Multi-agent systems are extensively used in several applications. An important class of applications involves the optimal spatial distribution of a group of mobile robots on a given area, where the optimality refers to the assignment of subregions to the robots, in such a way that a suitable coverage metric is maximized. Typically the coverage metric encodes a risk distribution defined on the area, and a measure of the performance of individual robots with respect to points inside the region of interest. The coverage metric will be maximized when the set of mobile robots configure themselves as the centroids of the Voronoi tessellation dictated by the risk density. In this work we advance on this result by considering a generalized area control problem in which the coverage metric is non-autonomous, that coverage metric is time varying independently of the states of the robots. This generalization is motivated by the study of coverage control problems in which the coordinated motion of a set of mobile robots accounts for the kinematics of objects penetrating from the outside. Asymptotic convergence and optimality of the non-autonmous system are studied by means of Barbalat's Lemma, and connections with the kinematics of the moving intruders is established. Several numerical simulation results are used to illustrate theoretical predictions.

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In mathematical epidemiology, disease transmission is commonly assumed to behave in accordance with the law of mass action; however, other disease incidence terms also exist in the literature. A homogeneous Susceptible-Infectious-Removed (SIR) model with a generalized incidence term is presented along with analytic and numerical results concerning effects of the generalization on the global disease dynamics. The spatial heterogeneity of the metapopulation with nonrandom directed movement between populations is incorporated into a heterogeneous SIR model with nonlinear incidence. The analysis of the combined effects of the spatial heterogeneity and nonlinear incidence on the disease dynamics of our model is presented along with supporting simulations. New global stability results are established for the heterogeneous model utilizing a graph-theoretic approach and Lyapunov functions. Numerical simulations confirm nonlinear incidence gives raise to rich dynamics such as synchronization and phase-lock oscillations.

