order necessary condition for optimal control and sufficient condition for local optimal control are all established finally a numerical example is offered to demonstrate the applications of the results obtained in this work

second order optimality conditions in sets of sf l infty functions with range in a polyhedron formal extensions of the general second order necessary conditions and sufficient conditions for local optimality in a polyhedral convex set u subset bbb r m are established for sf l infty local optimality and sf l 2 local optimality in the infinite dimensional nonpolyhedral convex set omega of sf l infty functions u cdot 0 1 rightarrow u a more refined analysis for nonconvex cost functions with specially structured differentials yields optimality conditions that apply to an important class of constrained input bolza optimal control problems the gap between the necessary conditions and sufficient conditions in this setting is uncharacteristically small for infinite dimensional problems in the control problem context the sf l infty local optimality conditions and sf l 2 local optimality conditions entail a mild strengthening of a pointwise strict complementarity condition and variants of the legendre clebsch condition and the pontryagin minimum principle in related recent studies similar second order sufficient conditions for the special case u 0 infty are the key hypotheses in corresponding local convergence theories for iterative constrained minimization algorithms

singularity complexity and quasi integrability of rational mappings we investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models by exploiting their nature of birational transformations of projective spaces we give an algorithmic analysis of the structure of invariants of such mappings we discuss some characteristic conditions for their quasi integrability and in particular its links with their singularities in the 2 plane finally we describe some of their propertiesqua dynamical systems making contact with arnol d s notion of complexity and exemplify remarkable behaviours

singularity analysis and integrability for discrete dynamical systems abstract evidence that suggests a necessary consequence of integrability for discrete dynamical systems is presented the evidence is analogous to that provided by the painleve test of integrability for continuous dynamical systems autonomous discretizations of several integrable continuous systems are studied it is shown that those discrete systems exhibiting no chaotic behaviour in the complex space of the iterates possess solutions that have a property analogous to the painleve property of their corresponding continuous dynamical system

the road to the discrete analogue of the painleve property nevanlinna meets singularity confinement the question of integrability of discrete systems is analysed in the light of the recent findings of ablowitz et al who have conjectured that a fast growth of the solutions of a differ ence equation is an indication of nonintegrability the study of the behaviour of the solutions of a mapping is based on the theory of nevanlinna in this paper we show how this approach can be implemented in the csse of second order mappings which include the discrete painlevk equations since the nevanlinna approach does offer only a necessary condition which is not restrictive enough we complement it by the singularity confinement requirement first in an autonomous setting and then for deautonomisation we believe that this three tiered approach is the closest one can get to a discrete analogue of the painlevb property 2003 elsevier science ltd all rights reserved

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

characterization and computation of the solution to the optimal l sub spl infin approximation problem the characterization of the solution to the problem of approximating a given stable proper rational transfer function in l sub spl infin norm sense by a rational function with prescribed number of stable and unstable modes is developed a simple state space suboptimal computational algorithm for the solution is presented

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

second order necessary and sufficient optimality conditions for optimization problems and applications to control theory this paper deals with a class of nonlinear optimization problems in a function space where the solution is restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type second order necessary and sufficient optimality conditions are established where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces the results are applied to some optimal control problems for ordinary and partial differential equations

h sub spl infin norm approximation of systems by constant matrices and related results in this work the h sub spl infin norm approximation of a given stable proper rational transfer function by a constant matrix is considered a new improved bound on h sub spl infin norm of a given stable transfer function is derived optimal approximation for relaxation systems is given

searching for integrable lattice maps using factorization we analyse the factorization process for lattice maps searching for integrable cases the maps were assumed to be at most quadratic in the dependent variables and we required minimal factorization one linear factor after two steps of iteration the results were then classified using the algebraic entropy some new models with polynomial growth strongly associated with integrability were found one of them is a nonsymmetric generalization of the homogeneous quadratic maps associated with kdv modified and schwarzian for this new model we have also verified the consistency around a cube

how to detect the integrability of discrete systems several integrability tests for discrete equations will be reviewed all tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as lax pairs specifically singularity confinement algebraic entropy nevanlinna theory diophantine integrability and discrete systems over finite fields will be described

do integrable cellular automata have the confinement property we analyse a criterion introduced by joshi and lafortune for the integrability of cellular automata obtained from discrete systems through the ultradiscretization procedure we show that while this criterion can be used in order to single out integrable ultradiscrete systems there do exist cases where the system is nonintegrable and still the criterion is satisfied conversely we show that for ultradiscrete systems that are derived from linearizable mappings the criterion is not satisfied we investigate this phenomenon further in the case of a mapping which includes a linearizable subcase and show how the violation of the criterion comes to be finally we comment on the growth properties of ultradiscrete systems

characterization of local quadratic growth for strong minima in the optimal control of semi linear elliptic equations in this article we consider an optimal control problem of a semi linear elliptic equation with bound constraints on the control our aim is to characterize local quadratic growth for the cost function j in the sense of strong solutions this means that the function j growths quadratically over all feasible controls whose associated state is close enough to the nominal one in the uniform topology the study of strong solutions classical in the calculus of variations seems to be new in the context of pde optimization our analysis based on a decomposition result for the variation of the cost combines pontryagin s principle and second order conditions while these two ingredients are known we use them in such a way that we do not need to assume that the hessian of lagrangian of the problem is a legendre form or that it is uniformly positive on an extended set of critical directions

growth and integrability in discrete systems we introduce a new discrete integrability criterion inspired from the recent findings of ablowitz and collaborators this criterion is based on the study of the growth of some characteristic of the solutions of a mapping using nevanlinna theory since the practical implementation of the latter does not always lead to a clear cut answer we complement the growth criterion by the singularity confinement property this combination turns out to be particularly efficient its application allows us to recover the known forms of the discrete painleve equations and to show that no new ones may exist within a given parametrization

finite order meromorphic solutions and the discrete painleve equations let w z be a finite order meromorphic solution of the second order difference equation w z 1 w z 1 r z w z where r z w z is rational in w z and meromorphic in z then either w z satisfies a difference linear or riccati equation or else equation can be transformed to one of a list of canonical difference equations this list consists of all known difference painleve equation of the form together with their autonomous versions this suggests that the existence of finite order meromorphic solutions is a good detector of integrable difference equations

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure