Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space $X$ which is acted on by any continuous semigroup $\{S(t)\}_{t \geq 0}$. Suppose that $\S(t)\}_{t \geq 0}$ possesses a global attractor $\mathcal{A}$. We show that, for any generalized Banach limit $\underset{T \rightarrow \infty}{\rm{LIM}}$ and any distribution of initial conditions $\mathfrak{m}_0$, that there exists an invariant probability measure $\mathfrak{m}$, whose support is contained in $\mathcal{A}$, such that $$\int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t,$$ for all observables $\phi$ living in a suitable function space of continuous mappings on $X$. This work is based on a functional analytic framework simplifying and generalizing previous works in this direction. In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when $\{S(t)\}_{t \geq 0}$ does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and limits the phase space $X$ to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic

An analysis of solutions to fractional neutral differential equations with delay

This paper discusses some properties of solutions to fractional neutral delay differential equations. By combining a new weighted norm, the Banach fixed point theorem and an elegant technique for extending solutions, results on existence, uniqueness, and growth rate of global solutions under a mild Lipschitz continuous condition of the vector field are first established. Be means of the Laplace transform the solution of some delay fractional neutral differential equations are derived in terms of three-parameter Mittag–Leffler functions; their stability properties are hence studied by using use Rouché’s theorem to describe the position of poles of the characteristic polynomials and the final value theorem to detect the asymptotic behavior. By means of numerical simulations the theoretical findings on the asymptotic behavior are verified

Stability in distribution of stochastic functional differential equations

In this paper we investigate the stability in distribution for a class of stochastic functional differential equations (SFDEs), which include stochastic differential delay equations (SDDEs). Although stability in distribution has been studied by several authors recently, there is so far no stability-in-distribution criterion on SFDEs where the terms involved the delay components are highly nonlinear (not bounded by linear functions). In this paper we will establish the sufficient criteria on the stability in distribution for a class of highly nonlinear SFDEs. Two examples will be given to illustrate our new results. We also explain the reason why the existing stability-in-distribution criteria are not applicable by these two examples

Stability analysis of coupled ordinary differential systems with a string equation: application to a drilling mechanism

Cette thèse porte sur l'analyse de stabilité de couplage entre deux systèmes, l'un de dimension finie et l'autre infinie. Ce type de systèmes apparait en physique car il est intimement lié aux modèles de structures. L'analyse générique de tels systèmes est complexe à cause des natures très différentes de chacun des sous-systèmes. Ici, l'analyse est conduite en utilisant deux méthodologies. Tout d'abord, la séparation quadratique est utilisée pour traiter le côté fréquentiel de ce système couplé. L'autre méthode est basée sur la théorie de Lyapunov pour prouver la stabilité asymptotique de l'interconnexion. Tous ces résultats sont obtenus en utilisant la méthode de projection de l'état de dimension infinie sur une base polynomiale. Il est alors possible de prendre en compte le couplage entre les deux systèmes et ainsi d'obtenir des tests numériques fiables, rapides et peu conservatifs. De plus, une hiérarchie de conditions est établie dans le cas de Lyapunov. L'application au cas concret du forage pétrolier est proposée pour illustrer l'efficacité de la méthode et les nouvelles perspectives qu'elle offre. Par exemple, en utilisant la notion de stabilité pratique, nous avons montré qu'une tige de forage controlée à l'aide d'un PI est sujette à un cycle limite et qu'il est possible d'estimer son amplitude.This thesis is about the stability analysis of a coupled finite dimensional system and an infinite dimensional one. This kind of systems emerges in the physics since it is related to the modeling of structures for instance. The generic analysis of such systems is complex, mainly because of their different nature. Here, the analysis is conducted using different methodologies. First, the recent Quadratic Separation framework is used to deal with the frequency aspect of such systems. Then, a second result is derived using a Lyapunov-based argument. All the results are obtained considering the projections of the infinite dimensional state on a basis of polynomials. It is then possible to take into account the coupling between the two systems. That results in tractable and reliable numerical tests with a moderate conservatism. Moreover, a hierarchy on the stability conditions is shown in the Lyapunov case. The real application to a drilling mechanism is proposed to illustrate the efficiency of the method and it opens new perspectives. For instance, using the notion of practical stability, we show that a PI-controlled drillstring is subject to a limit cycle and that it is possible to estimate its amplitude

Análisis de sistemas dinámicos infinito-dimensionales asociados a ecuaciones en derivadas parciales funcionales.

