singular perturbations for third order nonlinear multi point boundary value problem abstract this paper is devoted to study the following third order multi point singularly perturbed boundary value problem x t f t x t x t x t 0 0 t 1 0 1 x 0 0 ax 0 bx 0 i 1 n 2 i x i a cx 1 dx 1 i 1 n 2 i x i b where a b c d 0 a b r a b 0 c d 0 i 0 i 0 i 1 2 n 2 0 1 2 n 2 1 and 0 1 2 n 2 1 the existence uniqueness and asymptotic estimates of solutions of the boundary value problem are give by using priori estimates differential inequalities technique and leray schauder degree theory

on computing the stabilizing solution of the discrete time riccati equation abstract a necessary and sufficient condition is given for the existence of the stabilizing solution of the discrete time riccati equation without any invertibility or positivity assumptions the result involves a special associated matrix pencil

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

uniqueness and existence results for a third order nonlinear multi point boundary value problem this paper provides sufficient conditions for the existence and uniqueness of solutions to a nonlinear multi point boundary value problem for third order differentia equation the nonlinear term f in the differential equation under consideration may depend on second order derivative the emphasis here is the boundary conditions are nonlinear and this is where the main novelty of this work lies by applying the upper and lower solutions method as well as degree theory the existence and uniqueness results of the problem are established some examples are given to demonstrate the main results

a nested computational approach to the discrete time finite horizon lq control problem a new algorithmic setting is proposed for the discrete time finite horizon linear quadratic lq optimal control problem with constrained or unconstrained final state no matter whether the problem is cheap singular or regular the proposed solution based on matrix pseudoinversion is completed and made practically implementable by a nesting procedure for welding optimal subarcs that enables arbitrary enlargement of the control time interval

the generalized continuous algebraic riccati equation and impulse free continuous time lq optimal control abstract the purpose of this paper is to investigate the role that the so called constrained generalized riccati equation plays within the context of continuous time singular linear quadratic lq optimal control this equation has been defined following the analogy with the discrete time setting however while in the discrete time case the connections between this equation and the linear quadratic optimal control problem has been thoroughly investigated to date very little is known on these connections in the continuous time setting this note addresses this point we show in particular that when the continuous time constrained generalized riccati equation admits a solution the corresponding linear quadratic problem admits an impulse free optimal control we also address the corresponding infinite horizon lq problem for which we establish a similar result under the additional constraint that there exists a control input for which the cost index is finite

natural frequency information for circular and annular plates abstract this paper presents the results of an extensive literature search and review of available sources of numerical natural frequency information for stationary circular and annular elastic plates in addition to the source information is given regarding the specific plate theory boundary conditions geometric properties and material properties used to determine the natural frequency information sources of information are tabulated according to the above parameters for easy reference and selection of desired information a description of the various plate theories including the basic governing partial differential equation of motion is also given this information may be particularly useful to experimentalists and designers when natural frequency data are required without the need for detailed analysis

on the reduction of the continuous time generalized algebraic riccati equation an effective procedure for solving the singular lq problem with smooth solutions abstract this paper presents a reduction technique for the continuous time constrained generalized riccati equation arising in the context of the singular linear quadratic lq optimal control problem this technique allows to express the solutions of the constrained generalized riccati equation in terms of the solutions of a reduced order standard riccati equation this result is used to provide a solution to the singular lq problem with closed loop stability in the case when the allowed controls are restricted to be regular for any initial condition

existence and uniqueness results for third order nonlinear differential systems abstract sufficient conditions are given for the existence of solutions to third order boundary value problems for nonlinear differential systems some appropriate conditions are given to guarantee that the nagumo condition is satisfied by constructing appropriate a lower solution upper solution pair a concept that is defined in this paper the uniqueness result of the problem is also established the emphasis here is that the differential systems has nonlinear dependence on all over order derivatives and the boundary conditions are nonlinear

