constrained linear quadratic regulation the paper is a contribution to the theory of the infinite horizon linear quadratic regulator lqr problem subject to inequality constraints on the inputs and states extending an approach first proposed by sznaier and damborg 1987 a solution algorithm is presented which requires solving a finite number of finite dimensional positive definite quadratic programs the constrained lqr outlined does not feature the undesirable mismatch between open loop and closed loop nominal system trajectories which is present in the other popular forms of model predictive control mpc that can be implemented with a finite quadratic programming algorithm the constrained lqr is shown to be both optimal and stabilizing the solution algorithm is guaranteed to terminate in finite time with a computational cost that has a reasonable upper bound compared to the minimal cost for computing the optimal solution inherent to the approach is the removal of a tuning parameter the control horizon which is present in other mpc approaches and for which no reliable tuning guidelines are available two examples are presented that compare constrained lqr and two other popular forms of mpc the examples demonstrate that constrained lqr achieves significantly better performance than the other forms of mpc on some plants and the computational cost is not prohibitive for online implementation

generic cartan invariants for frobenius kernels and chevalley groups chastkofsky 2 described the generic behavior of the composition factor multiplicities cartan invariants of indecomposable projective modules for finite groups of lie type in the defining characteristic more recently ye lo provided a different approach along with interesting refinements in each case the reason for the existence of generic cartan invariants is somewhat hidden behind the complicated character manipulations involved here we offer a more conceptual treatment emphasizing the much simpler picture for the group schemes g t associated with the frobenius kernels g in the ambient algebraic group this is closer in spirit to lo than to 2 but avoids the messy explicit calculations found in both papers

extensions of simple modules for finite chevalley groups let g be an almost simple algebraic group over an algebraically closed field k of positive characteristic p we assume that g is defined and split over the prime field 5 for it 3 1 we denote by g n the finite group consisting of the points of g over the field with p elements our first result states that for a certain class of simple modules for g n the only extensions that can occur are those which are restrictions of g extensions see theorem 2 8 the full story about g extensions is not known yet except for a few rather small groups however much more is known about extensions for the algebraic group than about those for the finite group for instance the linkage principle l gives an upper bound for which g extensions may occur moreover a proof of the lusztig conjecture on the irreducible characters for g also implies strong statements about g extensions see 3 proposition 2 81 it is an open question whether all extensions between simple g n modules can be found via the extension theory for g we prove that in generic cases this is in fact so see theorem 3 2 moreover for the groups x 3 p and sp 4 p we are able to determine completely all extensions between simple modules from the corresponding known results for the algebraic groups x 3 k and sp 4 k our results extend previous results by cline parshall and scott s 91 who consider extensions of the trivial module by a minimal module and smith ls who considers extensions between certain minimal modules the method we use is a generalization of the one used by humphreys 131 who proves the existence of certain selfextensions for sp 4 p it should be pointed out that our methods do not work for p small on the other hand everything we do generalizes easily to the twisted chevalley groups see theorem 3 4 we demonstrate this by working out all extensions for su 3 p see section 4 3

general thermostatistical formalisms invariance under uniform spectrum translations and tsallis q additivity the recent proposal by tsallis of a generalized thermostatistics devised to treat systems endowed with long range interactions long range memory eects or a fractal like relevant phase space has raised interesting and profound issues concerning the properties of general thermostatistical formalisms in the present paper we identify families of thermostatistical formalisms that share some fundamental characteristics the canonical ensemble s invariance under uniform translations of the hamiltonian s energy spectrum is shown to be a universal property veried by any thermostatistical formalism based upon linear mean energy constraints we also provide multiparametric families of entropies exhibiting tsallis q additivity law c 1999 elsevier science b v all rights reserved

ordinary and modular characters of su 3 p abstract this article deals with the relationship between ordinary and p modular characters of the group su 3 p parallel to the treatment of sl 3 p in j e humphreys ordinary and modular characters of sl 3 p j algebra 72 1981 8 16 here we denote by su 3 q the special unitary subgroup of sl 3 q2 where q is a power of the prime p the notation su 3 q2 is perhaps more standard but we want to emphasize the comparison with sl 3 q our main conclusion is that the decomposition behavior is essentially the same for both groups in a sense to be made more precise below

