A Separation Principle for the Continuous-Time LQ-Problem With Markovian Jump Parameters

In this paper, we devise a separation principle for the finite horizon quadratic optimal control problem of continuous-time Markovian jump linear systems driven by a Wiener process and with partial observations. We assume that the output variable and the jump parameters are available to the controller. It is desired to design a dynamic Markovian jump controller such that the closed loop system minimizes the quadratic functional cost of the system over a finite horizon period of time. As in the case with no jumps, we show that an optimal controller can be obtained from two coupled Riccati differential equations, one associated to the optimal control problem when the state variable is available, and the other one associated to the optimal filtering problem. This is a separation principle for the finite horizon quadratic optimal control problem for continuous-time Markovian jump linear systems. For the case in which the matrices are all time-invariant we analyze the asymptotic behavior of the solution of the derived interconnected Riccati differential equations to the solution of the associated set of coupled algebraic Riccati equations as well as the mean square stabilizing property of this limiting solution. When there is only one mode of operation our results coincide with the traditional ones for the LQG control of continuous-time linear systems.

Disagreement between correlations of quantum mechanics and stochastic electrodynamics in the damped parametric oscillator

Intracavity and external third order correlations in the damped nondegenerate parametric oscillator are calculated for quantum mechanics and stochastic electrodynamics (SED), a semiclassical theory. The two theories yield greatly different results, with the correlations of quantum mechanics being cubic in the system's nonlinear coupling constant and those of SED being linear in the same constant. In particular, differences between the two theories are present in at least a mesoscopic regime. They also exist when realistic damping is included. Such differences illustrate distinctions between quantum mechanics and a hidden variable theory for continuous variables.

On S-asymptotically omega-periodic functions on Banach spaces and applications

This paper is devoted to the study of the class of continuous and bounded functions f : [0, infinity] -> X for which exists omega > 0 such that lim(t ->infinity) (f (t + omega) - f (t)) = 0 (in the sequel called S-asymptotically omega-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically omega-periodic functions. We also study the existence of S-asymptotically omega-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces. (C) 2008 Elsevier Inc. All rights reserved.

EXISTENCE OF S-ASYMPTOTICALLY omega-PERIODIC SOLUTIONS FOR ABSTRACT NEUTRAL EQUATIONS

A bounded continuous function it u : [0, infinity) -> X is said to be S-asymptotically omega-periodic if lim(t ->infinity)[u(t + omega) - u(t)] = 0. This paper is devoted to study the existence and qualitative properties of S-asymptotically omega-periodic mild solutions for some classes of abstract neutral functional differential equations with infinite delay, Furthermore, applications to partial differential equations are given.

Approximate controllability of second-order distributed implicit functional systems

In this paper we study the approximate controllability of control systems with states and controls in Hilbert spaces, and described by a second-order semilinear abstract functional differential equation with infinite delay. Initially we establish a characterization for the approximate controllability of a second-order abstract linear system and, in the last section, we compare the approximate controllability of a semilinear abstract functional system with the approximate controllability of the associated linear system. (C) 2008 Elsevier Ltd. All rights reserved.

Limiting conditional distributions for birth-death processes

In a recent paper [16], one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

Pulsed quadrature-phase squeezing of solitary waves in chi((2)) parametric waveguides

It is shown that coherent quantum simultons (simultaneous solitary waves at two different frequencies) can undergo quadrature-phase squeezing as they propagate through a dispersive chi((2)) waveguide. This requires a treatment of the coupled quantized fields including a quantized depleted pump field. A technique involving nonlinear stochastic parabolic partial differential equations using a nondiagonal coherent state representation in combination with an exact Wigner representation on a reduced phase space is outlined. We explicitly demonstrate that group-velocity matched chi((2)) waveguides which exhibit collinear propagation can produce quadrature-phase squeezed simultons. Quasi-phase-matched KTP waveguides, even with their large group-velocity mismatch between fundamental and second harmonic at 425 nm, can produce 3 dB squeezed bright pulses at 850 nm in the large phase-mismatch regime. This can be improved to more than 6 dB by using group-velocity matched waveguides.

Solution techniques for transport problems involving steep concentration gradients: application to noncatalytic fluid solid reactions

Some efficient solution techniques for solving models of noncatalytic gas-solid and fluid-solid reactions are presented. These models include those with non-constant diffusivities for which the formulation reduces to that of a convection-diffusion problem. A singular perturbation problem results for such models in the presence of a large Thiele modulus, for which the classical numerical methods can present difficulties. For the convection-diffusion like case, the time-dependent partial differential equations are transformed by a semi-discrete Petrov-Galerkin finite element method into a system of ordinary differential equations of the initial-value type that can be readily solved. In the presence of a constant diffusivity, in slab geometry the convection-like terms are absent, and the combination of a fitted mesh finite difference method with a predictor-corrector method is used to solve the problem. Both the methods are found to converge, and general reaction rate forms can be treated. These methods are simple and highly efficient for arbitrary particle geometry and parameters, including a large Thiele modulus. (C) 2001 Elsevier Science Ltd. All rights reserved.