computation of general inner outer and spectral factorizations in this paper we solve two problems in linear systems theory the computation of the inner outer and spectral factorizations of a continuous time system considered in the most general setting we show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic riccati equation of order usually much smaller than the mcmillan degree of the transfer function matrix of the system the proposed procedures are completely general being applicable for a polynomial proper improper system whose transfer function matrix could be rank deficient and could have poles zeros on the imaginary axis or at infinity as an application we discuss the extension to the case of rational matrices of the complete orthogonal decomposition of a constant matrix numerical refinements and examples illustrating the proposed approach are discussed in detail

existence of solutions for a class of third order nonlinear boundary value problems abstract in this paper we are concerned with the following third order ordinary differential equation x t f t x t x t x t 0 0 with the nonlinear boundary conditions x 0 0 g x 0 x 0 a h x 1 x 1 b where a b r f 0 1 r 3 r is continuous g h r 2 r are continuous the existence result is given by using a priori estimate nagumo condition upper and lower solutions and leray schauder degree and we give an example to demonstrate our result

spectral and inner outer factorizations for discrete time systems the spectral and inner outer factorization problems are solved for discrete time systems with transfer matrices of arbitrary rank simple and explicit state space formulas expressed in terms of the stabilizing solution to the discrete time algebraic riccati system are derived the results are numerically effective since a reliable algorithm for computing such a stabilizing solution is available

an asymptotic numerical method for singularly perturbed third order ordinary differential equations of convection diffusion type abstract singularly perturbed two point boundary value problems spbvps for third order ordinary differential equations odes with a small parameter multiplying the highest derivative are considered a numerical method is suggested in this paper to solve such problems in this method the given bvp is transformed into a weakly coupled system of two odes subject to suitable initial and boundary conditions then the computational method presented in this paper is applied to this system in this method we reduce the weakly coupled system into a decoupled system then to solve this decoupled system numerically we apply a boundary value technique bvt in which the domain of definition of the differential equation is divided into two nonoverlapping subintervals called inner and outer regions then we solve the decoupled system over these regions as two point boundary value problems an exponentially fitted finite difference scheme is used in the inner region and a classical finite difference scheme in the outer region the boundary conditions at the transition point are obtained using the zero order asymptotic expansion approximation of the solution of the problem this computational method is distinguished by the facts that the decoupling reduces the computational time very much and it is well suited for parallel computing this method can be extended to a system of two ordinary differential equations of which one is of first order and the other is of second order numerical examples are given to illustrate the method

optimal design of internal ring supports for vibrating circular plates this paper presents the optimality conditions for optimum locations of internal ring supports in circular plates for maximum frequency of axisymmetric vibration of a specified order the optimality conditions require both the reaction forces and moments at the internal supports to vanish for optimal solutions this means that the internal ring supports should be placed at the nodal rings of the appropriate higher order vibration mode of a corresponding unsupported circular plate for maximum effectiveness in raising the vibration frequency of the plate some design examples are given to verify the derived optimality conditions

continuous and discrete time riccati theory a popov function approach abstract necessary and sufficient conditions for the existence of the stabilizing solution of the algebraic riccati equation are derived for both the continuous and the discrete time case under the weakest possible assumptions imposed on the initial data both frequency domain conditions involving an associated popov function and time domain conditions involving an associated matrix pencil are investigated

spectral and inner outer factorizations through the constrained riccati equation the topic of the paper is the spectral factorization problem for a proper rational matrix function of constant rank but not necessarily maximal on the extended imaginary axis the problem is reduced to the computation of the stabilizing solution of a so called constrained riccati equation the proof of the main result suggests a schur like algorithm applied to a singular matrix pencil

the asymptotic solution of a class of third order boundary value problems arising in the theory of thin film flows this paper studies boundary and interior layer phenomena exhibited by solutions of certain singularly perturbed third order boundary value problems which govern the motion of thin liquid films subject to viscous capillary and gravitational forces precise conditions specifying where and when the third order derivative terms in the differential equations can be neglected are derived and improved estimates for the actual solutions in terms of solutions of the lower order models are constructed the paper also contains a technique for replacing a third order problem with an asymptotically equivalent second order one that may have wider applicability