variations on hulsurkar s matrix with applications to representations of algebraic chevalley groups let a be an indecomposable root system for a semi simple simply connected algebraic group defined over the field k if let x be the root lattice x the weight lattice x the set of dominant weights and w the weyl group of a let z x be the group ring whose canonical basis we write in the form e d a e x and let z x w denote the ring of invariants under the natural action of w on z xl in 3 certain elements of z xl were defined and it was shown that they were generically equal to generalizations given by verma i1 and jantzen 8 of invariants defined by hulsurkar in 4 in this paper we will define a matrix indexed by x and whose entries are integers which are closely related to these elements and from which in fact the latter can be found we will give some properties of these elements and show how they can be computed in particular we will show that the usual partial ordering of x makes this matrix upper triangular with ones on the diagonal we will also show how our methods give some easy derivations of results of jantzen and show how they can be used to compute generic patterns of the decomposition of weyl modules into irreducibles

controller design for markov jumping systems subject to actuator saturation in this paper the stochastic stabilization problem for a class of markov jumping linear systems mjls subject to actuator saturation is considered the concept of domain of attraction in mean square sense is used to analyze the closed loop stability when the jumping mode is available a mode dependent state feedback controller is developed otherwise we give a less conservative approach to design the mode independent state feedback controller both design procedures can be converted into a set of linear matrix inequalities lmis finally a numerical example is provided to show the effectiveness of the techniques

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

quantum corrections for general partition functions abstract the classical limit for quite general partition functions is obtained using coherent states in this framework a general procedure is presented to obtain all the quantum corrections to the classical limit in particular the first and second order corrections are worked out explicitly and the classical limit for the tsallis thermostatistics is determined the results of this work generalize the ones obtained by wigner phys rev 40 1932 749 for usual statistical mechanics

the cartan invariant matrix for the finite symplectic group sp 4 11 the cartan invariant matrix c c 1 x 1 t of the finite symplecticgroup sp 4 11 of type b 2 is determined in this paper

quantum statistical mechanics for nonextensive systems the traditional basis of description of many particle systems in terms of green functions is here generalized to the case when the system is nonextensive by incorporating the tsallis form of the density matrix indexed by a nonextensive parameter q this is accomplished by expressing the many particle q green function in terms of a parametric contour integral over a kernel multiplied by the usual grand canonical green function which now depends on this parameter we study one and two particle green functions in detail from the one particle green function we deduce some experimentally observable quantities such as the one particle momentum distribution function and the one particle energy distribution function special forms of the two particle green functions are related to physical dynamical structure factors some of which are studied here we deduce different forms of sum rules in the q formalism a diagrammatic representation of the q green functions similar to the traditional ones follows because the equations of motion for both of these are formally similar approximation schemes for one particle q green functions such as hartree and hartree fock schemes are given as examples this extension enables us to predict possible experimental tests for the validity of this framework by expressing some observable quantities in terms of the q averages

low dimensional projective indecomposable modules for chevalley groups in defining characteristic abstract the paper studies lower bounds for the dimensions of projective indecomposable modules for chevalley groups g in defining characteristic p the main result extending earlier one by malle and weigel 2008 determines the modules in question of dimension equal to the order of a sylow p subgroup of g we also substantially generalize a result by ballard 1978 on lower bounds for the dimensions of projective indecomposable modules and find lower bounds in some cases where ballard s bounds are vacuous

embeddings of hecke algebras in group algebras let g be a finite group and let b be a subgroup of g the endomorphism algebra of the induced permutation representation indg 1 is called the hee but the multiplying elements are different from those mentioned above even in characteristic zero since their definition involves only division by 1 h where h n w b in particular if r is a field of characteristic p the duals of our elements h turn out to be words in the steinberg idempotents of parabolic subgroups of g our results provide explicit formulae for an arbitrary product of these in rg see theorem 1 10 this modular case is discussed in section 3 together with related questions on the representation theory of g the modular case was partially treated by kuhn 6 and landrock 7 and we thank particularly the former for sharing his ideas with us 159 0021 8693 87 3 00

robust receding horizon control of constrained nonlinear systems we present a method for the construction of a robust dual mode receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error the controller is dual mode in a neighborhood of the origin the control action is generated by a linear feedback controller designed for the linearized system outside this neighborhood receding horizon control is employed existing receding horizon controllers for nonlinear continuous time systems which are guaranteed to stabilize the nonlinear system to which they are applied require the exact solution at every instant of an optimal control problem with terminal equality constraints these requirements are considerably relaxed in the dual mode receding horizon controller presented in this paper stability is achieved by imposing a terminal inequality rather than an equality constraint only approximate minimization is required a variable time horizon is permitted robustness is achieved by employing conservative state and stability constraint sets thereby permitting a margin of error the resultant dual mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear constrained systems