Based on the theory of functional diferential equations, theory of semigroup, theory of random dynamical systems and theory of in nite dimensional dynamical systems, this thesis studies the long time behavior of several kinds of in nite dimensional dynamical systems associated to partial diferential equations containing some kinds of hereditary characteristics (such as variable delay, distributed delay or memory, etc), including existence and upper semicontinuity of pullback/random attractors and the stability analysis of stationary (steady-state) solutions. Three important mathematical-phyiscal models are considered, namely, reaction-di usion equation, 2D-Navier-Stokes equation as well as in-compressible non-Newtonian uids. Chapter 1 is devoted to the dynamics of an integer order stochastic reaction-difusion equation with thermal memory when the nonlinear term is subcritical or critical. Notice that our model contains not only memory but also white noise, which means it is not easy to prove the existence and uniqueness of solutions directly. In order to deal with this problem, we need introduce a new variable to transform our model into a system with two equations, and we use the Ornstein-Uhlenbeck to transfer this system into a deterministic ones only with random parameter. Then a semigroup method together with the Lax-Milgram theorem is applied to prove the existence, uniqueness and continuity of mild solutions. Next, the dynamics of solutions is analyzed by a priori estimates, and the existence of pullback random attractors is established. Besides, we prove that this pullback random attractors cannot explode, a property known as upper semicontinuity. But the dimension of the random attractor is still unknown. On the other hand, it has been proved that sometimes, especially when self-orgnization phenomena, anisotropic di usion, anomalous difusion occurs, a fractional order diferential equation can model this phenomena more precisely than a integer one. Hence, in Chapter 2, we focus on the asymptotical behavior of a fractional stochastic reaction-difusion equation with memory, which is also called fractional integro-diferential equation. First of all, the Ornstein-Uhlenbeck is applied to change the stochastic reaction-difusion equation into a deterministic ones, which makes it more convenient to solve. Then existence and uniqueness of mild solutions is proved by using the Lumer-Phillips theorem. Next, under appropriate assumptions on the memory kernel and on the magnitude of the nonlinearity, the existence of random attractor is achieved by obtaining some uniform estimates and solutions decomposition. Moreover, the random attractor is shown to have nite Hausdorf dimension, which means the asymptotic behavior of the system is determined by only a nite number of degrees of freedom, though the random attractor is a subset of an in nite-dimensional phase space. But we still wonder whether this random attractor has inertial manifolds, which means this random attractor needs to be exponentially attracting. Besides, the long time behavior of time-fractional reaction-difusion equation and fractional Brownian motion are still unknown. The rst two chapters consider an important partial function diferential equations with in nite distributed delay. However, partial functional diferential equations include more than only distributed delays; for instance, also variable delay terms can be considered. Therefore, in the next chapter, we consider another signi cant partial functional diferential equation but with variable delay. In Chapter 3, we discuss the stability of stationary solutions to 2D Navier-Stokes equations when the external force contains unbounded variable delay. Notice that the classic phase space C which is used to deal with diferential equations with in nite delay does not work well for our unbounded variable delay case. Instead, we choose the phase space of continuous bounded functions with limits at1. Then the existence and uni- queness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solutions is established by means of the Lax-Milgram theorem and the Schauder xed point theorem. Afterward, the local stability analysis of stationary solutions is carried out by three diferent approaches: the classical Lyapunov function method, the Razumikhin-Lyapunov technique and by constructing appropriate Lyapunov functionals. It worths mentioning that the classical Lyapunov function method requires diferentiability of delay term, which in some extent is restrictive. Fortunately, we could utilize Razumikhin-Lyapunov argument to weak this condition, and only requires continuity of every operators of this equation but allows more general delay. Neverheless, by these methods, the best result we can obtain is the asymptotical stability of stationary solutions by constructing a suitable Lyapunov functionals. Fortunately, we could obtain polynomial stability of the steady-state in a particular case of unbounded variable delay, namely, the proportional delay. However, the exponential stability of stationary solutions to Navier-Stokes equation with unbounded variable delay still seems an open problem. We can also wonder about the stability of stationary solutions to 2D Navier-Stokes equations with unbounded delay when it is perturbed by random noise. Therefore, in Chapter 4, a stochastic 2D Navier-Stokes equation with unbounded delay is analyzed in the phase space of continuous bounded functions with limits at1. Because of the perturbation