First-order Linear Partial Differential Equations Research Articles

A New Perspective in Functional EIV Linear Models: Part II

In Error-in-Variables (EIV) models, all variables are subject to error and their estimators are heavily biased especially in the case of small sample sizes. This paper extends the results of Al-Sharadqah and proposes a new estimator for the problem of fitting 3D planes to data. Instead of minimizing an objective function obtained by the likelihood principles or others, a family of objective functions depending on an unknown smooth weight is considered. Then, the optimal weight function is derived so that the minimizer of the corresponding objective function has a zero second-order bias. To derive such a weight, a general form of the second-order bias is derived by applying our perturbation theory. This leads to a system of two first-order linear partial differential equations that admits a unique solution. The explicit formula of the weight is derived yielding the objective function. This turns to be a standard nonlinear least squares problem. Accordingly, the Levenberg-Marquardt (LM) algorithm is implemented. Finally, the effectiveness and superiority of our method are assessed by a series of numerical experiments.

Backstepping stabilization of a linearized ODE–PDE Rijke tube model

Abstract The problem of boundary stabilization of thermoacoustic oscillations in the Rijke tube is investigated using the backstepping method; as a first step, this work only considers the full-state design. This system consists of a vertical tube open at both ends and a heater placed in the lower half of the tube. To study this problem we consider that the mathematical model takes the form of 2 × 2 linear first-order hyperbolic partial differential equations (PDEs) with a point source term (induced by the Dirac delta distribution) on the right hand side, plus the coupling of an ordinary differential equation (ODE), and control input at one boundary condition. The presence of the Dirac delta distribution implies that the system solution has a discontinuity on a point of the domain, but is continuous everywhere else. We use a coordinate transformation to rewrite the equations into a system of four transport PDEs convecting in opposite directions and to translate the discontinuity to the boundary conditions. Then, a full state feedback backstepping controller is designed to exponentially stabilize the origin. However, the model is non-strict-feedback making unfeasible the use of standard backstepping designs. This issue is tackled by formulating a well-posed and invertible integral transformation with Volterra and Fredholm terms that maps the Rijke system into a target system with desirable stability properties. An exact piecewise-differentiable expression for the kernels of this transformation is found, allowing us in turn to derive an explicit feedback control law. Simulation results are presented to illustrate the effectiveness of the proposed control design.

Optimal control of univariate and multivariate population balance systems involving external fines removal

We propose an algorithm for optimal control of univariate and multivariate population balance systems governed by a class of first-order linear partial differential equation of hyperbolic type involving size dependent particle growth and filtering. In particular, we investigate the impact of filtering on the optimal control outcomes. To this end, we apply a recently developed polynomial method of moments and a standard steepest descent gradient-based optimization scheme to batch crystallization benchmark problems involving external fines removal. Numerical evaluations for various case-studies aiming at minimizing the mass of grown nuclei in the context of crystal shape manipulation demonstrate the effectiveness of fines dissolution. We also validate the accuracy of the proposed method of moments by utilizing a known numerical moving grid discretization method as a reference. Our results are of interest for the control of a wide range of population balance systems in various applications such as pharmaceutics, chemical engineering, particle shape engineering.

Determination of the basin of attraction of a periodic orbit in two dimensions
 using meshless collocation

A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of a periodic orbit without requiring information about its position or stability. Moreover, it is robust to small perturbations of the system. In two-dimensional systems, a contraction metric can be characterised by a scalar-valued function. In [9], the function was constructed as solution of a first-order linear Partial Differential Equation (PDE), and numerically constructed using meshless collocation. However, information about the periodic orbit was required, which needed to be approximated. In this paper, we overcome this requirement by studying a second-order PDE, which does not require any information about the periodic orbit. We show that the second-order PDE has a solution, which defines a contraction metric. We use meshless collocation to approximate the solution and prove error estimates. In particular, we show that the approximation itself is a contraction metric, if the collocation points are dense enough. The method is applied to two examples.

Distinguishing Lorenz and Chen Systems Based Upon Hamiltonian Energy Theory

Solving the linear first-order Partial Differential Equations (PDEs) derived from the unified Lorenz system, it is found that there is a unified Hamiltonian (energy function) for the Lorenz and Chen systems, and the unified energy function shows a hyperboloid of one sheet for the Lorenz system and an ellipsoidal surface for the Chen system in three-dimensional phase space, which can be used to explain that the Lorenz system is not equivalent to the Chen system. Using the unified energy function, we obtain two generalized Hamiltonian realizations of these two chaotic systems, respectively. Moreover, the energy function and generalized Hamiltonian realization of the Lü system and a four-dimensional hyperchaotic Lorenz-type system are also discussed.