extensions for finite chevalley groups i let g be a connected semisimple algebraic group defined and split over the field fp with p elements and k be the algebraic closure of fp assume further that g is almost simple and simply connected and let g fq be the finite chevalley group consisting of fq rational points of g where q pr for a non negative integer r in this paper formulas are found relating extensions between simple kg fq modules and extensions over g considered as an algebraic group over k one of these formulas which only holds for primes p 3 h 1 where h is the coxeter number of g is then used to show the vanishing of self extensions between simple kg fq modules except for certain simple modules when r 1 and the underlying root system is of type a1 or cn

discrete time stability with perturbations application to model predictive control exponential stability and asymptotic perturbed stability results are derived for nonlinear discrete time systems some related results are available in the literature but proofs for the discrete time case are difficult to find our motivation for obtaining these results was to determine conditions under which discrete time nonlinear model predictive control is stabilizing in the face of perturbations this problem has particular relevance when output rather than state feedback is used a nonlinear state estimator is then required to reconstruct the state and it is necessary to identify the class of reconstruction errors for which the cascaded estimator controller assembly is stabilizing

normalized tsallis entropy and its implications for the nonextensive thermostatistics abstract implications from a thermostatistics based on normalized tsallis entropy introduced by rajagopal and abe phys rev lett 83 1999 1911 are analyzed in particular this normalization gives the thermodynamic stability c q 0 for tsallis formalism leading to a natural extension to the fluctuation dissipation theorem furthermore an example is worked out in connection with a fractal spectrum of energy suggesting the relevance of the present formalism

enlarging the domain of attraction of mpc controllers this paper presents a method for enlarging the domain of attraction of nonlinear model predictive control mpc the usual way of guaranteeing stability of nonlinear mpc is to add a terminal constraint and a terminal cost to the optimization problem such that the terminal region is a positively invariant set for the system and the terminal cost is an associated lyapunov function the domain of attraction of the controller depends on the size of the terminal region and the control horizon by increasing the control horizon the domain of attraction is enlarged but at the expense of a greater computational burden while increasing the terminal region produces an enlargement without an extra cost in this paper the mpc formulation with terminal cost and constraint is modified replacing the terminal constraint by a contractive terminal constraint this constraint is given by a sequence of sets computed off line that is based on the positively invariant set each set of this sequence does not need to be an invariant set and can be computed by a procedure which provides an inner approximation to the one step set this property allows us to use one step approximations with a trade off between accuracy and computational burden for the computation of the sequence this strategy guarantees closed loop stability ensuring the enlargement of the domain of attraction and the local optimality of the controller moreover this idea can be directly translated to robust mpc

extensions for finite chevalley groups ii let g be a semisimple simply connected algebraic group defined and split over the field fp with p elements let g fq be the finite chevalley group consisting of the fq rational points of g where q p r and let g r be the rth frobenius kernel the purpose of this paper is to relate extensions between modules in mod g f q and mod g r with extensions between modules in mod g among the results obtained are the following for r 2 and p 3 h 1 the g f q extensions between two simple g f q modules are isomorphic to the g extensions between two simple p r restricted g modules with suitably twisted highest weights for p 3 h 1 we provide a complete characterization of h 1 g f q h0 where h 0 ind g b a and a is p r restricted furthermore for p 3 h 1 necessary and sufficient bounds on the size of the highest weight of a g module v are given to insure that the restriction map h 1 g v h 1 g f q v is an isomorphism finally it is shown that the extensions between two simple p r restricted g modules coincide in all three categories provided the highest weights are close together

brief nonlinear model predictive control with polytopic invariant sets ellipsoidal invariant sets have been widely used as target sets in model predictive control mpc these sets can be computed by constructing appropriate linear difference inclusions together with additional constraints to ensure that the ellipsoid lies within a given inclusion polytope the choice of inclusion polytope has a significant effect on the size of the target ellipsoid but the optimal inclusion polytope cannot in general be computed systematically this paper shows that use of polytopic invariant sets overcomes this difficulty allowing larger stabilizable sets without loss of performance in the interests of online efficiency consideration is focused on interpolation based mpc

perturbation and variational methods in nonextensive tsallis statistics a unified presentation of the perturbation and variational methods for the generalized statistical mechanics based on tsallis entropy is given here in the case of the variational method the bogoliubov inequality is generalized in a very natural way following the feynman proof for the usual statistical mechanics the inequality turns out to be form invariant with respect to the entropic index q the method is illustrated with a simple example in classical mechanics the formalisms developed here are expected to be useful in the discussion of nonextensive systems

