One-dimensional Parabolic Systems Research Articles

Boundary stabilisation of an unstable parabolic PDE with a time-varying domain and the external disturbance

This paper considers the stability of the one-dimensional parabolic system, where one end changes over time and the other is the control end with the external disturbance. Firstly, by the boundary immobilisation method, the displacement change of the system boundary is transferred into the equation so that the original system is transformed into a system with fixed boundaries. Secondly, by combining the backstepping transformation and the sliding mode control method, the feedback control is proposed to compensate the instability of the system itself and reject the matched disturbance. Then, the resulting closed-loop system will be in the form of , where generates a semigroup, and are bounded and unbounded operators respectively, and is the external input. The existence of the generalised solution to the closed-loop system is proved by using the eigenfunction expansion of the system solution. By the Lyapunov method, the closed-loop system is shown to be exponentially stable. Finally, some numerical simulations are presented to illustrate the effectiveness of the proposed controller.

Boundary null-controllability of 1-D coupled parabolic systems with Kirchhoff-type conditions

The main concern of this article is to investigate the boundary controllability of some $$2\times 2$$ one-dimensional parabolic systems with both the interior and boundary couplings: The interior coupling is chosen to be linear with constant coefficient while the boundary one is considered by means of some Kirchhoff-type condition at one end of the domain. We consider here the Dirichlet boundary control acting only on one of the two state components at the other end of the domain. In particular, we show that the controllability properties change depending on which component of the system the control is being applied. Regarding this, we point out that the choices of the interior coupling coefficient and the Kirchhoff parameter play a crucial role to deduce the positive or negative controllability results. Further to this, we pursue a numerical study based on the well-known penalized HUM approach. We make some discretization for a general interior-boundary coupled parabolic system, mainly to incorporate the effects of the boundary couplings into the discrete setting. This allows us to illustrate our theoretical results as well as to experiment some more examples which fit under the general framework, for instance a similar system with a Neumann boundary control on either one of the two components.

Smoothness in the Dini Space of a Single Layer Potential for a Parabolic System in the Plane

We prove estimates in the Dini space for the second order spatial derivative of a vector parabolic single layer potential. We obtain estimates for higher order derivatives of solutions to initial-boundary-value problems for one-dimensional parabolic systems with Dini-continuous coefficients in a curvilinear domain with nonsmooth lateral boundary.

Predictors for linear parabolic systems with input delay

In this paper, we study the problem of stabilizing one-dimensional parabolic systems with input delay. Especially, we consider the case of “distributed control”. The purpose is to derive the stabilizing controller of predictor type in an abstract space, by using a backstepping method combined with the semigroup theory. The use of the semigroup theory makes the proof of continuity of the inverse transformation easy. Also, it is shown that the abstract controller can be actually implemented by using a finite number of eigenvalues and eigenfunctions of the system operator. Finally, a numerical simulation result is presented to demonstrate our design method.

Feedback stabilization of one-dimensional parabolic systems related to formations

Abstract This paper is concerned with the problem of stabilizing one-dimensional parabolic systems related to formations by using finitedimensional controllers of a modal type. The parabolic system is described by a Sturm-Liouville operator, and the boundary condition is different from any of Dirichlet type, Neumann type, and Robin type, since it contains the time derivative of boundary values. In this paper, it is shown that the system is formulated as an evolution equation with unbounded output operator in a Hilbert space, and further that it is stabilized by using an RMF (residual mode filter)-based controller which is of finite-dimension. A numerical simulation result is also given to demonstrate the validity of the finite-dimensional controller

Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences

Let (A,D(A)) be a diagonalizable generator of a C0-semigroup of contractions on a complex Hilbert space X, B∈L(C,Y), Y being some suitable extrapolation space of X, and u∈L2(0,T;C). Under some assumptions on the sequence of eigenvalues Λ={λk}k≥1⊂C of (A,D(A)), we prove the existence of a minimal time T0∈[0,∞] depending on Bernstein's condensation index of Λ and on B such that y′=Ay+Bu is null-controllable at any time T>T0 and not null-controllable for T<T0. As a consequence, we solve controllability problems of various systems of coupled parabolic equations. In particular, new results on the boundary controllability of one-dimensional parabolic systems are derived. These seem to be difficult to achieve using other classical tools.

