We develop a new approach to the analysis of pseudodifferential operators with small parameter 'epsilon' in (0,1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on 'epsilon'. Instead we consider the cylinder [0,1] x X over X and study pseudodifferential operators on the cylinder which act, by the very nature, on functions depending on 'epsilon' as well. The action in 'epsilon' reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of 'epsilon'.

>Magister Scientiae - MSc / With the development of technology seen in the last few decades, numerous solvers have been developed to provide adequate solutions to the problems that model different aspects of science and engineering. Quite often, these solvers are tailor-made for specific classes of problems. Therefore, more of such must be developed to accompany the growing need for mathematical models that help in the understanding of the contemporary world. This thesis treats two point boundary value singularly perturbed problems. The solution to this type of problem undergoes steep changes in narrow regions (called boundary or internal layer regions) thus rendering the classical numerical procedures inappropriate. To this end, robust numerical methods such as finite difference methods, in particular fitted mesh and fitted operator methods have extensively been used. While the former consists of transforming the continuous problem into a discrete one on a non-uniform mesh, the latter involves a special discretisation of the problem on a uniform mesh and are known to be more accurate. Both classes of methods are suitably designed to accommodate the rapid change(s) in the solution. Quite often, finite difference methods on piece-wise uniform meshes (of Shishkin-type) are adopted. However, methods based on such non-uniform meshes, though layer-resolving, are not easily extendable to higher dimensions. This work aims at investigating the possibility of capitalising on the advantages of both fitted mesh and fitted operator methods. Theoretical results are confirmed by extensive numerical simulations.

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Persistence problem of compact invariant manifold under random perturbation is considered in this dissertation. Under uniformly small random perturbation and the condition of normal hyperbolicity, the original invariant manifold persists and becomes a random invariant manifold. The random counterpart has random local stable and unstable manifolds. They could be invariantly foliated thanks to the normal hyperbolicity. Those underlie an extension of the geometric singular perturbation theory to the random case which means the slow manifold persists and becomes a random manifold so that the local global structure near the slow manifold persists under singular perturbation. A normal form for a random differential equation is obtained and this helps to prove a random version of the exchange lemma.

A compact model representing the dynamics between piezoelectric voltage inputs and cantilever probe positioning, including nonlinear surface interaction forces, for atomic force microscopes (AFM) is considered. By considering a relatively large cantilever stiffness, singular perturbation methods reduce complexity in the model and allows for faster responses to Van der Waals interaction forces experienced by the cantilever's tip and measurement sample. In this study, we outline a nonlinear near-optimal feedback control approach for non-contact mode imaging designed to move the cantilever tip laterally about a desired trajectory and maintain the tip vertically about the equilibrium point of the attraction and repulsion forces. We also consider the universal instance when the tip-sample interaction force is unknown, and we construct cascaded high-gain observers to estimate these forces and multiple AFM dynamics for the purpose of output feedback control. Our proposed output feedback controller is used to accomplish the outlined control objective with only the piezotube position available for state feedback. / Master of Science / In this thesis, the idea of an atomic force microscope (AFM), specifically the applications of the non-contact mode, will be discussed. An atomic force microscope (AFM) is a tool that measures the surface height of nanometer sized samples. To improve the speed and precision of the machine under a non-contact mode objective, a controller is designed based on optimality and is applied to the system. The system contains a series of equations designed to steer the system towards a desired trajectory and minimal vibrations. Given the complexity of the system, resulting from nonlinearities, we will apply singular perturbation principles on the system's stiffness property to separate the larger problem into two smaller ones. These two problems are inserted into a near-optimal controller and a series of simulations are conducted to demonstrate performance. Alongside this, we will outline an observer to estimate the unknown dynamics of the system. These estimates are then applied to our controller to demonstrate that only the AFM's piezotube position is to be known in order to estimate and control the remaining dynamics of the system.

