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Stability of Neutral Delay Differential Equations and Their Discretizations / Stability of Neutral Delay Differential Equations and Their DiscretizationsDražková, Jana January 2014 (has links)
Disertační práce se zabývá asymptotickou stabilitou zpožděných diferenciálních rovnic a jejich diskretizací. V práci jsou uvažovány lineární zpožděné diferenciální rovnice s~konstantním i neohraničeným zpožděním. Jsou odvozeny nutné a postačující podmínky popisující oblast asymptotické stability jak pro exaktní, tak i diskretizovanou lineární neutrální diferenciální rovnici s konstantním zpožděním. Pomocí těchto podmínek jsou porovnány oblasti asymptotické stability odpovídajících exaktních a diskretizovaných rovnic a vyvozeny některé vlastnosti diskrétních oblastí stability vzhledem k měnícímu se kroku použité diskretizace. Dále se zabýváme lineární zpožděnou diferenciální rovnicí s neohraničeným zpožděním. Je uveden popis jejích exaktních a diskrétních oblastí asymptotické stability spolu s asymptotickým odhadem jejich řešení. V závěru uvažujeme lineární diferenciální rovnici s více neohraničenými zpožděními.
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Global finite-time observers for a class of nonlinear systemsLi, Yunyan January 2013 (has links)
The contributions of this thesis lie in the area of global finite-time observer design for a class of
nonlinear systems with bounded rational and mixed rational powers imposed on the incremental rate
of the nonlinear terms whose solutions exist and are unique for all positive time. In the thesis, two
different kinds of nonlinear global finite-time observers are designed by employing of finite-time
theory and homogeneity properties with different methods. The global finite-time stability of both
proposed observers is derived on the basis of Lyapunov theory.
For a class of nonlinear systems with rational and mixed rational powers imposed on the nonlinearities,
the first global finite-time observers are designed, where the global finite-time stability of the
observation systems is achieved from two parts by combining asymptotic stability and local finitetime
stability. The proposed observers can only be designed for the class of nonlinear systems with
dimensions greater than 3. The observers have a dynamic high gain and two homogenous terms, one
homogeneous of degree greater than 1 and the other of degree less than 1. In order to prove the global
finite-time stability of the proposed results, two homogeneous Lyapunov functions are provided, corresponding
with the two homogeneous items. One is homogeneous of degree greater than 1, which
makes the observation error systems converging into a spherical area around the origin, and the other
is of degree less than 1, which ensures local finite-time stability.
The second global finite-time observers are also proposed based on the high-gain technique, which
does not place any limitation on the dimension of the nonlinear systems. Compared with the first
global finite-time observers, the newly designed observers have only one homogeneous term and a
new gain update law where two new terms are introduced to dominate some terms in the nonlinearities
and ensure global finite-time stability as well. The global finite-time stability is obtained directly
based on a sufficient condition of finite-time stability and only one Lyapunov function is employed in
the proof.
The validity of the two kinds of global finite-time observers that have been designed is illustrated through some simulation results. Both of them can make the observation error systems converge to
the origin in finite-time. The parameters, initial conditions as well as the high gain do have some
impact on the convergence time, where the high gain plays a stronger role. The bigger the high gain
is, the shorter the time it needs to converge. In order to show the performance of the two kinds of
observers more clearly, two examples are provided and some comparisons are made between them.
