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

about derivatives of heun s functions from polynomial transformations of hypergeometric equations we discuss how the derivative of solutions of some heun s differential equations can be given in some particular cases from the solution of another heun s equation this work uses some algebraic links between heun and hypergeometric equations c 2008 elsevier inc all rights reserved

stochastic linear quadratic control with conic control constraints on an infinite time horizon this paper is concerned with a stochastic linear quadratic lq control problem in the infinite time horizon where the control is constrained in a given arbitrary closed cone the cost weighting matrices are allowed to be indefinite and the state is scalar valued first the mean square conic stabilizability of the system is defined which is then characterized by a set of simple conditions involving linear matrix inequalities lmis next the issue of well posedness of the underlying optimal lq control which is necessitated by the indefiniteness of the problem is addressed in great detail and necessary and sufficient conditions of the well posedness are presented on the other hand to address the lq optimality two new algebraic equations a la riccati called extended algebraic riccati equations eares along with the notion of their stabilizing solutions are introduced for the first time optimal feedback control as well as the optimal value are explicitly derived in terms of the stabilizing solutions to the eares moreover several cases when the stabilizing solutions do exist are discussed and algorithms of computing the solutions are presented finally numerical examples are provided to illustrate the theoretical results established

orbit inequivalent actions for groups containing a copy of mathbb f _ 2 we prove that if a countable group contains a copy of mathbb f 2 then it admits uncountably many non orbit equivalent actions

constrained stochastic lqc a tractable approach despite the celebrated success of dynamic programming for optimizing quadratic cost functions over linear systems such an approach is limited by its inability to tractably deal with even simple constraints in this paper we present an alternative approach based on results from robust optimization to solve the stochastic linear quadratic control slqc problem in the unconstrained case the problem may be formulated as a semidefinite optimization problem sdp we show that we can reduce this sdp to optimization of a convex function over a scalar variable followed by matrix multiplication in the current state thus yielding an approach that is amenable to closed loop control and analogous to the riccati equation in our framework we also consider a tight second order cone socp approximation to the sdp that can be solved much more efficiently when the problem has additional constraints both the sdp and socp are tractable in the presence of control and state space constraints moreover compared to the riccati approach they provide much greater control over the stochastic behavior of the cost function when the noise in the system is distributed normally

well posedness and attainability of indefinite stochastic linear quadratic control in infinite time horizon abstract this paper is concerned with a stochastic linear quadratic lq problem in an infinite time horizon with multiplicative noises both in the state and the control a distinctive feature of the problem under consideration is that the cost weighting matrices for the state and the control are allowed to be indefinite a new type of algebraic riccati equation called a generalized algebraic riccati equation gare is introduced which involves a matrix pseudo inverse and two additional algebraic equality inequality constraints it is then shown that the well posedness of the indefinite lq problem is equivalent to a linear matrix inequality lmi condition whereas the attainability of the lq problem is equivalent to the existence of a stabilizing solution to the gare moreover all possible optimal controls are identified via the solution to the gare finally it is proved that the solution to the gare can be obtained via solving a convex optimization problem called semidefinite programming

orbit equivalence rigidity and bounded cohomology we establish new results and introduce new methods in the theory of measurable orbit equivalence using bounded cohomology of group representations our rigidity statements hold for a wide uncountable class of groups arising from negative curvature geometry amongst our applications are a measurable mostow type rigidity theorems for products of negatively curved groups b prime factorization results for measure equivalence c superrigidity for orbit equivalence d the first examples of continua of type ii1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic

linear matrix inequalities riccati equations and indefinite stochastic linear quadratic controls this paper deals with an optimal stochastic linear quadratic lq control problem in infinite time horizon where the diffusion term in dynamics depends on both the state and the control variables in contrast to the deterministic case we allow the control and state weighting matrices in the cost functional to be indefinite this leads to an indefinite lq problem which may still be well posed due to the deep nature of uncertainty involved the problem gives rise to a stochastic algebraic riccati equation sare which is however fundamentally different from the classical algebraic riccati equation as a result of the indefinite nature of the lq problem to analyze the sare we introduce linear matrix inequalities lmis whose feasibility is shown to be equivalent to the solvability of the sare moreover we develop a computational approach to the sare via a semi definite programming associated with the lmis finally numerical experiments are reported to illustrate the proposed approach

a converse to dye s theorem every non amenable countable group induces orbit inequivalent ergodic equivalence relations on standard borel probability spaces not every free ergodic measure preserving action of f 2 on a standard borel probability space is orbit equivalent to an action of a countable group on an inverse limit of finite spaces there is a treeable non hyperfinite borel equivalence relation which is not universal for treeable in the b ordering

on positive controllers and linear quadratic optimal control problems an assumption on the non negative solutions of quadratic equations has been used to study linear quadratic optimal control problems with positive controllers this note investigates this assumption in more detail