Local uncontrollability for affine control systems with jumps

ABSTRACTThis paper investigates affine control systems with jumps for which the ideal If(g1, …, gm) generated by the drift vector field f in the Lie algebra L(f, g1, …, gm) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.

An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems

In this paper, we consider the problems of stability analysis and control synthesis for first-order hyperbolic linear Partial Differential Equations (PDEs) over a bounded interval with spatially varying coefficients. We propose Linear Matrix Inequalities (LMI) conditions for the stability and for the design of boundary and distributed control for the system. These conditions involve an infinite number of LMI to solve. Hence, we show how to overapproximate these constraints using polytopic embeddings to reduce the problem to a finite number of LMI. We show the effectiveness of the overapproximation with several examples and with the Saint-Venant equations with friction.

Solutions to an advanced functional partial differential equation of the pantograph type.

A model for cells structured by size undergoing growth and division leads to an initial boundary value problem that involves a first-order linear partial differential equation with a functional term. Here, size can be interpreted as DNA content or mass. It has been observed experimentally and shown analytically that solutions for arbitrary initial cell distributions are asymptotic as time goes to infinity to a certain solution called the steady size distribution. The full solution to the problem for arbitrary initial distributions, however, is elusive owing to the presence of the functional term and the paucity of solution techniques for such problems. In this paper, we derive a solution to the problem for arbitrary initial cell distributions. The method employed exploits the hyperbolic character of the underlying differential operator, and the advanced nature of the functional argument to reduce the problem to a sequence of simple Cauchy problems. The existence of solutions for arbitrary initial distributions is established along with uniqueness. The asymptotic relationship with the steady size distribution is established, and because the solution is known explicitly, higher-order terms in the asymptotics can be readily obtained.

On the summability of divergent power series solutions for certain first-order linear PDEs

This article is concerned with the study of the Borel summability of divergent power series solutions for certain singular first-order linear partial differential equations of nilpotent type. Our main purpose is to obtain conditions which coefficients of equations should satisfy in order to ensure the Borel summability of divergent solutions. We will see that there is a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients of equations.

Generalizing ODE modeling structure for multivariate systems with distributed parameters

We propose an approximate model in form of a system of ordinary differential equations (ODEs) for a class of first-order multivariate linear partial differential equations (PDEs) of the hyperbolic type. The resulting scheme utilizes the method of moments and least-square approximations over orthogonal polynomial bases for the factors of PDE depending on the spatial coordinate of the PDE. The class of examined PDEs appears typically in population balance systems with fines removal. The proposed modeling approach is generally of interest for control and optimization of multivariate systems with distributed parameters.

Approximate ODE models for population balance systems

We propose an approximate polynomial method of moments for a class of first-order linear PDEs (partial differential equations) of hyperbolic type, involving a filtering term with applications to population balance systems with fines removal terms. The resulting closed system of ODEs (ordinary differential equations) represents an extension to a recently published method of moments which utilizes least-square approximations of factors of the PDE over orthogonal polynomial bases. An extensive numerical analysis has been carried out for proof-of-concept purposes. The proposed modeling scheme is generally of interest for control and optimization of processes with distributed parameters.

Fuzzy Control Design for Nonlinear ODE-Hyperbolic PDE-Cascaded Systems: A Fuzzy and Entropy-Like Lyapunov Function Approach

This paper addresses the problem of fuzzy control design for a class of nonlinear distributed parameter systems represented by a cascaded model consisting of a Takagi-Sugeno (T-S) fuzzy ordinary differential equation and a linear first-order hyperbolic partial differential equation (PDE), where the control input affects the entire system through a boundary condition of the PDE. This characteristic makes the PDE subject to an inhomogeneous boundary condition. A state transformation is introduced to make the inhomogeneous boundary condition homogeneous, and a composite Lyapunov function that involves a fuzzy Lyapunov function and an entropy-like Lyapunov function is constructed for the transformed system. Based on this composite Lyapunov function, a sufficient condition for the closed-loop exponential stability of the cascaded system is presented in terms of a set of algebraic linear matrix inequalities in space. Using the sector bound approach and the finite spatial domain, a linear matrix inequality-based fuzzy control design procedure is developed from the obtained stability analysis result. Finally, simulation results on two numerical examples are provided to illustrate the effectiveness and merit of the proposed design method.