Through these, it can be seen that under the same parameters and initial conditions, although the
amplitude of the observation error curve is slightly greater, the global finite-time observers with a
new gain update law can make the observation error systems converge much more quickly than the
global finite-time observers with two homogeneous terms. In the simulation results, one can see that,
as a common drawback of high gain observers, they are noise-sensitive. Finding methods to improve
their robustness and adaptiveness will be quite interesting, useful and challenging. / Thesis (PhD)--University of Pretoria, 2013. / gm2014 / Electrical, Electronic and Computer Engineering / unrestricted
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The asymptotic stability of stochastic kernel operatorsBrown, Thomas John 06 1900 (has links)
A stochastic operator is a positive linear contraction, P : L1 --+ L1,
such that
llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of
each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy,
where K( ·, y) is a density, defines a stochastic kernel operator. A general probabilistic/
deterministic model for biological systems is considered. This leads to the
LMT operator
P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy,
where -H'(x) = h(x) is a density. Several particular examples of cell cycle models
are examined. An operator overlaps supports iffor all densities f,g, pn f APng of 0
for some n. If the operator is partially kernel, has a positive invariant density and
overlaps supports, it is asymptotically stable. It is found that if h( x) > 0 for
x ~ xo ~ 0 and
["'" x"h(x) dx < liminf(Q(A(x))" - Q(x)") for a E (0, 1] lo x-oo
then P is asymptotically stable, and an opposite condition implies P is sweeping.
Many known results for cell cycle models follow from this. / Mathematical Science / M. Sc. (Mathematics)
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Qualitative Properties of Stochastic Hybrid Systems and ApplicationsAlwan, Mohamad January 2011 (has links)
Hybrid systems with or without stochastic noise and with or without time delay are addressed and the qualitative properties of these systems are investigated. The main contribution of this thesis is distributed in three parts.
In Part I, nonlinear stochastic impulsive systems with time delay (SISD) with variable impulses are formulated and some of the fundamental properties of the systems, such as existence of local and global solution, uniqueness, and forward continuation of the solution are established. After that, stability and input-to-state stability (ISS) properties of SISD with fixed impulses are developed, where Razumikhin methodology is used. These results are then carried over to discussed the same qualitative properties of large scale SISD. Applications to automated control systems and control systems with faulty actuators are used to justify the proposed approaches.
Part II is devoted to address ISS of stochastic ordinary and delay switched systems. To achieve a variety stability-like results, multiple Lyapunov technique as a tool is applied. Moreover, to organize the switching among the system modes, a newly developed initial-state-dependent dwell-time switching law and Markovian switching are separately employed.
Part III deals with systems of differential equations with piecewise constant arguments with and without random noise. These systems are viewed as a special type of hybrid systems. Existence and uniqueness results are first obtained. Then, comparison principles are established which are later applied to develop some stability results of the systems.
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Qualitative Properties of Stochastic Hybrid Systems and ApplicationsAlwan, Mohamad January 2011 (has links)
Hybrid systems with or without stochastic noise and with or without time delay are addressed and the qualitative properties of these systems are investigated. The main contribution of this thesis is distributed in three parts.
In Part I, nonlinear stochastic impulsive systems with time delay (SISD) with variable impulses are formulated and some of the fundamental properties of the systems, such as existence of local and global solution, uniqueness, and forward continuation of the solution are established. After that, stability and input-to-state stability (ISS) properties of SISD with fixed impulses are developed, where Razumikhin methodology is used. These results are then carried over to discussed the same qualitative properties of large scale SISD. Applications to automated control systems and control systems with faulty actuators are used to justify the proposed approaches.
Part II is devoted to address ISS of stochastic ordinary and delay switched systems. To achieve a variety stability-like results, multiple Lyapunov technique as a tool is applied. Moreover, to organize the switching among the system modes, a newly developed initial-state-dependent dwell-time switching law and Markovian switching are separately employed.
Part III deals with systems of differential equations with piecewise constant arguments with and without random noise. These systems are viewed as a special type of hybrid systems. Existence and uniqueness results are first obtained. Then, comparison principles are established which are later applied to develop some stability results of the systems.
