Predictive control of uncertain nonlinear parabolic PDE systems using a Galerkin/neural-network-based model

In this paper, a model predictive control (MPC) scheme for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities, arising in the context of transport-reaction processes, is proposed. A spatial operator of a parabolic PDE system is characterized by a spectrum that can be partitioned into a finite slow and an infinite fast complement. In this view, first, Galerkin method is used to derive a set of finite dimensional slow ordinary differential equation (ODE) system that captures the dominant dynamics of the initial PDE system. Then, a Multilayer Neural Network (MNN) is employed to parameterize the unknown nonlinearities in the resulting finite dimensional ODE model. Finally, a Galerkin/neural-network-based ODE model is used to predict future states in the MPC algorithm. The proposed controller is applied to stabilize an unstable steady-state of the temperature profile of a catalytic rod subject to input and state constraints.     

Algebra with Quadratic Commutation Relations for an Axially Perturbed Coulomb–Dirac Field

The motion of a particle in the field of an electromagnetic monopole (in the Coulomb–Dirac field) perturbed by an axially symmetric potential after quantum averaging is described by an integrable system. Its Hamiltonian can be written in terms of the generators of an algebra with quadratic commutation relations. We construct the irreducible representations of this algebra in terms of second-order differential operators; we also construct its hypergeometric coherent states. We use these states in the first-order approximation with respect to the perturbing field to obtain the integral representation of the eigenfunctions of the original problem in terms of solutions of the model Heun-type second-order ordinary differential equation and present the asymptotic approximation of the corresponding eigenvalues.     

Template-Based Stabilization of Relative Equilibria in Systems with Continuous Symmetry

We present an approach to the design of feedback control laws that stabilize relative equilibria of general nonlinear systems with continuous symmetry. Using a template-based method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fixed points of these reduced equations. Our controller design methodology is based on the linearization of the reduced equations about such fixed points. We present two different approaches of control design. The first approach assumes that the closed loop system is affine in the control and that the actuation is equivariant. We derive feedback laws for the reduced system that minimize a quadratic cost function. The second approach is more general; here the actuation need not be equivariant, but the actuators can be translated in the symmetry direction. The controller resulting from this approach leaves the dynamics associated with the symmetry variable unchanged. Both approaches are simple to implement, as they use standard tools available from linear control theory. We illustrate the approaches on three examples: a rotationally invariant planar ODE, an inverted pendulum on a cart, and the Kuramoto-Sivashinsky equation with periodic boundary conditions.     

Some Facts about the Ramsey Model

In modeling the dynamics of capital, the Ramsey equation coupled with the Cobb–Douglas production function is reduced to a linear differential equation by means of the Bernoulli substitution. This equation is used in the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. This result enriches our understanding of the model analysis in the optimal control framework.     

Null controllability of a system of viscoelasticity with a moving control

In this paper, we consider the wave equation with both viscous Kelvin–Voigt and frictional damping as a model of viscoelasticity in which we incorporate an internal control with a moving support. We prove the null controllability when the control region, driven by the flow of an ODE, covers all the domain. The proof is based upon the interpretation of the system as, roughly, the coupling of a heat equation with an ordinary differential equation (ODE). The presence of the ODE for which there is no propagation along the space variable makes the controllability of the system impossible when the control is confined into a subset in space that does not move. The null controllability of the system with a moving control is established in using the observability of the adjoint system and some Carleman estimates for a coupled system of a parabolic equation and an ODE with the same singular weight, adapted to the geometry of the moving support of the control. This extends to the multi-dimensional case the results by P. Martin et al. in the one-dimensional case, employing 1-d Fourier analysis techniques.     

Output feedback vibration control of a string driven by a nonlinear actuator

In this paper, we design an observer-based output feedback controller to exponentially stabilize a system of nonlinear ordinary differential equation-wave partial differential equation-ordinary differential equation. An observer is designed to estimate the full states of the system using available boundary values of the partial differential equation. The output feedback controller is built via the combination of the ordinary differential equation backstepping which is applied to deal with the nonlinear ordinary differential equation, and the partial differential equation backstepping which is used for the wave partial differential equation-ordinary differential equation. The controller can be applied into vibration suppression of a string-payload system driven by an actuator with nonlinear characteristics. The global exponential stability of all states in the closed-loop system is proved by Lyapunov analysis. The numerical simulation illustrates the states of the actuator, string, payload and the observer errors are fast convergent to zero under the proposed output feedback controller.     

Dynamic feedback control and exponential stabilization of a compound system

We studied the exponential stabilization problem of a compounded system composed of a flow equation and an Euler–Bernoulli beam, which is equivalent to a cantilever Euler–Bernoulli beam with a delay controller. We designed a dynamic feedback controller that stabilizes exponentially the system provided that the eigenvalues of the free system are not the zeros of controller. In this paper we described the design detail of the dynamic feedback controller and proved its stabilization property.     

Algebra with polynomial commutation relations for the Zeeman effect in the Coulomb-Dirac field

We consider a model of particle motion in the field of an electromagnetic monopole (in the Coulomb-Dirac field) perturbed by homogeneous and inhomogeneous electric fields. After quantum averaging, we obtain an integrable system whose Hamiltonian can be expressed in terms of the generators of an algebra with polynomial commutation relations. We construct the irreducible representations of this algebra and its hypergeometric coherent states. We use these states to represent the eigenfunctions of the original problem in terms of the solutions of the model ordinary differential equation. We also present the asymptotic approximations of the eigenvalues in the leading term of the perturbation theory, where the degeneration of the spectrum is removed completely.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 127–147, January, 2005.     

