products of prime powers in binary recurrence sequences part ii the elliptic case with an application to a mixed quadratic exponential equation in part i the diophantine equation formula was studied where formula is a linear binary recurrence sequence with positive discriminant in this second part we extend this to negative discriminants we use the p adic and complex gelfond baker theory to find explicit upper bounds for the solutions of the equation we give algorithms to reduce those bounds based on diophantine approximation techniques thus we have a method to solve the equation completely for arbitrary values of the parameters we give an application to a quadratic exponential equation

geometric insight into discrete time cheap and singular linear quadratic riccati lqr problems the hamiltonian system related to discrete time cheap linear quadratic riccati lqr problems is analyzed in a purely geometric context with the twofold purpose of getting a useful insight into its structural features and deriving a numerically implementable solution for the infinite horizon case by only using the standard geometric approach routines that are available

an improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects in this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil and implicitly its kronecker structure the algorithm compares favorably to existing ones in terms of elegance versatility and complexity in particular the algorithm without any modification yields the structural invariants associated with a generalized state space system and its system pencil two related geometric aspects are also discussed we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form we extend the notion of deflating subspace to the singular pencil case c 1997 elsevier science b v

canonical form of symplectic matrix pencils abstract we obtain a real canonical form for real symplectic pencils it extends earlier results which were derived over the complex field so that the canonical form was complex and which were limited to the case where the elementary divisors of eigenvalues on the unit circle have an even degree

lq tracking controls with fixed terminal states and their application to receding horizon controls in this paper finite horizon linear quadratic tracking controls fhlqtc with fixed terminal states are proposed in recursive forms for linear time invariant lti discrete and continuous time systems the proposed fhlqtcs are to minimize finite horizon quadratic cost functions while driving the terminal state exactly to the given fixed reference value the fhlqtcs in this paper are applied to the receding horizon linear quadratic tracking controls rhlqtc with guaranteed stability it is shown through simulation that the proposed fhlqtc moves the terminal state exactly to the given destination and the proposed rhlqtc follows the given reference signal well while guaranteeing stability

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

on the solution of the riccati differential equation arising from the lq optimal control problem in this paper we consider the matrix riccati differential equation rde that arises from linear quadratic lq optimal control problems in particular we establish explicit closed formulae for the solution of the rde with a terminal condition using particular solutions of the associated algebraic riccati equation we discuss how these formulae change as assumptions are progressively weakened an application to lq optimal control is briefly analysed

arithmetical study of a certain ternary recurrence sequence and related questions the complete solution in n yi y2 e z3 of the diophantine equation bn 2y1 3y2 is given where bn nez is berstel s recurrence sequence defined by bo bl 0 b2 1 bn 3 2bn 24bn i 4bn

products of prime powers in binary recurrence sequences part i the hyperbolic case with an application to the generalized ramanujan nagell equation we show how the gelfond baker theory and diophantine approximation techniques can be applied to solve explicitly the diophantine equation formula where formula is a binary recurrence sequence with positive discriminant for arbitrary values of the parameters we apply this to the equation formula which is a generalization of the ramanujan nagell equation formula we present algorithms to reduce upper bounds for the solutions of these equations the algorithms are easy to translate into computer programs we present an example which shows that in practice the method works well

gmpls label space minimization through hypergraph layouts all optical label switching aols is a new technology that performs packet forwarding without any optical electrical optical conversions in this paper we study the problem of routing a set of requests in aols networks using gmpls technology with the aim of minimizing the number of labels required to ensure the forwarding we first formalize the problem by associating to each routing strategy a logical hypergraph called a hypergraph layout whose hyperarcs are dipaths of the physical graph called tunnels in gmpls terminology we define a cost function for the hypergraph layout depending on its total length plus its total hop count minimizing the cost of the design of an aols network can then be expressed as finding a minimum cost hypergraph layout we prove hardness results for the problem namely for general directed networks we prove that it is np hard to find a clogn approximation where c is a positive constant and n is the number of nodes of the network for symmetric directed networks we prove that the problem is apx hard these hardness results hold even if the traffic instance is a partial broadcast on the other hand we provide approximation algorithms in particular an o logn approximation for symmetric directed networks finally we focus on the case where the physical network is a directed path providing a polynomial time dynamic programming algorithm for a fixed number k of sources running in o n k 2 time

the extended symplectic pencil and the finite horizon lq problem with two sided boundary conditions this technical note introduces a new approach to the solution of a very general class of finite horizon optimal control problems for discrete time systems this approach provides a parametric expression for the optimal control sequences as well as the corresponding optimal state trajectories by exploiting a new decomposition of the so called extended symplectic pencil this decomposition provides an original strategy for a more direct solution of the problem with no need of the system theoretic hypotheses including regularity of the symplectic pencil that have always been assumed in the literature so far

relaxed logarithmic barrier function based model predictive control of linear systems in this paper we investigate the use of relaxed logarithmic barrier functions in the context of linear model predictive control we present results that allow to guarantee asymptotic stability of the corresponding closed loop system and discuss further properties like performance and constraint satisfaction in dependence of the underlying relaxation the proposed stabilizing mpc schemes are not necessarily based on an explicit terminal set or state constraint and allow to characterize the stabilizing control input sequence as the minimizer of a globally defined continuously differentiable and strongly convex function the results are illustrated by means of a numerical example