Un grand nombre de phénomènes nous entourant peuvent être décrit par des modèles hybrides, c'est-à-dire, mettant en jeu simultanément une dynamique continu et une dynamique discrète. Également, il n'est pas rare que ces dynamiques puissent évoluer dans des échelles de temps différentes. Dans cette thèse, nous nous intéressons à l'analyse de stabilité des systèmes à commutations singulièrement perturbés à temps continu. En présence de commutations, l'analyse de stabilité des systèmes singulièrement perturbés dite "classique" (séparation des échelles de temps) n'est plus valable. En nous plaçant en dimension deux et en considérant deux modes, nous donnons une caractérisation complète du comportement asymptotique de tels systèmes lorsque le paramètre de perturbation tend vers zéro. Ensuite, nous étudions la discrétisation des systèmes à commutations singulièrement perturbés, en portant un intérêt particulier aux méthodes de discrétisation permettant de préserver la stabilité et les fonctions de Lyapunov quadratiques communes / Many phenomena we encounter can be described by hybrid models, namely, consisting of one continuous dynamic and one discret dynamic at the same time. Moreover, these dynamics often evolves in different time scales. In this thesis, we deal with the stability analysis of singularly perturbed switched systems in continuous time. When we consider switchings, the "classical" approach (decoupling fast and slow dynamics) allowing to analyse stability of singularly perturbed systems doesn't hold anymore. Considering second order singularly perturbed switched systems woth two modes, we completely characterize de stability behavior of such systems when the perturbation parameter goes to zero. Then, we study the discretization of singularly perturbed switched systems. In particular, we focus on methods allowing to preserve stability and common quadratic Lyapunov functions

<p>U disertaciji su razvijeni kolokacioni postupci sa C¹- splajnovima&nbsp;proizvoljnog stepena za re&scaron;avanje singularno-perturbovanih problema&nbsp;reakcije-difuzije u jednoj i dve dimenzije. U 1D, pokazano je da kolokacioni&nbsp;postupak sa kvadratnim C¹-&nbsp;splajnom na modifikovanoj &Scaron;i&scaron;kinovoj mreži,&nbsp;konvergira uniformno, sa redom konvergencije skoro dva. Takođe, na gradiranim mrežama, ovaj metod ima red konvergencije dva &ndash; uniformno do na logaritamski faktor. Aposterirona ocena je postignuta za kolokacione postupke sa C¹- splajnovima proizvoljnog stepena na proizvoljnoj mreži. Ova ocena je iskori&scaron;ćena i za kreiranje adaptivnih mreža. Numerički rezultati povtrđuju dobijene ocene. U 2D su razmatrane kolokacije sa bikvadratnim splajnovima. Aposterirona ocena gre&scaron;ke je postignuta. Numerički rezultati potvrđuju dobijene teorijske rezultate.<br />&nbsp;</p> / <p>Collocations with arbitrary order C¹-splines for a singularly perturbed reaction-diffusion problem in one dimension and two dimensions are studied. In 1D, collocation with quadratic C¹-splines is shown to be almost second order accurate on modified Shishkin mesh in the maximum norm, uniformly in the perturbation parameter. Also, we establish a second-order maximum norm a priori estimate on recursively graded mesh uniformly up to a logarithmic factor in the singular perturbation parameter. A posteriori error bounds are derived for the collocation method with arbitrary order C¹-splines on arbitrary meshes. These bounds are used to drive an adaptivemeshmoving algorithm. An adaptive algorithm is devised&nbsp;to resolve the boundary layers. Numerical results are presented. In 2D, collocation with biquadratic C¹-spline is studied. Robust a posteriori error bounds are derived for the collocation method on arbitrary meshes. Numerical experiments completed our theoretical results.</p>

The study of cardiac action potentials has many medical applications. Dr. Dennis Noble first used mathematical models to study cardiac action potentials in the 1960s. We begin our study of cardiac action potentials with one form of the Fitzhugh-Nagumo partial differential equation. We use the non-classical method to produce a closed form solution for the decoupled Fitzhugh Nagumo equation. Using voltage recording data of action potentials in a cardiac myocyte and in purkinje fibers, we estimate parameter values for the closed form solution with standard linear and non-linear regression methods. Results are limited, thus leading us to perturb the solution to obtain a better fit. We turn to singular perturbation theory to justify our pole-based approach. Finally, we test our model on independent action potential data sets to evaluate our model and to draw conclusions on how our model can be applied.