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Aplicação do método de linearização de Lyapunov na análise de uma dinâmica não linear para controle populacional do mosquito Aedes aegypti / Application of the Lyapunov linearization method in the analysis of a nonlinear dynamics for mosquito control population Aedes aegyptiMaranho, Luiz Cesar 20 August 2018 (has links)
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Previous issue date: 2018-08-20 / O mosquito Aedes aegypti é o principal vetor responsável por diversas arboviroses como a dengue, a febre amarela, o vírus zika e a febre chikungunya. Devido a sua resistência, adaptabilidade e proximidade ao homem, o Aedes aegypti é atualmente um dos maiores problemas de saúde pública no Brasil e nas Américas. Mesmo com os avanços e investimentos em pesquisas com vacinas, monitoramento, campanhas educativas e diversos tipos de controle deste vetor, ainda não existe um método eficaz para controlar e erradicar o mosquito. Portanto, esse trabalho destina-se ao auxílio na criação de estratégias para controlar esse agente transmissor, mediante a análise do espaço de estados e a estabilidade assintótica de uma dinâmica não linear para controle populacional do Aedes aegypti via a técnica de linearização de Lyapunov, além de apresentação de formas de prevenção e combate aos criadouros do mosquito. A dinâmica não linear proposta é uma dinâmica simplificada obtida de um modelo não linear existente na literatura, proposto por Esteva e Yang em 2005 e se baseia no ciclo de vida do mosquito, que é dividido em duas fases: fase imatura ou aquática (ovos, larvas e pupas) e fase alada (mosquitos adultos). Na fase adulta, os mosquitos são divididos em machos, fêmeas imaturas e fêmeas fertilizadas, sendo que a dinâmica proposta nesta dissertação de mestrado é baseada nos estudos efetuados por Reis desde 2016, obtendo um modelo simplificado no qual a soma das densidades das populações de fêmeas imaturas e fêmeas fertilizadas serão consideradas como fêmeas adultas. / The mosquito Aedes aegypti is the main vector responsible for several arboviruses such as dengue fever, yellow fever, zika virus and chikungunya fever. Due to its resistance, adaptability and proximity to humans, Aedes aegypti is currently one of the major public health problems in Brazil and the Americas. Even with the advances and investments in research with vaccines, monitoring, educational campaigns and various types of control of this vector, there is still no effective method to control and eradicate the mosquito. Therefore, this work is intended to aid in the creation of strategies to control this transmitting agent by analyzing the state space and the asymptotic stability of a nonlinear dynamics for population control of Aedes aegypti via the Lyapunov linearization technique to present ways of preventing and combating mosquito breeding sites. The proposed nonlinear dynamics is a simplified dynamics obtained from a nonlinear model existing in the literature, proposed by Esteva and Yang in 2005 and based on the life cycle of the mosquito, which is divided into two phases: immature or aquatic phase (eggs, larvae and pupae) and winged phase (adult mosquitoes). In the adult phase, mosquitoes are divided into males, immature females and fertilized females, and the dynamics proposed in this dissertation is based on studies carried out by Reis since 2016, obtaining a simplified model in which the sum of the densities of the populations of females immature and fertilized females will be considered as adult females.
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The asymptotic stability of stochastic kernel operatorsBrown, Thomas John 06 1900 (has links)
A stochastic operator is a positive linear contraction, P : L1 --+ L1,
such that
llPfII2 = llfll1 for f > 0. It is called asymptotically stable if the iterates pn f of
each density converge in the norm to a fixed density. Pf(x) = f K(x,y)f(y)dy,
where K( ·, y) is a density, defines a stochastic kernel operator. A general probabilistic/
deterministic model for biological systems is considered. This leads to the
LMT operator
P f(x) = Jo - Bx H(Q(>.(x)) - Q(y)) dy,
where -H'(x) = h(x) is a density. Several particular examples of cell cycle models
are examined. An operator overlaps supports iffor all densities f,g, pn f APng of 0
for some n. If the operator is partially kernel, has a positive invariant density and
overlaps supports, it is asymptotically stable. It is found that if h( x) > 0 for
x ~ xo ~ 0 and
["'" x"h(x) dx < liminf(Q(A(x))" - Q(x)") for a E (0, 1] lo x-oo
then P is asymptotically stable, and an opposite condition implies P is sweeping.