of random noise, the classical Galerkin approximations alone is not enough to prove the existence and uniqueness of weak solutions. By combing a technical lemma and Faedo-Galerkin approach, the existence and uniqueness of weak solutions is obtained. Next, the local stability analysis of constant solutions (equilibria) is carried out by exploiting two methods. Namely, the Lyapunov function method and by constructing appropriate Lyapunov functionals. Although it is not possible, in general, to establish the exponential convergence of the stationary solutions, the polynomial convergence towards the stationary solutions, in a particular case of unbounded variable delay can be proved. We would like to point out that the Razumikhin argument cannot be applied to analyze directly the stability of stationary solutions to stochastic equations as we did to deterministic equations. Actually, we need more technical, and this will be our forthcoming paper. We also would like to mention that exponential stability of other special cases of in nite delay remains as an open problem for both the deterministic and stochastic cases. Especially, we are interested in the pantograph equation, which is a typical but simple unbounded variable delayed diferential equation.We believe that the study of pantograph equation can help us to improve our knowledge about 2D{Navier-Stokes equations with unbounded delay. Notice that Chapter 3 and Chapter 4 are both concerned with delayed Navier-Stokes equations, which is a very important Newtonian uids, and it is extensively applied in physics, chemistry, medicine, etc. However, there are also many important uids, such as blood, polymer solutions, and biological uids, etc, whose motion cannot be modeled pre- cisely by Newtonian uids but by non-Newtonian uids. Hence, in the next two chapters, we are interested in the long time behavior of an incompressible non-Newtonian uids ith delay. In Chapter 5, we study the dynamics of non-autonomous incompressible non-Newtonian uids with nite delay. The existence of global solution is showed by classical Galerkin approximations and the energy method. Actually, we also prove the uniqueness of solutions as well as the continuous dependence of solutions on the initial value. Then, the existence of pullback attractors for the non-autonomous dynamical system associated to this problem is established under a weaker condition in space C([h; 0];H2) rather than space C([h; 0];L2), and this improves the available results that worked on non-Newtonian uids. However, we still would like to analyze the Hausdor dimension or fractal dimension of the pullback attractor, as well as the existence of inertial manifolds and morsedecomposition. Finally, in Chapter 6, we consider the exponential stability of an incompressible non Newtonian uids with nite delay. The existence and uniqueness of stationary solutions are rst established, and this is not an obvious and straightforward work because of the nonlinearity and the complexity of the term N(u). The exponential stability of steady state solutions is then analyzed by means of four diferent approaches. The rst one is the classical Lyapunov function method, which requires the diferentiability of the delay term. But this may seem a very restrictive condition. Luckily, we could use a Razumikhin type argument to weaken this condition, but allow for more general types of delay. In fact, we could obtain a better stability result by this technique. Then, a method relying on the construction of Lyapunov functionals and another one using a Gronwall-like lemma are also exploited to study the stability, respectively. We would like to emphasize that by using a Gronwall-like lemma, only the measurability of delay term is demanded, but still ensure the exponential stability. Furthermore, we also would like to discuss the dynamics of stochastic non-Newtonian uids with both nite delay and in nite delay. All the problems deserve our attraction, and actually, these are our forthcoming work

Qualitative analysis of solutions to mixed-order positive linear coupled systems with bounded or unbounded delays

This paper addresses the qualitative theory of mixed-order positive linear coupled systems with bounded or unbounded delays. First, we introduce a general result on the existence and uniqueness of solutions to mixed-order linear coupled systems with time-varying delays. Next, we obtain the necessary and sufficient criteria which characterize the positivity of a mixed-order delay linear coupled system. Our main contribution is in Section 5. More precisely, by using a smoothness property of solutions to fractional differential equations and developing a new appropriated comparison principle for solutions to mixed-order delayed positive systems, we prove the attractivity of mixed-order non-homogeneous linear positive coupled systems with bounded or unbounded delays. We also establish a necessary and sufficient condition to characterize the stability of homogeneous systems. As a consequence of these results, we show the smallest asymptotic bound of solutions to mixed-order delay non-homogeneous linear positive coupled systems where disturbances are continuous and bounded. Finally, we provide numerical simulations to illustrate the proposed theoretical results

Proceedings of Seminar on Partial Differential Equations in Osaka 2012 : in honor of Professor Hiroki Tanabe’s 80th birthday

Osaka University, August 20‐24, 2012Edited by Atsushi Yagi and Yoshitaka Yamamot