Modelling the activated sludge flocculation process combining laser light diffraction particle sizing and population balance modelling (PBM)

A technique based on laser light diffraction is shown to be successful in collecting on-line experimental data. Time series of floc size distributions (FSD) under different shear rates (G) and calcium additions were collected. The steady state mass mean diameter decreased with increasing shear rate G and increased when calcium additions exceeded 8 mg/l. A so-called population balance model (PBM) was used to describe the experimental data, This kind of model describes both aggregation and breakage through birth and death terms. A discretised PBM was used since analytical solutions of the integro-partial differential equations are non-existing. Despite the complexity of the model, only 2 parameters need to be estimated: the aggregation rate and the breakage rate. The model seems, however, to lack flexibility. Also, the description of the floc size distribution (FSD) in time is not accurate.

Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem. Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense. An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented. (C) 2003 Elsevier Science Ltd. All rights reserved.

Large amplitude vibration of thermo-electro-mechanically stressed FGM laminated plates

This paper presents a large amplitude vibration analysis of pre-stressed functionally graded material (FGM) laminated plates that are composed of a shear deformable functionally graded layer and two surface-mounted piezoelectric actuator layers. Nonlinear governing equations of motion are derived within the context of Reddy's higher-order shear deformation plate theory to account for transverse shear strain and rotary inertia. Due to the bending and stretching coupling effect, a nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the plate that is subjected to uniform temperature change, in-plane forces and applied actuator voltage. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed FGM laminated plates are derived. A semi-analytical method that is based on one-dimensional differential quadrature and Galerkin technique is proposed to predict the large amplitude vibration behavior of the laminated rectangular plates with two opposite clamped edges. Linear vibration frequencies and nonlinear normalized frequencies are presented in both tabular and graphical forms, showing that the normalized frequency of the FGM laminated plate is very sensitive to vibration amplitude, out-of-plane boundary support, temperature change, in-plane compression and the side-to-thickness ratio. The CSCF and CFCF plates even change the inherent hard-spring characteristic to soft-spring behavior at large vibration amplitudes. (C) 2003 Elsevier B.V. All rights reserved.

Aproximação numérica – analítica para a modelagem da conversão termoquímica de combustíveis sólidos

Um algoritmo numérico foi criado para apresentar a solução da conversão termoquímica de um combustível sólido. O mesmo foi criado de forma a ser flexível e dependente do mecanismo de reação a ser representado. Para tanto, um sistema das equações características desse tipo de problema foi resolvido através de um método iterativo unido a matemática simbólica. Em função de não linearidades nas equações e por se tratar de pequenas partículas, será aplicado o método de Newton para reduzir o sistema de equações diferenciais parciais (EDP’s) para um sistema de equações diferenciais ordinárias (EDO’s). Tal processo redução é baseado na união desse método iterativo à diferenciação numérica, pois consegue incorporar nas EDO’s resultantes funções analíticas. O modelo reduzido será solucionado numericamente usando-se a técnica do gradiente bi-conjugado (BCG). Tal modelo promete ter taxa de convergência alta, se utilizando de um número baixo de iterações, além de apresentar alta velocidade na apresentação das soluções do novo sistema linear gerado. Além disso, o algoritmo se mostra independente do tamanho da malha constituidora. Para a validação, a massa normalizada será calculada e comparada com valores experimentais de termogravimetria encontrados na literatura, , e um teste com um mecanismo simplificado de reação será realizado.

An Extension of Gompertzian Growth Dynamics Weibull and Frechet Models

In this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by Beta* (p, q), which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for p = 2, the investigation is extended to the extreme value models of Weibull and Frechet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the Beta* (2, q) densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.