Many known results for cell cycle models follow from this. / Mathematical Science / M. Sc. (Mathematics)
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M?todos num?ricos para resolu??o de equa??es diferenciais ordin?rias lineares baseados em interpola??o por splineAraujo, Thiago Jefferson de 13 August 2012 (has links)
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Previous issue date: 2012-08-13 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this work we have elaborated a spline-based method of solution of inicial value
problems involving ordinary differential equations, with emphasis on linear equations.
The method can be seen as an alternative for the traditional solvers such as Runge-Kutta,
and avoids root calculations in the linear time invariant case.
The method is then applied on a central problem of control theory, namely, the step
response problem for linear EDOs with possibly varying coefficients, where root calculations
do not apply. We have implemented an efficient algorithm which uses exclusively
matrix-vector operations. The working interval (till the settling time) was determined
through a calculation of the least stable mode using a modified power method.
Several variants of the method have been compared by simulation. For general linear
problems with fine grid, the proposed method compares favorably with the Euler method.
In the time invariant case, where the alternative is root calculation, we have indications
that the proposed method is competitive for equations of sifficiently high order. / Neste trabalho desenlvolvemos um m?todo de resolu??o de problemas de valor inicial
com equa??es diferenciais ordin?rias baseado em splines, com ?nfase em equa??es lineares.
O m?todo serve como alternativa para os m?todos tradicionais como Runge-Kutta e no
caso linear com coeficientes constantes, evita o c?lculo de ra?zes de polin?mios. O m?todo
foi aplicado para um problema central da teoria de controle, o problema de resposta a
degrau para uma EDO linear, incluindo o caso de coeficientes n?o-constantes, onde a alternativa
pelo c?lculo de ra?zes n?o existe. Implementamos um algoritmo eficiente que usa
apenas opera??es tipo matriz-vetor. O intervalo de trabalho (at? o tempo de acomoda??o)
para as equa??es est?veis com coeficientes constantes ?e determinado pelo c?lculo da raiz
menos est?vel do sistema, a partir de uma adapta??o do m?todo da pot?ncia. Atrav?s de
simula??es, comparamos algumas variantes do m?todo. Em problemas lineares gerais com
malha suficientemente fina, o novo m?todo mostra melhores resultados em compara??o
com o m?todo de Euler. No caso de coeficientes constantes, onde existe a alternativa baseada
em c?lculo das ra?zes, temos indica??es que o novo m?todo pode ficar competitivo
para equa??es de grau bastante alto
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Comportement asymptotique des systèmes de fonctions itérées et applications aux chaines de Markov d'ordre variable / Asymptotic behaviour of iterated function systems and applications to variable length Markov chainsDubarry, Blandine 14 June 2017 (has links)
L'objet de cette thèse est l'étude du comportement asymptotique des systèmes de fonctions itérées (IFS). Dans un premier chapitre, nous présenterons les notions liées à l'étude de tels systèmes et nous rappellerons différentes applications possibles des IFS telles que les marches aléatoires sur des graphes ou des pavages apériodiques, les systèmes dynamiques aléatoires, la classification de protéines ou encore les mesures quantiques répétées. Nous nous attarderons sur deux autres applications : les chaînes de Markov d'ordre infini et d'ordre variable. Nous donnerons aussi les principaux résultats de la littérature concernant l'étude des mesures invariantes pour des IFS ainsi que ceux pour le calcul de la dimension de Hausdorff. Le deuxième chapitre sera consacré à l'étude d'une classe d'IFS composés de contractions sur des intervalles réels fermés dont les images se chevauchent au plus en un point et telles que les probabilités de transition sont constantes par morceaux. Nous donnerons un critère pour l'existence et pour l'unicité d'une mesure invariante pour l'IFS ainsi que pour la stabilité asymptotique en termes de bornes sur les probabilités de transition. De plus, quand il existe une unique mesure invariante et sous quelques hypothèses techniques supplémentaires, on peut montrer que la mesure invariante admet une dimension de Hausdorff exacte qui est égale au rapport de l'entropie sur