Application of real and functional analysis to solve boundary value problems

This thesis is about using appropriate tools in functional analysis arid classical analysis to tackle the problem of existence and uniqueness of nonlinear partial differential equations. There being no unified strategy to deal with these equations, one approaches each equation with an appropriate method, depending on the characteristics of the equation. The correct setting of the problem in appropriate function spaces is the first important part on the road to the solution. Here, we choose the setting of Sobolev spaces. The second essential part is to choose the correct tool for each equation. In the first part of this thesis (Chapters 3 and 4) we consider a variety of nonlinear hyperbolic partial differential equations with mixed boundary and initial conditions. The methods of compactness and monotonicity are used to prove existence and uniqueness of the solution (Chapter 3). Finding a priori estimates is the main task in this analysis. For some types of nonlinearity, these estimates cannot be easily obtained, arid so these two methods cannot be applied directly. In this case, we first linearise the equation, using linear recurrence (Chapter 4). In the second part of the thesis (Chapter 5), by using an appropriate tool in functional analysis (the Sobolev Imbedding Theorem), we are able to improve previous results on a posteriori error estimates for the finite element method of lines applied to nonlinear parabolic equations. These estimates are crucial in the design of adaptive algorithms for the method, and previous analysis relies on, what we show to be, unnecessary assumptions which limit the application of the algorithms. Our analysis does not require these assumptions. In the last part of the thesis (Chapter 6), staying with the theme of choosing the most suitable tools, we show that using classical analysis in a proper way is in some cases sufficient to obtain considerable results. We study in this chapter nonexistence of positive solutions to Laplace's equation with nonlinear Neumann boundary condition. This problem arises when one wants to study the blow-up at finite time of the solution of the corresponding parabolic problem, which models the heating of a substance by radiation. We generalise known results which were obtained by using more abstract methods.

Obtaining functional parametric models using active vision strategies

This paper addresses the problem of determining which 3D shape is present, and more importantly, the dimensions of the shape in a scene. This is performed in an active vision system because it reduces the complexity of the problem through the use of gaze stabilization, choice of foveation point, and selective processing by adaptively processing regions of interest. In our case, only a small number of equations and parameters are needed for each shape and these are incorporated into functional descriptions of the shapes.

The functional significance of ventilation frequency, and its relationship to oxygen demand in the resting brown long-eared bat, Plecotus auritus

Mean oxygen consumption and simultaneous ventilation frequency of nine non-reproductive brown long-eared bats (body mass 8.53–13.33 g) were measured on 159 occasions. Ambient (chamber) temperature at which the measurements were made ranged from 10.8 to 41.1°C. Apneic ventilation occurred in 22 of the 59 measurements made when mean oxygen consumption was less than 0.5 ml·min-1. No records of apneic ventilation were obtained when it was over 0.5 ml·min-1. The relationship between ventilation frequency and mean oxygen consumption depended on whether ventilation was apneic or non-apneic. When ventilation was non-apneic the relationship was positive and log-linear. When ventilation was apneic the relationship was log-log. Within the thermoneutral zone ventilation frequency was not significantly different from that predicted from allometric equations for a terrestrial mammal of equivalent body mass, but was significantly greater than that predicted for a bird. A reduction in the amount of oxygen consumed per breath occurred at ambient temperatures above the upper critical temperature (39°C).

A new algorithm to design minimal multi-functional observers for linear systems

Designing minimum possible order (minimal) observers for Multi-Input Multi-Output (MIMO) linear systems have always been an interesting subject. In this paper, a new methodology to design minimal multi-functional observers for Linear Time-Invariant (LTI) systems is proposed. The approach is applicable, and it also helps in regulating the convergence rate of the observed functions. It is assumed that the system is functional observable or functional detectable, which is less conservative than assuming the observability or detectability of the system. To satisfy the minimality of the observer, a recursive algorithm is provided that increases the order of the observer by appending the minimum required auxiliary functions to the desired functions that are going to be estimated. The algorithm increases the number of functions such that the necessary and sufficient conditions for the existence of a functional observer are satisfied. Moreover, a new methodology to solve the observer design interconnected equations is elaborated. Our new algorithm has advantages with regard to the other available methods in designing minimal order functional observers. Specifically, it is compared with the most common schemes, which are transformation based. Using numerical examples it is shown that under special circumstances, the conventional methods have some drawbacks. The problem partly lies in the lack of sufficient numerical degrees of freedom proposed by the conventional methods. It is shown that our proposed algorithm can resolve this issue. A recursive algorithm is also proposed to summarize the observer design procedure. Several numerical examples and simulation results illustrate the efficacy, superiority and different aspects of the theoretical findings.

