Capítulos (Ecuaciones Diferenciales y Análisis Numérico) https://hdl.handle.net/11441/10835 2019-08-22T19:56:12Z A random model for immune response to virus in fluctuating environments https://hdl.handle.net/11441/58710 A random model for immune response to virus in fluctuating environments Sadovnichiy, Victor A.; Zgurovsky, Mikhail Z. In this work we study a model for virus dynamics with a random immune response and a random production rate of susceptible cells from cell proliferation. In traditional models for virus dynamics, the rate at which the viruses are cleared by the immune system is constant, and the rate at which susceptible cells are provided is constant or a function depending on the population of all cells. However, the human body in general is never stationary, and thus these rates can barely be constant. Here we assume that the human body is a random environment and models the rates by random processes, which result in a system of random differential equations. We then analyze the long term behavior of the random system, in particular the existence and geometric structure of the random attractor, by using the theory of random dynamical systems. Numerical simulations are provided to illustrate the theoretical result. 2016-01-01T00:00:00Z Some aspects concerning the dynamics of stochastic chemostats https://hdl.handle.net/11441/58688 Some aspects concerning the dynamics of stochastic chemostats Sadovnichiy, Victor A.; Zgurovsky, Mikhail Z. In this paper we study a simple chemostat model influenced by white noise which makes this kind of models more realistic. We use the theory of random attractors and, to that end, we first perform a change of variable using the OrnsteinUhlenbeck process, transforming our stochastic model into a system of differential equations with random coefficients. After proving that this random system possesses a unique solution for any initial value, we analyze the existence of random attractors. Finally we illustrate our results with some numerical simulations. 2016-01-01T00:00:00Z Control of weakly blowing up semilinear heat equations https://hdl.handle.net/11441/55140 Control of weakly blowing up semilinear heat equations Berestycki, Henri; Pomeau, Yves In these notes we consider a semilinear heat equation in a bounded domain of IRd , with control on a subdomain and homogeneous Dirichlet boundary conditions. We consider nonlinearities for which, in the absence of control, blow up arises. We prove that when the nonlinearity grows at infinity fast enough, due to the local (in space) nature of the blow up phenomena, the control may not avoid the blow up to occur for suitable initial data. This is done by means of localized energy estimates. However, we also show that when the nonlinearity is weak enough, and provided the system admits a globally defined solution (for some initial data and control), the choice of a suitable control guarantees the global existence of solutions and moreover that the solution may be driven in any finite time to the globally defined solution. In order for this to be true we require the nonlinearity f to satisfy at infinity the growth condition f(s) |s| log3/2 (1 + |s|) → 0 as |s| → ∞. This is done by means of a fixed point argument and a careful analysis of the control of linearized heat equations relying on global Carleman estimates. The problem of controlling the blow up in this sense remains open for nonlinearities growing at infinity like f(s) ∼ |s|logp (1 + |s|) with 3/2 ≤ p ≤ 2. 2002-01-01T00:00:00Z The control of PDEs: some basic concepts, recent results and open problems https://hdl.handle.net/11441/54055 The control of PDEs: some basic concepts, recent results and open problems Kaplický, Petr These Notes deal with the control of systems governed by some PDEs. I will mainly consider time-dependent problems. The aim is to present some fundamental results, some applications and some open problems related to the optimal control and the controllability properties of these systems. In Chapter 1, I will review part of the existing theory for the optimal control of partial differential systems. This is a very broad subject and there have been so many contributions in this field over the last years that we will have to limit considerably the scope. In fact, I will only analyze a few questions concerning some very particular PDEs. We shall focus on the Laplace, the stationary Navier-Stokes and the heat equations. Of course, the existing theory allows to handle much more complex situations. Chapter 2 is devoted to the controllability of some systems governed by linear time-dependent PDEs. I will consider the heat and the wave equations. I will try to explain which is the meaning of controllability and which kind of controllability properties can be expected to be satisfied by each of these PDEs. The main related results, together with the main ideas in their proofs, will be recalled. Finally, Chapter 3 is devoted to present some controllability results for other time-dependent, mainly nonlinear, parabolic systems of PDEs. First, we will revisit the heat equation and some extensions. Then, some controllability results will be presented for systems governed by stochastic PDEs. Finally, I will consider several nonlinear systems from fluid mechanics: Burgers, NavierStokes, Boussinesq, micropolar, etc. Along these Notes, a set of questions (some of them easy, some of them more intrincate or even difficult) will be stated. Also, several open problems will be mentioned. I hope that all this will help to understand the underlying basic concepts and results and to motivate research on the subject. 2012-01-01T00:00:00Z