Mathematical analysis and stability of a chemotaxis model with logistic term

In this paper we study a non‐linear system of differential equations arising in chemotaxis. The system consists of a PDE that describes the evolution of a population and an ODE which models the concentration of a chemical substance. We study the number of steady states under suitable assumptions, the existence of one global solution to the evolution problem in terms of weak solutions and the stability of the steady states. Copyright © 2004 John Wiley & Sons, Ltd.     

Construction of ODE Curves

This paper deals with a construction problem of free-form curves from data constituted by some approximation points and a boundary value problem for an ordinary differential equation (ODE). The solution of this problem is called an ODE curve. We discretize the problem in a space of B-spline functions. Finally, we analyze a graphical example in order to illustrate the validity and effectiveness of our method.     

A Dynamic Model for a Hybrid Articulated Robot

The dynamic modeling of hybrid systems, consisting of flexible and rigid parts results in large partial differential equation systems (PDE). With the assumption of small deflections and the Ritz expansion the PDE can be approximated by an ordinary differential equation system (ODE) but the number of degrees of freedom is generally high. In this paper a hybrid articulated robot with 2 flexible links and 6 joints is under consideration. The joints are equipped with Harmonic Drive gears with high gear ratio but relative low stiffness. Therefore additionally degrees of freedom are introduced for the elastic deflection of the gears. The links are modeled with flexibility in two bending directions and in torsional sense. To be able to achieve structured equations the projection equation in subsystem representation is used. The projection equation is based on the momentum and the angular momentum equations of each single finite or infinitesimal body which are projected into the space of minimal coordinates and subsequently are summed up. Groups of bodies are collected to the so called subsystems with separated describing velocities. These subsystems are linked together with the kinematical chain. Because the robot is tree structured it is possible to obtain an explicit expression for the second derivatives of the minimal coordinates with a recursive scheme (O(n) efficiency). The robot is controlled with a feed forward controller and a linear PD joint controller. Simulation results and measured data are presented and compared. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modelling,simulation and optimization of an elastic structure under moving loads

We consider an elastic structure that is subject to moving loads representing e.g. heavy trucks on a bridge or a trolley on a crane beam. A model for the quasi-static mechanical behaviour of the structure is derived, yielding a coupled problem involving partial differential equations (PDE) and ordinary differential equations (ODE). The problem is simulated numerically and validated by comparison with a standard formula used in engineering. We derive an optimal policy for passing over potentially fragile bridges. In general, our problem class leads to optimal control problems subject to coupled ODE and PDE. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal control of an elastic crane-trolley-load system - a case study for optimal control of coupled ODE-PDE systems

We present a mathematical model of a crane-trolley-load model, where the crane beam is subject to the partial differential equation (PDE) of static linear elasticity and the motion of the load is described by the dynamics of a pendulum that is fixed to a trolley moving along the crane beam. The resulting problem serves as a case study for optimal control of fully coupled partial and ordinary differential equations (ODEs). This particular type of coupled systems arises from many applications involving mechanical multi-body systems. We motivate the coupled ODE-PDE model, show its analytical well-posedness locally in time and examine the corresponding optimal control problem numerically by means of a projected gradient method with Broyden-Fletcher-Goldfarb-Shanno (BFGS) update.     

Bernoulli substitution in the Ramsey model: Optimal trajectories under control constraints

We consider a neoclassical (economic) growth model. A nonlinear Ramsey equation, modeling capital dynamics, in the case of Cobb-Douglas production function is reduced to the linear differential equation via a Bernoulli substitution. This considerably facilitates the search for a solution to the optimal growth problem with logarithmic preferences. The study deals with solving the corresponding infinite horizon optimal control problem. We consider a vector field of the Hamiltonian system in the Pontryagin maximum principle, taking into account control constraints. We prove the existence of two alternative steady states, depending on the constraints. A proposed algorithm for constructing growth trajectories combines methods of open-loop control and closed-loop regulatory control. For some levels of constraints and initial conditions, a closed-form solution is obtained. We also demonstrate the impact of technological change on the economic equilibrium dynamics. Results are supported by computer calculations.     

Scattering of a neutral fermion with anomalous magnetic moment by a charged straight thin thread

We consider the scattering of a massive neutral fermion with an anomalous magnetic moment in the electric field of a homogeneously charged straight thin thread from the standpoint of the quantum mechanical problem of constructing a self-adjoint Hamiltonian for the nonrelativistic Dirac-Pauli equation. Using the solutions obtained for the self-adjoint Hamiltonian, we investigate the scattering of the neutral fermion in the electric field of a thread oriented perpendicular to the plane of fermion motion (the Aharonov-Casher effect). We find expressions for the scattering amplitude and cross section of neutral fermions in the electric field of the thread. We show that the scattering amplitude and cross section depend both on the direct interaction between the fermion anomalous magnetic moment and the electric field and on the polarization of the fermionic beam in the initial state.     

An ornstein-uhlenbeck model for random oscillations

We consider the effect of a random "noise" on an n-dimensional simple harmonic oscillator with time-dependent damping. The noise in the system is modelled by incorporating a Brownian motion term in the equation for the velocity process of the simple harmonic oscillator, giving a stochastic differential equation similar to that of an Ornstein-Uhlenbeck proces. Necessary and sufficient conditions for the convergence of the solution of this SDE to an orbit of simple harmonic motion (satisfying the usual ODE) are then obtained