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

the gln q module structure of the symmetric algebra around the steinberg module abstract we determine the graded composition multiplicity in the symmetric algebra s v of the natural gl n q module v or equivalently in the coinvariant algebra of v for a large class of irreducible modules around the steinberg module this was built on a computation via connections to algebraic groups of the steinberg module multiplicity in a tensor product of s v with other tensor spaces of fundamental weight modules

generalized distribution functions and an alternative approach to generalized planck radiation law abstract in this study recently introduced generalized distribution functions are summarized and by using one of these distribution functions namely generalized planck distribution an alternative approach to the generalized planck law for the blackbody radiation has been tackled the expression obtained is compared with the expression given by c tsallis et al phys rev e 52 1995 1447 and it is found that this approximate scheme provides bounds to the exact result depending on the values of q index

variational method in generalized statistical mechanics concavity properties of a recently generalized not necessarily extensive entropy enable among others the generalization of the bogolyubov inequality hence of the variational method in equilibrium statistical mechanics

non extensive statistical mechanics and generalized fokker planck equation a non linear generalized fokker planck gfp equation is studied whose exact stationary solutions are the maximum entropy distributions introduced by tsallis in his generalization of statistical mechanics in the case of a constant or linearly varying drift the time dependent solutions of the gfp equation are seen to obey a suitable form of the celebrated h theorem the temporal changes of tsallis entropies are seen to be given in terms of the fisher s information measure particular time dependent solutions of the gfp equation of the maximum tsallis entropy form are obtained

path integral approach to the nonextensive canonical density matrix abstract feynman s path integral is herein generalized to the nonextensive canonical density matrix based on tsallis entropy this generalization is done in two ways by using unnormalized and normalized constraints firstly we consider the path integral formulation with unnormalized constraints and this generalization is worked out through two different ways which are shown to be equivalent these formulations with unnormalized constraints are solutions to the two generalized bloch equations proposed in this work the first form of the generalized bloch equation is linear but with a temperature dependent effective hamiltonian the second form is nonlinear and resembles the anomalous correlated diffusion equation porous medium equation furthermore we can extend these results to the prescription of field theory using integral representations the second development is dedicated to analyzing the path integral formulation with normalized constraints to illustrate the methods introduced here we analyze the free particle case and a non interacting scalar field the results herein obtained are expected to be useful in the discussion of generic nonextensive contexts

tsallis nonextensive statistics with normalized q expectation values thermodynamical stability and simple illustrations abstract we prove that the specific heat c q q is nonnegative for q 0 1 in the tsallis generalized canonical ensemble with normalized q expectation values in this way we show the thermodynamic stability for this ensemble we also analyze the behavior of some thermodynamic functions such as internal energy and specific heat in the classical and quantum cases considering a spectrum of energy e n e n r e r 0

tensor product markov chains abstract we analyze families of markov chains that arise from decomposing tensor products of irreducible representations this illuminates the burnside brauer theorem for building irreducible representations the mckay correspondence and pitman s 2 m x theorem the chains are explicitly diagonalizable and we use the eigenvalues eigenvectors to give sharp rates of convergence for the associated random walks for modular representations the chains are not reversible and the analytical details are surprisingly intricate in the quantum group case the chains fail to be diagonalizable but a novel analysis using generalized eigenvectors proves successful

a quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability we present in this paper a novel nonlinear model predictive control scheme that guarantees asymptotic closed loop stability the scheme can be applied to both stable and unstable systems with input constraints the objective functional to be minimized consists of an integral square error ise part over a finite time horizon plus a quadratic terminal cost the terminal state penalty matrix of the terminal cost term has to be chosen as the solution of an appropriate lyapunov equation furthermore the setup includes a terminal inequality constraint that forces the states at the end of the finite prediction horizon to lie within a prescribed terminal region if the jacobian linearization of the nonlinear system to be controlled is stabilizable we prove that feasibility of the open loop optimal control problem at time t 0 implies asymptotic stability of the closed loop system the size of the region of attraction is only restricted by the requirement for feasibility of the optimization problem due to the input and terminal inequality constraints and is thus maximal in some sense

