Semilinear Hyperbolic Partial Differential Equations Research Articles

Observer Design for a Class of Semilinear Hyperbolic PDEs With Distributed Sensing and Parametric Uncertainties

We study a class of heterodirectional semilinear hyperbolic partial differential equations, where the distributed state vector is partially measured. An observer estimating the unmeasured part of the state vector is designed. The design is, for instance, applicable to multiphase 1-D fluid models, where the pressure is measured, but the distributed flow and phase concentrations are not. Furthermore, the observer is extended to systems with parametric uncertainties appearing in the dynamics of the unmeasured part of the state. While required to be linear in the uncertain parameter, the uncertain term may be nonlinear in the state, even in the unmeasured part of the state. Terms of this type appear often in applications and cover, for instance, viscous drag in fluid flow systems. A noteworthy property of the design is that convergence of the state estimate is achieved without requiring persistent excitation. Two example applications are presented, and the design is illustrated in a simulation.

Local Exponential Stabilization of Semi-Linear Hyperbolic Systems by Means of a Boundary Feedback Control

This letter investigates the boundary feedback control for a class of semi-linear hyperbolic partial differential equations with nonlinear relaxation, which is local Lipschitz continuous with a stable matrix structure. A sufficient condition in terms of linear inequalities is developed for the existence of global Cauchy solutions and the exponential stability by seeking a balance between the relaxation term and the boundary condition. These results are illustrated with an application to the boundary feedback control for a class of hyperbolic Lotka–Volterra models.

Lyapunov-based Boundary Control for a MIMO Counter-Propagating Raman Amplifier

This paper considers a boundary feedback control problem for a MIMO counter-propagating Raman amplifier. The system is modeled as a set of coupled semilinear hyperbolic partial differential equations with Lotka-Volterra type nonlinearity. The system is linearized about the steady-state solution, and a boundary controller is designed based on a Lyapunov functional whose time derivative is made strictly negative by an appropriate choice of boundary conditions. As a result, exponential convergence to the steady-state solution is shown in the L2-norm. The results are extended to the nonlinear system under a key assumption.

Order reduction and control of hyperbolic, countercurrent distributed parameter systems using method of characteristics

The aim of this paper is to develop a model reduction technique based on method of characteristics (MOC) for control of counter-current distributed parameter systems that are modeled by semi-linear hyperbolic partial differential equations (PDEs). In our previous work, MOC was shown to be a suitable model reduction technique for a class of hyperbolic PDEs. This concept is extended to counter-current systems, wherein the so-called characteristic lines have slopes with opposite signs. Two different approximations are proposed that allow the use of MOC as a model reduction technique. The open-loop results from MOC are compared with a large dimensional model, based on the method of lines. The MOC-based models are used for closed loop simulations within the approximate dynamic programming (ADP) framework. Two case studies are considered: a non-adiabatic plug flow reactor (having characteristics with two different slopes) and a non-adiabatic fixed bed reactor (having characteristics with three different slopes). We demonstrate that using the MOC-based model in an ADP controller results in a significant improvement in computational time, along with a slight improvement in controller performance.

Lymphome et taches café-au lait

The aim of this paper is to develop a model reduction technique based on method of characteristics (MOC) for control of counter-current distributed parameter systems that are modeled by semi-linear hyperbolic partial differential equations (PDEs). In our previous work, MOC was shown to be a suitable model reduction technique for a class of hyperbolic PDEs. This concept is extended to counter-current systems, wherein the so-called characteristic lines have slopes with opposite signs. Two different approximations are proposed that allow the use of MOC as a model reduction technique. The open-loop results from MOC are compared with a large dimensional model, based on the method of lines. The MOC-based models are used for closed loop simulations within the approximate dynamic programming (ADP) framework. Two case studies are considered: a non-adiabatic plug flow reactor (having characteristics with two different slopes) and a non-adiabatic fixed bed reactor (having characteristics with three different slopes). We demonstrate that using the MOC-based model in an ADP controller results in a significant improvement in computational time, along with a slight improvement in controller performance.

Optimality Conditions for Semilinear Hyperbolic Equations with Controls in Coefficients

An optimal control problem for semilinear hyperbolic partial differential equations is considered. The control variable appears in coefficients. Necessary conditions for optimal controls are established by method of two-scale convergence and homogenized spike variation. Results for problems with state constraints are also stated.

Trajectory planning for semilinear PDEs modeling a countercurrent heat exchanger

The trajectory planning and feedforward control design for a countercurrent heat exchanger modeled by a set of coupled semilinear hyperbolic partial differential equations is considered. The solution approach is based on a subdivision of the relevant time–space region along the characteristic curves of the system and a parameterization of the system state and input in terms of formal power series. The summability of the series on all subregions assures the applicability of the approach for a given desired output trajectory. Simulation results illustrate the benefits of the approach.

Mathematical modelling of fluidized bed combustion: 1. Combustion of carbon in bubbling beds

A three-phase model for carbon combustion in fluidized beds has been developed to take into account the effects of gas backmixing and solids circulation. Inclusion of these phenomena is expected ultimately to be important in extending the fluidized bed combustion model to predict N 2O emissions. Transient mass and energy balance equations are presented in this paper for the fast bubble regime; these equations can be easily modified for the slow bubble operating regime. The system of nine first-order semilinear hyperbolic partial differential equations has been solved numerically. The transient solution permits comparison of the predictive capability of the model with published data on combustion rates, burnout times and particle temperatures obtained from batch experiments. The comparisons reveal that the agreement with data on combustion rates and burnout times is in most cases within ∼30%. In addition, these comparisons show that improvement in the predictive capabilities of the model depends on ability to predict the product CO CO 2 ratio.

Finite-amplitude love-waves on an isotropic layered half-space

A pair of semi-linear hyperbolic partial differential equations governing the slow variations in amplitude and phase of a quasi-monochromatic finite-amplitude Love-wave on an isotropic layered half-space is derived using the method of multiple-scales. The analysis of the exact solution of these equations for a signalling problem reveals that the amplitude of the wave remains constant along its characteristic and that the phase of the wave increases linearly behind the wave-front.

Nonlinear mode coupling of surface acoustic waves on an isotropic solid

The nonlinear mode coupling between two co-directional quasi-harmonic Rayleigh surface waves on an isotropic solid is analysed using the method of multiple scales. This procedure yields a system of six semi-linear hyperbolic partial differential equations with the same principal part governing the slow variations in the (complex) amplitudes of the two fundamental, the two second harmonic and the two combination frequency waves at the second stage of the perturbation expansion. A numerical solution of these equations for excitation by monochromatic signals at two arbitrary frequencies, indicates that there is a continuous transfer of energy back and forth among the fundamental, second harmonic and combination frequency waves due to mode coupling. The mode coupling tends to be more pronounced as the frequencies of the interacting waves approach each other.

Normal forms for systems of semilinear hyperbolic partial differential equations

Normal forms for systems of semilinear hyperbolic partial differential equations

Mathematical Theory of Optimal Control for Semilinear Hyperbolic Systems in Two Independent Variables

Optimal solutions to mixed initial boundary value problems for control processes described by semilinear hyperbolic partial differential equations in two independent variables