Robôs quadrúpedes: geração de trajectórias em tempo real usando sistemas dinâmicos não-lineares

A geração de trajectórias de robôs em tempo real é uma tarefa muito complexa, não existindo ainda um algoritmo que a permita resolver de forma eficaz. De facto, há controladores eficientes para trajectórias previamente definidas, todavia, a adaptação a variações imprevisíveis, como sendo terrenos irregulares ou obstáculos, constitui ainda um problema em aberto na geração de trajectórias em tempo real de robôs. Neste trabalho apresentam-se modelos de geradores centrais de padrões de locomoção (CPGs), inspirados na biologia, que geram os ritmos locomotores num robô quadrúpede. Os CPGs são modelados matematicamente por sistemas acoplados de células (ou neurónios), sendo a dinâmica de cada célula dada por um sistema de equações diferenciais ordinárias não lineares. Assume-se que as trajectórias dos robôs são constituídas por esta parte rítmica e por uma parte discreta. A parte discreta pode ser embebida na parte rítmica, (a.1) como um offset ou (a.2) adicionada às expressões rítmicas, ou (b) pode ser calculada independentemente e adicionada exactamente antes do envio dos sinais para as articulações do robô. A parte discreta permite inserir no passo locomotor uma perturbação, que poderá estar associada à locomoção em terrenos irregulares ou à existência de obstáculos na trajectória do robô. Para se proceder á análise do sistema com parte discreta, será variado o parâmetro g. O parâmetro g, presente nas equações da parte discreta, representa o offset do sinal após a inclusão da parte discreta. Revê-se a teoria de bifurcação e simetria que permite a classificação das soluções periódicas produzidas pelos modelos de CPGs com passos locomotores quadrúpedes. Nas simulações numéricas, usam-se as equações de Morris-Lecar e o oscilador de Hopf como modelos da dinâmica interna de cada célula para a parte rítmica. A parte discreta é modelada por um sistema inspirado no modelo VITE. Medem-se a amplitude e a frequência de dois passos locomotores para variação do parâmetro g, no intervalo [-5;5]. Consideram-se duas formas distintas de incluir a parte discreta na parte rítmica: (a) como um (a.1) offset ou (a.2) somada nas expressões que modelam a parte rítmica, e (b) somada ao sinal da parte rítmica antes de ser enviado às articulações do robô. No caso (a.1), considerando o oscilador de Hopf como dinâmica interna das células, verifica-se que a amplitude e frequência se mantêm constantes para -50.2. A extensão do movimento varia de forma directamente proporcional à amplitude. No caso das equações de Morris-Lecar, quando a componente discreta é embebida (a.2), a amplitude e a frequência aumentam e depois diminuem para - 0.170.5 Pode concluir-se que: (1) a melhor forma de inserção da parte discreta que menos perturbação insere no robô é a inserção como offset; (2) a inserção da parte discreta parece ser independente do sistema de equações diferenciais ordinárias que modelam a dinâmica interna de cada célula. Como trabalho futuro, é importante prosseguir o estudo das diferentes formas de inserção da parte discreta na parte rítmica do movimento, para que se possa gerar uma locomoção quadrúpede, robusta, flexível, com objectivos, em terrenos irregulares, modelada por correcções discretas aos padrões rítmicos.

Constraint reasoning for differential models

The basic motivation of this work was the integration of biophysical models within the interval constraints framework for decision support. Comparing the major features of biophysical models with the expressive power of the existing interval constraints framework, it was clear that the most important inadequacy was related with the representation of differential equations. System dynamics is often modelled through differential equations but there was no way of expressing a differential equation as a constraint and integrate it within the constraints framework. Consequently, the goal of this work is focussed on the integration of ordinary differential equations within the interval constraints framework, which for this purpose is extended with the new formalism of Constraint Satisfaction Differential Problems. Such framework allows the specification of ordinary differential equations, together with related information, by means of constraints, and provides efficient propagation techniques for pruning the domains of their variables. This enabled the integration of all such information in a single constraint whose variables may subsequently be used in other constraints of the model. The specific method used for pruning its variable domains can then be combined with the pruning methods associated with the other constraints in an overall propagation algorithm for reducing the bounds of all model variables. The application of the constraint propagation algorithm for pruning the variable domains, that is, the enforcement of local-consistency, turned out to be insufficient to support decision in practical problems that include differential equations. The domain pruning achieved is not, in general, sufficient to allow safe decisions and the main reason derives from the non-linearity of the differential equations. Consequently, a complementary goal of this work proposes a new strong consistency criterion, Global Hull-consistency, particularly suited to decision support with differential models, by presenting an adequate trade-of between domain pruning and computational effort. Several alternative algorithms are proposed for enforcing Global Hull-consistency and, due to their complexity, an effort was made to provide implementations able to supply any-time pruning results. Since the consistency criterion is dependent on the existence of canonical solutions, it is proposed a local search approach that can be integrated with constraint propagation in continuous domains and, in particular, with the enforcing algorithms for anticipating the finding of canonical solutions. The last goal of this work is the validation of the approach as an important contribution for the integration of biophysical models within decision support. Consequently, a prototype application that integrated all the proposed extensions to the interval constraints framework is developed and used for solving problems in different biophysical domains.