Cette thèse est consacrée à l'étude de certains aspects du comportement en temps longs des solutions de deux équations de Schrödinger non-linéaires en dimension trois dans des régimes perturbatives convenables. Le premier modèle consiste en une équation de Schrödinger avec une non-linéarité concentrée obtenue en considérant une interaction ponctuelle de force $alpha$, c'est-à-dire une perturbation singulière du Laplacien décrite par un opérateur autoadjoint $H_{alpha}$, où la force $alpha$ dépend de la fonction d'onde : $ifrac{du}{dt}= H_alpha u$, $alpha=alpha(u)$. Il est bien connu que les éléments du domaine d'une interaction ponctuelle en trois dimensions peuvent être décrits comme la somme d'une fonction régulière et d'une fonction ayant une singularité proportionnelle à $|x - x_0|^{-1}$, où $x_0$ est l'emplacement du point d'interaction. Si $q$ est la charge d'un élément du domaine $u$, c'est-à-dire le coefficient de sa partie singulière, alors pour introduire une non-linéarité, on fait dépendre la force $alpha$ de $u$ selon la loi $alpha=-nu|q|^sigma$, avec $nu > 0$. Ce modèle est défini comme une équation de Schrödinger non-linéaire focalisant de type puissance avec une non-linéarité concentrée en $x_0$. Notre étude regarde la stabilité orbitale et asymptotique des ondes stationnaires de ce modèle. Nous prouvons l'existence d'ondes stationnaires de la forme $u (t)=e^{iomega t}Phi_{omega}$, qui soient orbitalement stables pour $sigma in (0,1)$ et orbitalement instables quand $sigma geq 1.$ De plus nous montrons que si $sigma in (0,frac{1}{sqrt 2}) cup (frac{1}{sqrt 2}, 1)$, alors chaque onde stationnaire est asymptotiquement stable, à savoir que pour des données initiales proches d'un état stationnaire dans la norme d'énergie et appartenant à un espace $L^p$ pondéré où les estimations dispersives sont valides, l'affirmation suivante est vérifiée : il existe $omega_{infty} > 0$ et $psi_{infty} in L^2(R^3)$ tel que $psi_{infty} = O_{L^2}(t^{-p})$ quand $t rightarrow +infty$, tel que $u(t) = e^{iomega_{infty} t +il(t)} Phi_{omega_{infty}} +U_t*psi_{infty} +r_{infty}$, où $U_t$ est le propagateur de Schrödinger libre, $p = frac{5}{4}$, $frac{1}{4}$ respectivement en fonction de $sigma in (0, 1/sqrt{2})$, $sigma in left( frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right)$, et $l(t)$ est une fonction à croissance logarithmique qui apparaît quand $sigma in (frac{1}{sqrt{2}}, sigma^*)$, où $sigma^* in left( frac{1}{sqrt{2}},frac{sqrt{3} +1}{2sqrt{2}} right]$. Notons que dans ce modèle les non-linéarités pour lesquelles on a la stabilité asymptotique sont sous-critiques dans le sens où quelle que soit la donnée initiale il n'y a pas de solutions explosives. Quant au deuxième modèle, il s'agit de l'équation de Schrödinger non-linéaire focalisant à énergie critique : $i frac{du}{dt}=-Delta u-|u|^4 u$. Pour ce cas, nous prouvons, pour tout $nu$ et $alpha_0$ suffisamment petits, l'existence de solutions radiales à énergie finie de la forme $u(t,x)=e^{ialpha(t)}lambda^{1/2}(t)W(lambda(t)x)+e^{iDelta t}zeta^*+o_{dot H^1} (1)$ tout $trightarrow +infty$, où $alpha(t)=alpha_0ln t$, $lambda(t)=t^{nu}$, $W(x)=(1+frac13|x|^2)^{-1/2}$ est l'état stationnaire et $zeta^*$ est arbitrairement petit en $dot H^1$ / The present thesis is devoted to the investigation of certain aspects of the large time behavior of the solutions of two nonlinear Schrödinger equations in dimension three in some suitable perturbative regimes. The first model consist in a Schrödinger equation with a concentrated nonlinearity obtained considering a {point} (or contact) interaction with strength $alpha$, which consists of a singular perturbation of the Laplacian described by a self adjoint operator $H_{alpha}$, and letting the strength $alpha$ depend on the wave function: $ifrac{du}{dt}= H_alpha u$, $alpha=alpha(u)$.It is well-known that the elements of the domain of a point interaction in three dimensions can be written as the sum of a regular function and a function that exhibits a singularity proportional to $|x - x_0|^{-1}$, where $x_0$is the location of the point interaction. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of itssingular part, then, in order to introduce a nonlinearity, we let the strength $alpha$ depend on $u$ according to the law $alpha=-nu|q|^sigma$, with $nu > 0$. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. In particular, we study orbital and asymptotic stability of standing waves for such a model. We prove the existence of standing waves of the form $u (t)=e^{iomega t}Phi_{omega}$, which are orbitally stable in the range $sigma in (0,1)$, and orbitally unstable for $sigma geq 1.$ Moreover, we show that for $sigma in(0,frac{1}{sqrt 2}) cup left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right)$ every standing wave is asymptotically stable, in the following sense. Choosing an initial data close to the stationary state in the energy norm, and belonging to a natural weighted $L^p$ space which allows dispersive stimates, the following resolution holds: $u(t) =e^{iomega_{infty} t +il(t)} Phi_{omega_{infty}}+U_t*psi_{infty} +r_{infty}$, where $U_t$ is the free Schrödinger propagator,$omega_{infty} > 0$ and $psi_{infty}$, $r_{infty} inL^2(R^3)$ with $| r_{infty} |_{L^2} = O(t^{-p}) quadtextrm{as} ;; t right arrow +infty$, $p = frac{5}{4}$,$frac{1}{4}$ depending on $sigma in (0, 1/sqrt{2})$, $sigma in (1/sqrt{2}, 1)$, respectively, and finally $l(t)$ is a logarithmic increasing function that appears when $sigma in (frac{1}{sqrt{2}},sigma^*)$, for a certain $sigma^* in left(frac{1}{sqrt{2}}, frac{sqrt{3} +1}{2sqrt{2}} right]$. Notice that in the present model the admitted nonlinearities for which asymptotic stability of solitons is proved, are subcritical in the sense that it does not give rise to blow up, regardless of the chosen initial data. The second model is the energy critical focusing nonlinear Schrödinger equation $i frac{du}{dt}=-Delta u-|u|^4 u$. In this case we prove, for any $nu$ and $alpha_0$ sufficiently small, the existence of radial finite energy solutions of the form$u(t,x)=e^{ialpha(t)}lambda^{1/2}(t)W(lambda(t)x)+e^{iDeltat}zeta^*+o_{dot H^1} (1)$ as $tright arrow +infty$, where$alpha(t)=alpha_0ln t$, $lambda(t)=t^{nu}$,$W(x)=(1+frac13|x|^2)^{-1/2}$ is the ground state and $zeta^*$is arbitrarily small in $dot H^1$