singularly perturbed third order boundary value problem for nonlinear systems abstract this paper deals with the existence of solutions to a singularly perturbed boundary value problem for third order nonlinear differential systems the authors construct an appropriate generalized lower solution upper solution pair a concept that is defined in this paper and employ the method of descent nagumo conditions algebraic boundary layer functions and volterra type integral operator to guarantee the existence of solutions of the problem the uniformly valid asymptotic estimate of the solutions is given as well the differential systems has nonlinear dependence on all over order derivatives of the unknown

generalized continuous time riccati theory abstract a riccati like equation which incorporates as special cases the standard continuous time algebraic riccati equation and the constrained continuous time algebraic riccati equation is introduced and studied a characterization of the conditions under which an equation of general form has a stabilizing solution are investigated in terms of the so called proper deflating subspace of the extended hamiltonian pencil

singular perturbations of boundary value problems for a class of third order nonlinear ordinary differential equations abstract this article shows the existence uniqueness and asymptotic estimates of solutions of the singularly perturbed boundary value problems for a class of the third order nonlinear ordinary differential equations with a small parameter e 0 ey x y y e satisfying the condition y 0 0 y 1 0 y 1 0 or y 0 0 y 1 0 y 1 0

continuous time singular linear quadratic control necessary and sufficient conditions for the existence of regular solutions abstract the purpose of this paper is to provide a full understanding of the role that the constrained generalized continuous algebraic riccati equation plays in singular linear quadratic lq optimal control indeed in spite of the vast literature on lq problems only recently a sufficient condition for the existence of a non impulsive optimal control has for the first time connected this equation with the singular lq optimal control problem in this paper we establish four equivalent conditions providing a complete picture that connects the singular lq problem with the constrained generalized continuous algebraic riccati equation and with the geometric properties of the underlying system

generalized discrete time popov yakubovich theory abstract this paper deals with the problem of general conditions for the existence of a stabilizing solution to the so called kalman szego popov yakubovich system the well known popov yakubovich positivity condition is replaced with a more general one expressed in terms of the invertibility of an adequate operator this allows one to consider also game theoretical situations

singular perturbations of third order nonlinear differential equations with full nonlinear boundary conditions in this paper we discuss singular perturbations of third order nonlinear ordinary differential equations with full nonlinear boundary conditions the emphasis here is that the nonlinear term depends on the first second order derivatives and the boundary conditions are full nonlinear that is where the main novelty of this work lies by applying the upper and lower solutions method as well as analysis technique the existence uniqueness results for the singularly perturbed boundary value problem are established and asymptotic estimates of solutions is also obtained

a soft table for the natural frequencies and modal parameters of uniform circular plates with elastic edge support abstract the paper introduces a soft table for the natural frequencies and modal parameters of uniform circular plates with elastic edge support in contrast to conventional tabulations on paper the soft table allows to save space and display parameters for required number of vibration modes the soft table is realized in a collection of matlab codes that calculate natural frequencies and mode shape parameters for a given set of translational and rotational constraints and modifiable poisson ratio calculations of natural frequencies and modal parameters for a wide range of constraints revealed distinctive regions where the frequencies and parameters could change considerably knowing position and extension of these regions could be essential for design and vibration control of structures incorporating circular plates

generalized discrete time riccati theory a riccati like equation termed the generalized discrete time algebraic riccati equation which incorporates as special cases both the standard and the constrained discrete time algebraic riccati equations is introduced and investigated under the weakest possible assumptions imposed on the initial data a complete characterization of the conditions under which such an equation of general form has a stabilizing solution is presented in terms of the so called proper deflating subspace of the extended hamiltonian pencil an evaluation of an associated quadratic index along constrained stable trajectories is given in terms of the stabilizing solution to the generalized riccati equation possible applications of the developed theory range from nonstandard spectral and inner outer factorizations to h 2 and h infty control in singular cases the results exposed in the present paper are the discrete time counterpart of those stated in the authors previous paper concerning the generalized singular continuous time riccati theory the results could be also seen as an extension to singular cases of the usual discrete time algebraic riccati equation theory of indefinite sign