Simultaneous and non-simultaneous quenching for coupled parabolic system

In this paper, we study the quenching phenomena for a one-dimensional parabolic system coupled by non-linear boundary flux. First, we identify simultaneous and non-simultaneous quenching, then establish four kinds of simultaneous quenching rates, which are represented via the characteristic algebraic system. A precise classification of parameters is given for the multiple simultaneous quenching rates. For example, we distinguish between two kinds of subcases, with or without particular requirements on initial data, realizing a common simultaneous quenching rate. On the other hand, it is interesting to find that there are two simultaneous quenching rates which share a common subregion of parameters, determined by the initial data assumed.

NEW L1-GRADIENT TYPE ESTIMATES OF SOLUTIONS TO ONE-DIMENSIONAL QUASILINEAR PARABOLIC SYSTEMS

We consider a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of the degenerate type. We derive some new L1-gradient type estimates for its solutions which are uniform in the sense that they do not depend on the coefficients nor on the size of the spatial domain. We also give some applications of such estimates to gas dynamics, filtration problems, a p-Laplacian parabolic type equation and some first order systems of Hamilton–Jacobi or conservation laws type.

On gradient estimates and other qualitative properties of solutions of nonlinear non autonomous parabolic systems

We prove several uniform $$ L^1 $$ -estimates on solutions of a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of degenerate type. They are uniform in the sense that they don’t depend on the coefficients, nor on the size of the spatial domain. The estimates concern the own solution or/and its spatial gradient. This paper extends some previous results by the authors to the case of nonautonomous coefficients and possibly non homogeneous boundary conditions. Moreover, an application to the asymptotic decay of the $$ L^1 $$ -norm of solutions, as t → +∞, is also given.

Asymptotic analysis of a parabolic system arising in combustion theory

We consider a one-dimensional parabolic system which describes the propagation of a flame with Lewis number close to one. We first prove estimates on the solutions that are independent of the activation energy. Next, we investigate the high activation energy asymptotics and derive the limiting free boundary problem. Our results rigorously justify the models currently used by physicists.

An L1 bound for |ux (t,x) | in a one-dimensional parabolic conservation system

This paper reports an L1 bound for |uxt,x | in a one dimensional parabolic system of conservation laws that is used elsewhere 3,4 when establishing global existence and optimal decay estimates by means of entropy flux pairs for the system where u = u1,...,un T and f u = f1u,...,fnu T are n ×1 column vectors, with f u a smooth function defined in a ball of radius r centered at a fixed vector ūε Rn, and ε >0 is a constant.

Comparison of adaptive methods for one-dimensional parabolic systems

We describe two adaptive space-time hp-refinement algorithms for solving one-dimensional vector systems of parabolic partial differential equations. Solutions for both methods are calculated using Galerkin's method with a piecewise polynomial hierarchical basis in space. The first method, hpsirk, uses singly-implicit Runge-Kutta methods in time and integrates the spatial and temporal discretization strategies. The second, hpdassl, is a method-of-lines approach using a BDF code. The two methods are compared along with the algorithm epdcol as a benchmark on four nonlinear test problems.

A posteriori error estimation with finite element methods of lines for one-dimensional parabolic systems

Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewiseP th -degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degreep+1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.

Nonlocal Nonlinear Problems for One-Dimensional Parabolic System

In the paper two nonlocal, nonlinear problems for a system of parabolic equations are considered: to find a solution of the system $$ \vec u\_t (x,t) = D\vec u\_{xx}(x, t) + \vec f (x, t, \vec u (x, t)) $$ subject to the conditions $$ \vec u (0, t) = \vec \varphi (t), \quad t \in (0, T), $$ $$ \vec u (x, 0) = \vec \psi (x), \quad x \in (0, 1), $$ $$ \vec u (1, t) - \vec u (x\_0, t) = \vec h (x\_0, t, \vec u (x\_0, t)) $$ or $$ \int ^1\_0 \vec u (x, t) dx = \vec g (t). $$ For this an operator $L: C(\bar \Omega) \to C(\bar \Omega)$ being a sum of four potentials is constructed. It is shown that the operator $L$ has only one fixed point. Moreover it is proved that the fixed point is the only solution of the considered problem.