Neste texto é feita uma discussão sobre alguns resultados que fornecem condições suficientes para que um difeomorfismo local, do espaço euclideano n-dimensional nele próprio, seja injetivo. Dentro deste cenário, são exploradas as contribuições destes resultados na tentativa de solucionar conhecidas conjecturas no meio científico como a Conjectura Jacobiana e a Conjectura de Ponto Fixo. Do ponto de vista dinâmico, existem relações entre injetividade global e estabilidade assintótica global. Neste sentido, os resultados também são contextualizados com respeito a importantes conjecturas de estabilidade assintótica: Conjectura de Markus-Yamabe e o Problema de LaSalle / We present some results which give suficient conditions for a local diffeomorphism from the n-dimensional Euclidean space into itself be globally injective. Within this context, we consider some partial results addressed to solve the well known Fixed Point Conjecture and Jacobian Conjecture. From the dynamical point of view, there are connections between global injectivity and global asymptotic stability. In this way, we present a solution of the Markus-Yamabe Conjecture and of the LaSalle Problem

The concept of LR-nets was introduced and investigated firstly by H.P. Lotz in [27] and by F. Raebiger in [30]. Therefore we call such nets Lotz-Raebiger nets, shortly LR-nets. In this thesis we treat two problems on asymptotic behavior of these operator nets. First problem is to generalize well known theorems for Ces`aro averages of a single operator to LR-nets, namely to generalize the Eberlein and Sine theorems. The second problem is related to the strong convergence of Markov LR-nets on L1-spaces. We prove that the existence of a lower-bound functions is necessary and sufficient for asymptotic stability of LR-nets of Markov operators.

The technological changes and educational expansion have created the heterogeneity in the human species. Clearly, this heterogeneity generates a structure in the population dynamics, namely: citizen, permanent resident, visitor, and etc. Furthermore, as the heterogeneity in the population increases, the human mobility between meta-populations patches also increases. Depending on spatial scales, a meta-population patch can be decomposed into sub-patches, for examples: homes, neighborhoods, towns, etc. The dynamics of human mobility in a heterogeneous and scaled structured population is still its infancy level. We develop and investigate (1) an algorithmic two scale human mobility dynamic model for a meta-population. Moreover,the two scale human mobility dynamic model can be extended to multi-scales by applying the algorithm. The subregions and regions are interlinked via intra-and inter regional transport network systems. Under various types of growth order assumptions on the intra and interregional residence times of the residents of a sub region, different patterns of static behavior of the mobility process are studied. Furthermore, the human mobility dynamic model is applied to a two-scale population dynamic exhibiting a special real life human transportation network pattern. The static evolution of all categories of residents of a given site ( homesite, visiting sites within the region, and visiting sites in other regions) over continuous changes in the intra and inter-regional visiting times is also analyzed. The development of the two scale human mobility dynamic model provides a suitable approach to undertake the study of the non-uniform global spread of emergent infectious diseases of humans in a systematic and unified way. In view of this, we derive (2) a SIRS stochastic epidemic dynamic process in a two scale structured population. By defining a positively self invariant set for the dynamic model the stochastic asymptotic stability results of the disease free equilibrium are developed(2). Furthermore, the significance of the stability results are illustrated in a simple real life scenario that is under controlled quarantine disease strategy. In addition, the epidemic dynamic model (2) is applied to a SIR influenza epidemic in a two scale population that is under the influence of a special real life human mobility pattern. The simulated trajectories for the different states (susceptible, Infective, Removal) with respect to current location in the two-scale population structure are presented. The simulated findings reveal comparative evolution patterns for the different states and current locations over time. The SIRS stochastic epidemic dynamic model (2) is extended to a SIR delayed stochastic epidemic dynamic model(3). The delay effects in the dynamic model (3) is temporary and account for natural or infection acquired immunity conferred by the disease after disease recovery. Again, we justify the model validation as a prerequisite for the dynamic modeling. Moreover, we also exhibit the real life scenario under controlled quarantine disease strategy.In addition, the developed delayed SIR dynamic model is also applied to SIR influenza epidemic with temporary immunity to an influenza disease strain. The simulated results reveal an oscillatory effect in the trajectory of the naturally immune population. Moreover, the oscillations are more significant at the homesite. We further extended the stochastic temporary delayed epidemic dynamic model (3) into a stochastic delayed epidemic dynamic model with varying immunity period(4). The varying immunity period accounts for the varying time lengths of natural immunity against the infectious agent exhibited within the naturally immune population. Obviously, the stochastic dynamic model with varying immunity period generalizes the SIR temporary delayed dynamic.