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

explicit solution for constrained scalar state stochastic linear quadratic control with multiplicative noise we study in this paper a class of constrained linear quadratic lq optimal control problem formulations for the scalar state stochastic system with multiplicative noise which has various applications especially in the financial risk management the linear constraint on both the control and state variables considered in our model destroys the elegant structure of the conventional lq formulation and has blocked the derivation of an explicit control policy so far in the literature we successfully derive in this paper the analytical control policy for such a class of problems by utilizing the state separation property induced from its structure we reveal that the optimal control policy is a piecewise affine function of the state and can be computed offline efficiently by solving two coupled riccati equations under some mild conditions we also obtain the stationary control policy for an infinite time horizon we demonstrate the implementation of our method via some illustrative examples and show how to calibrate our model to solve dynamic constrained portfolio optimization problems

solvability and asymptotic behavior of generalized riccati equations arising in indefinite stochastic lq controls the optimal control problem in a finite time horizon with an indefinite quadratic cost function for a linear system subject to multiplicative noise on both the state and control can be solved via a constrained matrix differential riccati equation in this paper we provide general necessary and sufficient conditions for the solvability of this generalized differential riccati equation furthermore its asymptotic behavior is investigated along with its connection to the generalized algebraic riccati equation associated with the linear quadratic control problem in finite time horizon examples are presented to illustrate the results established

novel relations and new properties of confluent heun s functions and their derivatives of arbitrary order the present paper reveals important properties of the confluent heun s functions we derive a set of novel relations for confluent heun s functions and their derivatives of arbitrary order specific new subclasses of confluent heun s functions are introduced and studied a new alternative derivation of confluent heun s polynomials is presented

stochastic linear quadratic control via semidefinite programming we study stochastic linear quadratic lq optimal control problems over an infinite time horizon allowing the cost matrices to be indefinite we develop a systematic approach based on semidefinite programming sdp a central issue is the stability of the feedback control and we show this can be effectively examined through the complementary duality of the sdp furthermore we establish several implication relations among the sdp complementary duality the generalized riccati equation and the optimality of the lq control problem based on these relations we propose a numerical procedure that provides a thorough treatment of the lq control problem via primal dual sdp it identifies a stabilizing feedback control that is optimal or determines that the problem possesses no optimal solution for the latter case we develop an epsilon approximation scheme that is asymptotically optimal

physical applications of second order linear differential equations that admit polynomial solutions in this paper conditions for the second order linear differential equation to have polynomial solutions are given several applications of these results to schrodinger s equation are discussed conditions under which the confluent biconfluent and general heun equation yields polynomial solutions are explicitly given some new classes of exactly solvable differential equations are also discussed the results of this work are expressed in such a way as to allow direct use without preliminary analysis

error estimates for the logarithmic barrier method in linear quadratic stochastic optimal control problems we consider a linear quadratic stochastic optimal control problem whith non negativity control constraints the latter are penalized with the classical logarithmic barrier using a duality argument and the stochastic minimum principle we provide an error estimate for the solution of the penalized problem which is the natural extension of the well known estimate in the deterministic framework

constrained stochastic lq control with random coefficients and application to portfolio selection this paper is devoted to the study of a stochastic linear quadratic lq optimal control problem where the control variable is constrained in a cone and all the coefficients of the problem are random processes employing tanaka s formula optimal control and optimal cost are explicitly obtained via solutions to two extended stochastic riccati equations esres the esres introduced for the first time in this paper are highly nonlinear backward stochastic differential equations bsdes whose solvability is proved based on a truncation function technique and kobylanski s results the general results obtained are then applied to a mean variance portfolio selection problem for a financial market with random appreciation and volatility rates and with short selling prohibited feasibility of the problem is characterized and efficient portfolios and efficient frontier are presented in closed forms

mean variance portfolio selection with random parameters in a complete market this paper concerns the continuous time mean variance portfolio selection problem in a complete market with random interest rate appreciation rates and volatility coef cients the problem is tackled using the results of stochastic linear quadratic lq optimal control and backward stochastic differential equations bsdes two theories that have been extensively studied and developed in recent years specifically the mean variance problem is formulated as a linearly constrained stochastic lq control problem solvability of this lq problem is reduced in turn to proving global solvability of a stochastic riccati equation the proof of existence and uniqueness of this riccati equation which is a fully nonlinear and singular bsde with random coefficients is interesting in its own right and relies heavily on the structural properties of the equation efficient investment strategies as well as the mean variance efficient frontier are then analytically derived in terms of the solution of this equation in particular it is demonstrated that the efficient frontier in the mean standard deviation diagram is still a straight line or equivalently risk free investment is still possible even when the interest rate is random finally a version of the mutual fund theorem is presented