With the arrival of modern component-based modeling tools for dynamic systems, the differential-algebraic equation form is increasing in popularity as it is general enough to handle the resulting models. However, if uncertainty is allowed in the equations — no matter how small — this thesis stresses that such equations generally become ill-posed. Rather than deeming the general differential-algebraic structure useless up front due to this reason, the suggested approach to the problem is to ask what assumptions that can be made in order to obtain well-posedness. Here, “well-posedness” is used in the sense that the uncertainty in the solutions should tend to zero as the uncertainty in the equations tends to zero. The main theme of the thesis is to analyze how the uncertainty in the solution to a differential-algebraic equation depends on the uncertainty in the equation. In particular, uncertainty in the leading matrix of linear differential-algebraic equations leads to a new kind of singular perturbation, which is referred to as “matrix-valued singular perturbation”. Though a natural extension of existing types of singular perturbation problems, this topic has not been studied in the past. As it turns out that assumptions about the equations have to be made in order to obtain well-posedness, it is stressed that the assumptions should be selected carefully in order to be realistic to use in applications. Hence, it is suggested that any assumptions (not counting properties which can be checked by inspection of the uncertain equations) should be formulated in terms of coordinate-free system properties. In the thesis, the location of system poles has been the chosen target for assumptions. Three chapters are devoted to the study of uncertain differential-algebraic equations and the associated matrix-valued singular perturbation problems. Only linear equations without forcing function are considered. For both time-invariant and time-varying equations of nominal differentiation index 1, the solutions are shown to converge as the uncertainties tend to zero. For time-invariant equations of nominal index 2, convergence has not been shown to occur except for an academic example. However, the thesis contains other results for this type of equations, including the derivation of a canonical form for the uncertain equations. While uncertainty in differential-algebraic equations has been studied in-depth, two related topics have been studied more passingly. One chapter considers the development of point-mass filters for state estimation on manifolds. The highlight is a novel framework for general algorithm development with manifold-valued variables. The connection to differential-algebraic equations is that one of their characteristics is that they have an underlying manifold-structure imposed on the solution. One chapter presents a new index closely related to the strangeness index of a differential-algebraic equation. Basic properties of the strangeness index are shown to be valid also for the new index. The definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments.

Abstract In this dissertation, the eigenvalue analysis and decentralized robust controller design of uncertain multi-time-scale system with parametrical perturbations are considered. Because the eigenvalues of the multi-time-scale systems cluster in some difference regions of the complex plane, we can use the singular perturbation method to separate the systems into some subsystems. These subsystems are independent to each other. We can discuss the properties of eigenvalues and design controller for these subsystem respectively, then we composite these controllers to a decentralized controller. The eigenvalue positions dominate the stability and the performance of the dynamic system. However, we cannot obtain the precise position of the eigenvalues from the influence of parametrical perturbations. The sufficient conditions of the eigenvalues clustering for the multi-time-scale systems will be discussed. The uncertainties consider as unstructured and structured perturbations are taken into considerations. The design algorithm provides for designing a decentralized controller that can assign the poles to our respect regions. The specified regions are half-plane and circular disk. Furthermore, the concepts of decentralized control and optimal control are used to design the linear quadratic regulator (LQR) controller and linear quadratic Gaussian (LQG) controller for the perturbed multi-time-scale systems. That is, the system can get the optimal robust performance. The bound of the singular perturbation parameter would influence the robust stability of the multi-time-scale systems. Finally, the sufficient condition to obtain the upper bound of the singular perturbation parameter presented by the Lyapunov method and matrix norm. The condition also extends for the pole assignment in the specified regions of each subsystem respectively. The illustrative examples are presented behind each topic. They show the applicability of the proposed theorems, and the results are satisfactory.