structure preserving numerical algorithm for solving discrete time lmi and dars abstract we present a numerical algorithm to solve a discrete time linear matrix inequality lmi and discrete time algebraic riccati system dars with a given system a b c d we associate a para hermitian matrix pencil then we transform it by an orthogonal transformation matrix into a block triangular para hermitian form under either of the two assumptions 1 matrix pair a b is controllable or 2 matrix pair a b is reachable and a b c d is a left invertible system we extract the solution of lmi and dars by the entries of the orthogonal transformation matrix

a unified approach to finite horizon generalized lq optimal control problems for discrete time systems a closed form expression parameterizing the solutions of the extended symplectic difference equation over a finite time interval is given under the mild assumption of modulus controllability this representation is expressed in terms of the strongly unmixed solution of a discrete are and of an algebraic stein equation the most important application of this result is a generalized version of the finite horizon lq regulator in particular our framework enables different kind of boundary conditions to be treated in a unified fashion without resorting to the riccati difference equation for the computation of the optimal control function

employing the algebraic riccati equation for a parametrization of the solutions of the finite horizon lq problem the discrete time case in this paper a new methodology is developed for the closed form solution of a generalized version of the finite horizon linear quadratic regulator problem for lti discrete time systems the problem considered herein encompasses the classical version of the lq problem with assigned initial state and weighted terminal state as well as the so called fixed end point version in which both the initial and the terminal states are sharply assigned the present approach is based on a parametrization of all the solutions of the extended symplectic system in this way closed form expressions for the optimal state trajectory and control law may be determined in terms of the boundary conditions by taking advantage of standard software routines for the solution of the algebraic riccati and stein equations our results lead to a simple and computationally attractive approach for the solution of the considered optimal control problem without the need of iterating the riccati difference equation

relatively optimal control and its linear implementation motivated by the fact that determining a feedback solution for the optimal control problem under constraints is a hard task we introduce the concept of relative optimality roughly optimality for a specific nominal plant initial condition we consider a generic discrete time finite horizon constrained optimal control problem for linear systems and we seek for a state feedback possibly dynamic controller as a fundamental requirement we do not admit preactions or controller state initialization based on the plant initial state and we assume our controller to be time invariant in particular we do not consider controllers simply achieved by the feedforward and tracking of the optimal trajectory a relatively optimal control is a stabilizing controller such that if initialized at its zero state produces the optimal constrained trajectory for the nominal initial condition of the plant we show that one of such controllers is linear dead beat and its order is equal to the length of the horizon minus the plant order thus of complexity which is known a priori some additional features such as the assignment of the compensator poles to achieve strong stabilization are proposed we show that by means of the proposed approach we can face several problems such as optimal point to point operations optimal impulse response and optimal tracking

comments on structural invariant subspaces of singular hamiltonian systems and nonrecursive solutions of finite horizon optimal control problems in this note it is shown that the main result of concerning a characterization of a pair of structural invariant subspaces associated with the extended symplectic system is a particular case of a result presented in and within a more general and rigorous context we also analyse the proof of the main result of and the way such result is used to accomodate the boundary conditions in the solution of a finite horizon linear quadratic optimal control problem

lq optimal control for 2d roesser models of finite extent abstract this paper investigates several aspects of linear quadratic lq optimal control for roesser models over a two dimensional 2d signal index set of finite extent first we consider the characterisation and computation of open loop control laws when constraints on the system semi states are imposed at both the south west and north east boundaries of the frame i e signal index set of interest by virtue of the quarter plane causal structure of the roesser model the south west and north east boundary conditions are analogous to initial conditions and terminal constraints respectively a necessary and sufficient characterisation of optimality is obtained and explicitly computable formulae are derived to characterise the corresponding control inputs and performance index under reasonable assumptions on the problem data in the second part of the paper the problem of optimal lq control via semi state feedback is considered a 2d riccati like difference equation is introduced to characterise in a sufficient sense a solution to this problem

the solution of triangularly connected decomposable form equations an algorithm is given to solve the equations of the title it generalizes an earlier algorithm to solve discriminant form equations an application is given to finding curves of genus 2 with good reduction outside a given finite set of primes and weierstrass points in given number fields

biconnectivity approximations and graph carvings a spanning tree in a graph is the smallest connected spanning subgraph given a graph how does one find the smallest i e least number of edges 2 connected spanning subgraph connectivity refers to both edge and vertex connectivity if not specified unfortunately the problem is known to be np hard we consider the problem of finding a better approximation to the smallest 2 connected subgraph by an efficient algorithm for 2 edge connectivity our algorithm guarantees a solution that is no more than 3 2 times the optimal for 2 vertex connectivity our algorithm guarantees a solution that is no more than 5 3 times the optimal the previous best approximation factor is 2 for each of these problems the new algorithms and their analyses depend upon a structure called a carving of a graph which is of independent interest we show that approximating the optimal solution to within an additive constant is np hard as well we also consider the case where the graph has edge weights for this case we show that an approximation factor of 2 is possible in polynomial time for finding a k edge connected spanning subgraph this improves an approximation factor of 3 for k 2 due to frederickson and ja ja 1981 and extends it for any k with an increased running time though