on continuous time constrained stochastic linear quadratic control this paper studies a class of continuous time scalar state stochastic linear quadratic lq optimal control problems with the linear control constraints using the state separation theorem induced from its special structure we derive the analytical solution for this class of problems the revealed optimal control policy is a piece wise affine function of system state this control policy can be computed efficiently by solving two riccati equations off line under some mild conditions the stationary optimal control policy can be also achieved for this class of problems over an infinite horizon examples are presented to shed light on the theoretical results established

exact solution to the schrodinger s equation with pseudo gaussian potential we consider the radial schrodinger equation with the pseudo gaussian potential by making an ansatz to the solution of the eigenvalue equation for the associate hamiltonian we arrive at the general exact eigenfunction the values of energy levels for the bound states are calculated along with their corresponding normalized wave functions the case of positive energy levels known as meta stable states is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated

the coulomb problem on a 3 sphere and heun polynomials the paper studies the quantum mechanical coulomb problem on a 3 sphere we present a special parametrization of the ellipto spheroidal coordinate system suitable for the separation of variables after quantization we get the explicit form of the spectrum and present an algebraic equation for the eigenvalues of the runge lentz vector we also present the wave functions expressed via heun polynomials

extension of nikiforov uvarov method for the solution of heun equation we report an alternative method to solve second order differential equations which have at most four singular points this method is developed by changing the degrees of the polynomials in the basic equation of nikiforov uvarov nu method this is called extended nu method for this paper the eigenvalue solutions of heun equation and confluent heun equation are obtained via extended nu method some quantum mechanical problems such as coulomb problem on a 3 sphere two coulombically repelling electrons on a sphere and hyperbolic double well potential are investigated by this method

t property and nonisomorphic full factors of types ii and iii abstract the actions of t groups on von neumann s hyperfinite algebras are studied it is proved that there exist factors of type ii with different countable fundamental groups and hence different actions it is also proved that there exist nonisomorphic full factors of type iii 1 with any fixed invariant sd

stochastic linear quadratic regulators with indefinite control weight costs ii in part i of this paper s chen x li and x zhou siam j control optim 36 1998 pp 1685 1702 an optimization model of stochastic linear quadratic regulators lqrs with indefinite control cost weighting matrices is proposed and studied in this sequel the problem of solving lqr models with system diffusions dependent on both state and control variables which is left open in part i is tackled first the solvability of the associated stochastic riccati equations sres is studied in the normal case namely all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite with at least one positive definite which in turn leads to an optimal state feedback control of the lqr problem in the general indefinite case the problem is decomposed into two optimal lqr problems one with a forward dynamics and the other with a backward dynamics the well posedness and solvability of the original lqr problem are then obtained by solving these two subproblems and an optimal control is explicitly constructed examples are presented to illustrate the results

linear quadratic regulator problem with positive controls in this paper the linear quadratic regulator problem with a positivity constraint on the admissible control set is addressed necessary and sufficient conditions for optimality are presented in terms of inner products projections on closed convex sets pontryagin s maximum principle and dynamic programming the main results are concerned with smoothness of the optimal control and the value function the maximum principle will be extended to the infinite horizon case based on these analytical methods we propose a numerical algorithm for the computation of the optimal controls for the finite and infinite horizon problem the numerical methods will be justified by convergence properties between the finite and infinite horizon case on one side and discretized optimal controls and the true optimal control on the other

solution of schrodinger equation for two different potentials using extended nikiforov uvarov method and polynomial solutions of biconfluent heun equation exact solutions of the schrodinger equation for two different potentials are presented by using the extended nikiforov uvarov method the first one is the inverse square root potential which is a long range potential and the second one is a combination of coulomb linear and harmonic potentials which is often used to describe quarkonium eigenstate solutions are obtained in a systematic way without using any ansatz or transformation eigenfunctions for considered potentials are given in terms of biconfluent heun polynomials

hyperfinite and ii1 actions for nonamenable groups abstract the actions of certain nonamenable groups on the lebesgue space are studied an example is constructed of a group which has a continuum of weakly nonequivalent actions of type ii 1 it is also proved that if a free group with two generators has hyperfinite action on the lebesgue space then at least one generator acts dissipatively a hyperfinite action is constructed for any nonamenable group