Isotherm, kinetic, and thermodynamic equations for cefalexin removal from liquids using activated carbon synthesized from loofah sponge

Activated carbon (AC) developed from loofah sponge with phosphoric acid activation was applied to absorb cefalexin (CEX) in aqueous solution. AC was characterized by N2 adsorption–desorption isotherms and Fourier transform infrared spectroscopy (FTIR). Factors influencing the adsorption process were investigated. The equilibrium adsorption isotherms and kinetics of CEX were also studied. The results showed that AC prepared from loofah sponge had rough surface and abundant pores. The determination results of specific surface area (810.12 m²/g) and average pore size (5.28 nm) suggested the high adsorption capability. At low concentration, the AC could adsorb about 95% of CEX. The adsorption effect was independent of the temperature and pH. The maximum adsorption amount of CEX was about 55.11 mg/g at 308 K. The equilibrium data agreed well with Freundlich isotherm equation (R² = 0.9957) at 308 K, which indicated multilayer adsorption. FTIR analysis suggested the existence of phosphorus-containing functional groups, C–O bond, and C=C bond on the surface of AC of which the peak intensity of AC after adsorption was slightly lower after adsorption, indicating that the AC surface groups interacted with or were covered by the CEX species.

Minimal unknown-input functional observers for multi-input multi-output LTI systems

Designing minimum possible order (minimal) disturbance-decoupled proper functional observers for multi-input multi-output (MIMO) linear time-invariant (LTI) systems is studied. It is not necessary that a minimum-order unknown-input functional observer (UIFO) exists in our proposed design procedure. If the minimum-order observer cannot be attained, the observer's order is increased sequentially through a recursive algorithm, so that the minimal order UIFO can be obtained. To the best of our knowledge, this is the first time that this specific problem is addressed. It is assumed that the system is unknown-input functional detectable, which is the least requirement for the existence of a stable UIFO. This condition also is a certificate for the convergence of our observer's order-increase algorithm. Two methodologies are demonstrated to solve the observer design equations. The second presented scheme, is a new design method that based on our observations has a better numerical performance than the first conventional one. Numerical examples and simulation results in the MATLAB/Simulink environment describe the overall observer design procedure, and highlight the efficacy of our new methodology to solve the observer equations in comparison to the conventional one.

Minimal multi-functional observers for linear systems using a direct approach

The direct approach in designing functional observers was first presented in [1] for estimating a single function of the states of a Linear Time-Invariant (LTI) system. One of the benefits of the direct scheme is that it does not require solving the interconnected Sylvester equations that appear in the other observer design approaches. In the present paper, the direct approach is extended to reconstruct multiple functions of the states in such a way that the minimum possible order of the observer is achieved. The observer is designed so that an asymptotic functional observer can be obtained with arbitrary convergence rate. In the proposed methodology, it is not necessary that a reduced order observer exists for the desired functions to be estimated. To release this limitation, an algorithm is employed to find some auxiliary functions in the minimum required number to be appended to the desired functions. This method assumes that the system is functional observable. This assumption however is less restrictive than the observability and detectability conditions of the system. A numerical example and simulation results explain the efficacy and the benefits of the proposed algorithm.

Functional observer-based fault detection of time-delay systems via an LMI approach

This paper presents a novel residual generator that uses minimum-order functional observers to trigger actuator and component faults in time-delay systems. We first present a fault detection scheme and derive existence conditions of the residual generator and functional observer. The observer and residual parameters are then systematically determined via solving some coupled generalized Sylvester matrix equations. To deal with the time-delay issue, a stabilizability condition expressed in terms of linear matrix inequality (LMI) is derived to ensure the time-delay observer error system converges to zero with a prescribed convergence rate. Our design approach has the advantage that the designed fault detection scheme has lower order than existing results in the literature. Two numerical examples are given to illustrate the effectiveness of our results.

Exponential stabilization of non-autonomous delayed neural networks via Riccati equations

This paper concerns with the problem of exponential stabilization for a class of non-autonomous neural networks with mixed discrete and distributed time-varying delays. Two cases of discrete time-varying delay, namely (i) slowly time-varying; and (ii) fast time-varying, are considered. By constructing an appropriate Lyapunov-Krasovskii functional in case (i) and utilizing the Razumikhin technique in case (ii), we establish some new delay-dependent conditions for designing a memoryless state feedback controller which stabilizes the system with an exponential convergence of the resulting closed-loop system. The proposed conditions are derived through solutions of some types of Riccati differential equations. Applications to control a class of autonomous neural networks with mixed time-varying delays are also discussed in this paper. Some numerical examples are provided to illustrate the effectiveness of the obtained results.