Method of moments over orthogonal polynomial bases

A method for the design of approximate models in the form of a system of ordinary differential equations (ODE) for a class of first-order linear partial differential equations of the hyperbolic type with applications to monovariate and multivariate population balance systems is proposed in this work. We develop a closed moment model by utilizing a least square approximation of spatial-dependent factors over an orthogonal polynomial basis. A bounded hollow shaped interval of convergence with respect to the order of the approximate ODE model arises as a consequence of the structural and finite precision computation numerical errors. The proposed modeling scheme is of interest in model-based control and optimization of processes with distributed parameters.

Active Disturbance Rejection Control for a 2×2 Hyperbolic System with an Input Disturbance

In this paper, active disturbance rejection control (ADRC) approach is used to stabilize a 2×2 system of first-order linear hyperbolic partial differential equations (PDEs) subject to a boundary input disturbance. Disturbance attenuation is achieved with the designed controller, and the resulting closed-loop control system admits a unique solution, which could tend to any arbitrary vicinity of zero.

First-order linear partial differential equations in the class of separately differentiable functions

It is obtained a general solution of first-order linear partial differential equations in the class of separately differentiable functions.

The Goursat problem for the Einstein–Vlasov system: (II) The evolution of initial data

We solve, locally in time, the evolution problem associated with the Einstein–Vlasov (EV) system, the initial data being specified on two intersecting smooth null hypersurfaces. The proof of the obtained result relies heavily on a fixed point method deployed in appropriate weighted Sobolev spaces. The main tools of this method consist of adequate Sobolev inequalities and Moser estimates combined with energy inequalities for first-order and second-order linear hyperbolic partial differential equations.

An efficient method for solving nonlocal initial-boundary value problems for linear and nonlinear first-order hyperbolic partial differential equations

In this paper, we present a new approach to solve nonlocal initial-boundary value problems of linear and nonlinear hyperbolic partial differential equations of first-order subject to initial and nonlocal boundary conditions of integral type. We first transform the given nonlocal initial-boundary value problems into local initial-boundary value problems. Then we apply a modified Adomian decomposition method, which permits convenient resolution of these problems. Moreover, we prove this decomposition scheme applied to such nonlocal problems is convergent in a suitable Hilbert space, and then extend our discussion to include systems of first-order linear equations and other related nonlocal initial-boundary value problems.

Identification problems in Banach spaces for linear first-order partial differential equations in one space dimension and applications

Abstract. We prove existence and uniqueness results for mild solutions to direct and inverse Cauchy problems related to linear first-order partial differential equations in Banach spaces. Some applications are given to ultraparabolic differential and integrodifferential problems.

Stochastically exponential stability and stabilization of uncertain linear hyperbolic PDE systems with Markov jumping parameters

This paper is concerned with the problem of robustly stochastically exponential stability and stabilization for a class of distributed parameter systems described by uncertain linear first-order hyperbolic partial differential equations (FOHPDEs) with Markov jumping parameters, for which the manipulated input is distributed in space. Based on an integral-type stochastic Lyapunov functional (ISLF), the sufficient condition of robustly stochastically exponential stability with a given decay rate is first derived in terms of spatial differential linear matrix inequalities (SDLMIs). Then, an SDLMI approach to the design of robust stabilizing controllers via state feedback is developed from the resulting stability condition. Furthermore, using the finite difference method and the standard linear matrix inequality (LMI) optimization techniques, recursive LMI algorithms for solving the SDLMIs in the analysis and synthesis are provided. Finally, a simulation example is given to demonstrate the effectiveness of the developed design method.

Numerical determination of the basin of attraction for asymptotically autonomous dynamical systems

We develop a method to numerically analyse asymptotically autonomous systems of the form x ̇ = f ( t , x ) , where f ( t , x ) tends to g ( x ) as t → ∞ . The rate of convergence is not limited to exponential, but may be polynomial, logarithmic or any other rate. For these systems, we propose a transformation of the infinite time interval to a finite, compact one, which reflects the rate of convergence of f to g . In the transformed system, the origin is an asymptotically stable equilibrium, which is exponentially stable in x -direction. We consider a Lyapunov function in this transformed system as a solution of a suitable linear first-order partial differential equation and approximate it using Radial Basis Functions.