[pt] O Objetivo deste trabalho é prover os fundamentos necessários ao desenvolvimento de uma metodologia voltada para a análise e desenho das estratégias, regras e regulamentos associados ao setor elétrico, sob o contexto da teoria dos jogos evolutivos. A importância da escolha de estratégias eficientes, que formem perfis de estratégias com melhores payoffs traz a necessidade de uma abordagem que leve em conta as interações entre os agentes, submersos às incertezas regulatórias, hidrológicas e mercadológicas, existentes no setor elétrico, que geram superfícies de payoff descontínuas e ruidosas. É demonstrado como tais superfícies descontínuas podem ser desmembradas em um hiperespaço de estratégias mistas, onde órbitas regidas por dinâmicas baseadas em equações diferenciais convergirão para os perfis de equilíbrios atratores estáveis no sentido assintótico. Para a modelagem é sugerida a utilização de estratégias comportamentais, que possuem a propriedade de gerar perfis em equilíbrio mais robustos às constantes mudanças, assim como propiciar a análise entre os ambientes cooperativos e competitivos. / [en] The objective of this thesis is to provide the crucial points to the development of a methodology focused on the analysis of strategies, rules and regulations connected with the electrical sector, under the context of the evolutionary game theory. The importance of choosing efficient strategies responsible for profiles, with better payoffs, displays the approach regarding the interactions among agents under regulatory, hydrological and market uncertainties, which are present in the electrical sector, resulting in noncontinuous and noisy payoffs surfaces. It is demonstrated that the already mentioned non-continuous surfaces can be expanded in a hyper-space of mixed strategies, where orbits governed by the dynamics based on differential equations, will converge to profiles of stable attractive equilibrium, in an asymptotic meaning. In order to achieve the modeling, is suggested the employment of behavioral strategies, which possess the role of creating equilibrium profiles, immune to the frequently changes, as well as to propitiate the analysis in cooperative and competitive scenarios.

Neste texto é feita uma discussão sobre alguns resultados que fornecem condições suficientes para que um difeomorfismo local, do espaço euclideano n-dimensional nele próprio, seja injetivo. Dentro deste cenário, são exploradas as contribuições destes resultados na tentativa de solucionar conhecidas conjecturas no meio científico como a Conjectura Jacobiana e a Conjectura de Ponto Fixo. Do ponto de vista dinâmico, existem relações entre injetividade global e estabilidade assintótica global. Neste sentido, os resultados também são contextualizados com respeito a importantes conjecturas de estabilidade assintótica: Conjectura de Markus-Yamabe e o Problema de LaSalle / We present some results which give suficient conditions for a local diffeomorphism from the n-dimensional Euclidean space into itself be globally injective. Within this context, we consider some partial results addressed to solve the well known Fixed Point Conjecture and Jacobian Conjecture. From the dynamical point of view, there are connections between global injectivity and global asymptotic stability. In this way, we present a solution of the Markus-Yamabe Conjecture and of the LaSalle Problem

Dizertační práce se zabývá procesy, které jsou řízeny systémy zpožděných diferenciálních rovnic $$x'(t) =f(t,x_t,u),\,\,\,\, t\ge t_{0}$$ kde $t_0 \in \mathbb{R}$, funkce $f$ je definována v jistém podprostoru množiny $[t_0,\infty)\times {C}_{\tau}^{m}\times {\mathbb{R}}^r$, $m,r \in \mathbb{N}$, ${C}_{\tau}^{m}=C([-\tau,0],{\mathbb{R}}^{m})$, $\tau>0$, $x_t(\theta):=x(t+\theta)$, $\theta\in[-\tau,0]$, $x\colon [t_0-\tau,\infty)\to \mathbb{R}^{m}$. Za předpokladu $f(t,\theta_m^*,\theta_r)=\theta_m$, kde ${\theta}_m^*\in {C}_{\tau}^{m}$ je nulová vektorová funkce, $\theta_r$ a $\theta_m$ jsou $r$ a $m$-dimenzionální nulové vektory, je říd\'cí funkce $u=u(t,x_t)$, $u\colon[t_0,\infty)\times _^\to \mathbb^$, $u(t,_m^*)=\theta_r$ určena tak, že nulové řešení $x(t)=\theta_m$, $t\ge t_-\tau$ systému je asymptoticky stabilní a pro libovolné řešení $x=x(t)$ integrál $$\int _{t_}^\omega \left(t,x_t,u(t,x_t)\right)\diff t,$$ kde $\omega$ je pozitivně definitní funkcionál, existuje a nabývá své minimální hodnoty v daném smyslu. Pro řešení tohoto problému byla Malkinova metoda pro obyčejné diferenciální systémy rozšířena na zpožděné funkcionální diferenciální rovnice a byla použita druhá metoda Lyapunova. Výsledky jsou ilustrovány příklady a aplikovány na některé třídy zpožděných lineárních diferenciálních rovnic.