l'exposant de Lyapunov. Ce résultat étend la formule, établie dans la littérature pour des probabilités de transition continues, au cas considéré ici des probabilités de transition constantes par morceaux. Le dernier chapitre de cette thèse est, quant à lui, consacré à un cas particulier d'IFS : les chaînes de Markov de longueur variable (VLMC). On démontrera que sous une condition de non-nullité faible et de continuité pour la distance ultramétrique des probabilités de transitions, elles admettent une unique mesure invariante qui est attractive pour la convergence faible. / The purpose of this thesis is the study of the asymptotic behaviour of iterated function systems (IFS). In a first part, we will introduce the notions related to the study of such systems and we will remind different applications of IFS such as random walks on graphs or aperiodic tilings, random dynamical systems, proteins classification or else $q$-repeated measures. We will focus on two other applications : the chains of infinite order and the variable length Markov chains. We will give the main results in the literature concerning the study of invariant measures for IFS and those for the calculus of the Hausdorff dimension. The second part will be dedicated to the study of a class of iterated function systems (IFSs) with non-overlapping or just-touching contractions on closed real intervals and adapted piecewise constant transition probabilities. We give criteria for the existence and the uniqueness of an invariant probability measure for the IFSs and for the asymptotic stability of the system in terms of bounds of transition probabilities. Additionally, in case there exists a unique invariant measure and under some technical assumptions, we obtain its exact Hausdorff dimension as the ratio of the entropy over the Lyapunov exponent. This result extends the formula, established in the literature for continuous transition probabilities, to the case considered here of piecewise constant probabilities. The last part is dedicated to a special case of IFS : Variable Length Markov Chains (VLMC). We will show that under a weak non-nullness condition and continuity for the ultrametric distance of the transition probabilities, they admit a unique invariant measure which is attractive for the weak convergence.
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Nonlinear and Hybrid Feedbacks with Continuous-Time Linear Systems / Rétroactions non linéaires et hybrides avec systèmes linéaires à temps continuCocetti, Matteo 21 May 2019 (has links)
Dans cette thèse, nous étudions la rétroaction de systèmes linéaires invariants dans le temps reliés entre eux par trois blocs non linéaires spécifiques : un opérateur de lecture/arrêt, un mécanisme de réinitialisation de commutation et une zone morte adaptative. Cette configuration ressemble au problème de Lure étudié dans le cadre de stabilité absolue, mais les types de non-linéarités considérés ici ne satisfont pas (en général) une condition sectorielle. Ces blocs non linéaires donnent lieu à toute une série de phénomènes intéressants, tels que des ensembles compacts d’équilibres, des ensembles hybrides oméga-limites et des contraintes d’état. Tout au long de la thèse, nous utilisons le formalisme des systèmes hybrides pour décrire ces phénomènes et analyser ces boucles. Nous obtenons des conditions de stabilité très précises qui peuvent être formulées sous forme d’inégalités matricielles linéaires, donc vérifiables avec des solveurs numériques efficaces. Enfin, nous appliquons les résultats théoriques à deux applications automobiles. / In this thesis we study linear time-invariant systems feedback interconnected with three specific nonlinear blocks; a play/stop operator, a switching-reset mechanism, and an adaptive dead-zone. This setup resembles the Lure problem studied in the absolute stability framework, but the types of nonlinearities considered here do not satisfy (in general) a sector condition. These nonlinear blocks give rise to a whole range of interesting phenomena, such as compact sets of equilibria, hybrid omega-limit sets, and state constraints. Throughout the thesis, we use the hybrid systems formalism to describe these phenomena and to analyze these loops. We obtain sharp stability conditions that can be formulated as linear matrix inequalities, thus verifiable with numerically efficient solvers. Finally, we apply the theoretical findings to two automotive applications.
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