weight elimination in serre type conjectures we prove the weight elimination direction of the serre weight conjectures as formulated by herzig for forms of u n which are compact at infinity and split at places dividing p in generic situations that is we show that all modular weights for a mod p galois representation are contained in the set predicted by herzig under some additional hypotheses we also show modularity of all the obvious weights

technical communique null controllable region of lti discrete time systems with input saturation we present a formula for the extremes of the null controllable region of a general lti discrete time system with bounded inputs for an nth order system with only real poles not necessarily distinct the formula is simplified to an elementary matrix function which in turn shows that the set of the extremes coincides with a set of trajectories of the time reversed system under bang bang controls with n 2 or less switches

some general relations in arbitrary thermostatistics abstract the fundamental thermodynamic relation and the connection between the entropy maximum principle and the energy minimum principle are analyzed in arbitrary thermostatistics in the canonical ensemble it is shown that the relation and connection can occur in very general circumstances furthermore the relation is applied in a nonextensive context and the stability of the tsallis thermostatistics is analyzed

suboptimal model predictive control feasibility implies stability practical difficulties involved in implementing stabilizing model predictive control laws for nonlinear systems are well known stabilizing formulations of the method normally rely on the assumption that global and exact solutions of nonconvex nonlinear optimization problems are possible in limited computational time in the paper we first establish conditions under which suboptimal model predictive control mpc controllers are stabilizing the conditions are mild holding out the hope that many existing controllers remain stabilizing even if optimality is lost second we present and analyze two suboptimal mpc schemes that are guaranteed to be stabilizing provided an initial feasible solution is available and for which the computational requirements are more reasonable

modular representations as a possible basis of finite physics the approach in which physical systems are described by the elements of a linear space over a finite field and operators of physical quantities by linear operators in this space is discussed the mathematical formulation of the correspondence between such a description and the conventional one is given for a case when the characteristic of the finite field is sufficiently large the above correspondence is considered in the examples of finite analogs of representations of so 2 3 so 1 4 and osp 1 4 algebras finite analogs of representations of infinite dimensional algebras are briefly discussed

brief paper enlarging the domain of attraction of stable mpc controllers maintaining the output performance this work presents an alternative way to formulate the stable model predictive control mpc optimization problem that allows the enlargement of the domain of attraction while preserving the controller performance based on the dual mpc that uses the null local controller it proposed the inclusion of an appropriate set of slacked terminal constraints into the control problem as a result the domain of attraction is unlimited for the stable modes of the system and the largest possible for the non stable modes although this controller does not achieve local optimality simulations show that the input and output performances may be comparable to the ones obtained with the dual mpc that uses the lqr as a local controller

robust control of uncertain discrete time markovian jump systems with actuator saturation in this paper the robust stochastic stabilization problem for the class of discrete time uncertain markovian jump linear systems mjls with actuator saturation is considered the definition of domain of attraction in mean square sense doa mss is introduced to analyze the stochastic stability of the closed loop system by using a class of stochastic lyapunov function which is dependent on the jump mode and saturation function design procedures for both the mode dependent and mode independent state feedback controllers are developed based on the linear matrix inequality lmi approach finally a numerical example is provided to show the usefulness of the proposed techniques

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

an efficient and stabilizing model predictive control of switched systems model predictive control mpc of switched systems typically requires an on line solution of a mixed integer program mip since the worst case complexity of the optimization problem increases exponentially with respect to the number of integer variables an on line implementation of the mpc for problems with large number of sub systems and or large horizons is usually expensive in this technical note we propose a stabilizing mpc formulation for state dependent switched systems that enables a tradeoff between the computational complexity of the mpc controller and the optimal performance of the closed loop system the proposed approach uses a pre terminal set in addition to the positively invariant terminal set which aids in reducing the on line complexity although at the expense of optimality examples are presented to illustrate the computational benefits of the proposed mpc strategy over existing mpc for switched systems

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators

generic patterns for extensions of simple modules for finite chevalley groups abstract this paper describes generic patterns for the extensions between simple modules of a finite chevalley group a one to one correspondence between these extensions and the extensions between certain simple modules of the ambient algebraic group are established it is shown that an extension appears in a principal series representation of the finite chevalley group if and only if the corresponding extension for the algebraic group appears in a weyl module