After giving some motivation for studying the controllability properties of Schroedinger equations, we will briefly review some of the known results (focusing on exact control). We will then discuss some of the obstacles to controllability and techniques for showing non-controllability. (This latter part is based on work in progress and somewhat speculative.)
Hysteresis is typically displayed by systems that have multiple stable equilibrium points and dynamics that are faster than the rate at which inputs are varied. One such model is the Landau-Lifshitz equation, a nonlinear PDE which describes the behaviour of magnetization inside a magnetic object. It is known that the Landau-Lifshitz equation has an infinite number of stable equilibrium points. To control the hysteresis arising in this magnetization model requires finding a control that moves the system from one equilibrium to another. The initial equilibrium is no longer an equilibrium of the controlled system and the second point is an asymptotically stable equilibrium point of the controlled system. Such a control for the linear Landau-Lifshitz equation will be presented. In addition, the controlled system will be shown to be well-posed.
The mathematical models of many industrial transport-reaction processes are obtained from conservation laws, such as mass, momentum and/or energy in the form of partial differential equations (PDEs). In some applications, such processes involve the change in the shape of the material and domain of analysis as a result of phase change, chemical reaction, external forces, and mass transfer. It is well established that, the dominant behavior of dissipative PDE systems can be approximately described by finite-dimensional systems. The reduced-order model of a parabolic PDE system can be obtained by Galerkin’s method with the use of the eigenfunctions of the spatial differential operator. However, the analytic solution to the operator eigenvalue problem does not exist in general, examples are nonlinear operators or PDEs with nontrivial geometry. A well-known methodology in the extraction of eigenfunctions of these types of problems is the use of Karhunen-Loeve (KL) decomposition on an ensemble of solutions obtained from numerical or experimental resolution of the system. These modes, known as empirical eigenfunctions, are used in the derivation of accurate reduced-order approximations of many diffusion-reaction systems and fluid flows.
There are few studies on the order-reduction and control of PDE systems with spatially time-varying domain compared to the extensive research efforts on the order-reduction of distributed parameter systems with fixed domains. Assuming that the evolution of domain is known (which can be measured in many processes) KL decomposition cannot be directly applied to the solutions of PDEs with time-varying domain. Armaou and Christofides used a mathematical transformation to represent the nonlinear PDE on an appro- priate time-invariant domain and applied KL decomposition to obtain the set of eigenfunctions on the fixed domain [1,2]. In the study of the internal combustion engine flows by Fogleman et al., the velocity fields are stretched in one dimension to obtain data on a fixed grid such that the divergence of the original velocity field (continuity) is preserved . Following these contributions, one way to deal with the aforementioned problem is to map the set of the solutions on a time-invariant domain and then apply KL decomposition, however, different mappings could change the energy content of the solutions. The idea here is to map the solutions of the PDE system on a fixed reference geometry while preserving the invariance of physical properties (energy) of the solutions.
In this work, we find the control law to stabilize the heat-equation on a two-dimensional spatially time-varying domain. The set of the solutions of the PDE describing the system behavior is mapped to a fixed reference configuration while preserving the invariant property of thermal energy. A function basis can be found by using the KL decomposition on the mapped solutions, and by applying the inverse mapping, a set of time-varying empirical eigenfunctions are obtained that capture the most energy of the system. Subsequently, the empirical eigenfunctions are used as a basis for Galerkin’s method to derive the reduced-order ODE model that accurately captures the dominant dynamics of the PDE system. The reduced-order model is in the form of a linear time-varying system and the LQR control synthesis is considered.
. A. Armaou, P. D. Christofides, Nonlinear Feedback Control of Parabolic Partial Differential Equation Systems with Time-dependent Spatial Domains, J. Math. Anal. Appl. 239 (1), 1999, 124-157.
. A. Armaou, P. D. Christofides, Finite-Dimensional Control of Nonlinear Parabolic PDE Systems With Time-Dependent Spatial Domains Using Empirical Eigenfunctions, Int. J. Appl. Math. Comput. Sci. 11(2), 2001, 287-317.
. M. Fogleman, J. Lumley, D. Rempfer, D. Haworth, Application of the Proper Orthogonal Decomposition to Datasets of Internal Combustion Engine Flows, J. Turbul. 5, 2004, 023.
Cardiac alternans are defined as the beat-to-beat oscillations of the action potential duration (APD) in paced cardiac cells and have been linked to the onset of ventricular arrythmias and even sudden cardiac death. The annihilation of these alternans is therefore a promising antiarrhythmic strategy that requires more exploration within a realm of cardiac implantable devices. In this work, a model predictive control techniques are implemented on the small amplitude of alternans parabolic partial differential equation (PDE) used to describe the alternans in a cable of cardiac cells. In our proposed control strategy, both boundary and spatially distributed actuators are applied in order to suppress the alternans along the length of the cable. We explore the optimal control strategy in which low dimensional optimal controller can successfully annihilate cardiac alternans. However, optimal controllers might violate naturally present physiological constraints and in order to address the issue of constraints present in the system, we explore the model predictive control framework that explicitly accounts for constraints. We also demonstrate an important issue of input constraints satisfaction arising from the actuator limitations (pacing limitations) and the state constraints satisfaction which are naturally present in cardiac systems in the resulting closed loop system.
Hydrogen fuel cells (HFCs) are devices used to generate electricity from the electrochemical reaction between air and the fuel (hydrogen gas). An attractive advantage of these devices is that their by-product is water vapour, which is very safe to the environment. However, hydrogen fuel cells still lack some improvements in terms of increasing their life time and electricity production, and optimizing their operating conditions.
In this talk, we consider the cathode part of the fuel cell. Our Objective is to improve the HFC’s efficiency through optimizing the classical geometry of the cathode air channel.
To tackle this problem, we first introduce and validate a mathematical model describing the physics taking place in the cathode part. The cathode involves the air channel, gas diffusion layer (GDL, a porous medium) and catalyst layer (modeled as an interface). The mathematical model is based on conservation laws of mass, momentum and electrical charges. We assume the system to be a single gas-phase, isothermal, and at steady state.
Next, we introduce a shape optimization problem, which is defined as minimization of a cost functional subjected to the state equations. The cost functional concerns about the following three efficiency objectives:
· maximizing the total the current density over the catalyst layer
· minimizing the total variance of the current density over the catalyst layer
· minimizing the pressure drop used to deliver the air in the air channel
We present optimal shape designs for the cathode air channel to meet the individual and mixed objectives, and discuss the numerical methods as well as the existence and uniqueness of the solution.