Boundary control of a hyperbolic system of heat equations

We consider a boundary value problem modeling heat propagation in a rod in the framework of a hyperbolic thermal conduction law. We compute the temperature mode (control) at one end of the rod, the temperature at the other end being given, so as to ensure the desired rod temperature at a given time.     

Modeling and analysis of a contact problem for a viscoelastic rod

We consider a nonlinear viscoelastic rod which is in contact with a foundation along its length and is in contact with an obstacle at its end. The rod is acted up by body forces and, as a result, its mechanical state evolves. Our aim in this paper is twofold. The first one is to construct an appropriate mathematical model which describes the evolution of the rod. The second one is to prove the weak solvability of the problem. To this end, we use arguments on second-order inclusions with multivalued pseudomonotone operators.     

Optimization of the boundary displacement control of vibrations of a rod consisting of two dissimilar parts

We optimize the boundary displacement control that is applied at one end of a rod consisting of two dissimilar parts and brings the rod vibrations from a given initial state to a given terminal state for the case in which the other end of the rod is fixed.     

Optimal control of the end-temperature in a semi-infinite rod

The temperature distribution in a semi-infinite rod is controlled by heat input at one end. The objective is to keep the temperature at this end of the rod close to a given value, without excessive heat input. This distributed parameter optimal control problem is reformulated as a calculus of variations problem for the optimal end-temperature. An explicit solution is derived, and its general properties are examined. Two example cases are provided to illustrate the results.

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Zusammenfassung Die Wärmeverteilung in einem einseitig unendlichen Stab wird durch Wärmezufuhr an einem Ende gesteuert. Man versucht, die Temperatur an diesem Ende ohne übermässige Wärmezufuhr möglichst nahe bei einem vorgegebenen Wert zu halten. Das optimale Steuerungsproblem des Systems, dessen mathematisches Modell durch eine partielle Differentialgleichung gegeben ist, wird in ein Variationsrechnungsproblem für die optimale Endtemperatur transformiert. Eine explizite Lösung wird hergeleitet, ihre allgemeinen Eigenschaften werden untersucht und an zwei Beispielen illustriert.

     

Optimal boundary control of heat transfer in a one-dimensional material: A hyperbolic model

We consider a boundary value problem describing heat propagation in a rod in the framework of a hyperbolic model of heat transfer. We construct a class, depending on a function parameter, of boundary data (controls) ensuring a given rod temperature distribution at a given time; by using the Lagrange method, from this class, we single out a unique control minimizing a given loss function.     

Boundary control by an elastic force at one end of an inhomogeneous rod with the other end free for the case in which the wave travels either of the homogeneous parts in the same time

We study the boundary control by an elastic force at one end of an inhomogeneous rod that has two parts of different densities and elasticities and whose other end is free. The case in which the wave travels either of the homogeneous parts in the same time is considered. We present a closed-form analytical expression for the boundary control by an elastic force that brings the rod from the initial quiescent state to a given terminal state specified by given terminal displacement and terminal velocity.     

Determining conductivity by boundary measurements

In [2], A. P. Caldéron posed the following question: can one determine the heat conductivity of an object from static temperature and heat flux measurements at the boundary? We show that such measurements uniquely determine the conductivity and all of its derivatives at the boundary.     

Existence of Solutions for Impulsive Parabolic Partial Differential Equations

We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Green's function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions.     

Continuous dependence results for a class of nonlinear heat equations in a cylindrical region

In this paper, we derive bounds for the solutions of a quasilinear heat equation in a finite cylindrical region if the far end and the lateral surface are held at zero temperature, and a nonzero temperature is applied at the near end. Some continuous dependence inequalities are also obtained. We also investigate the case in which a given heat flux is prescribed at the near end, instead of a given temperature. Copyright © 2008 John Wiley & Sons, Ltd.     

Dynamic equations of thermoelastic Cosserat rods

A thermoelastic Cosserat rod with a heat flux along its length is modeled after reviewing a simple Cosserat rod model. Extended Kirchhoff constitutive relations that include thermal effects, and the associated heat conduction equation, are derived using the first law of thermodynamics. The rate of internal dissipation of the Cosserat rod is estimated by the Clausius–Duhem inequality. Nonlinear dynamic equations of the thermoelastic Cosserat rod, which extend the simple Cosserat rod model, are obtained. Dynamic equations of a planar thermoelastic Cosserat rod, the Timoshenko thermoelastic beam, and the planar Euler–Bernoulli thermoelastic beam are derived as a special case within the framework of the thermoelastic Cosserat rod.     

A procedure for determining a spacewise dependent heat source and the initial temperature

The inverse problem of determining a spacewise dependent heat source, together with the initial temperature for the parabolic heat equation, using the usual conditions of the direct problem and information from two supplementary temperature measurements at different instants of time is studied. These spacewise dependent temperature measurements ensure that this inverse problem has a unique solution, despite the solution being unstable, hence the problem is ill-posed. We propose an iterative algorithm for the stable reconstruction of both the initial data and the source based on a sequence of well-posed direct problems for the parabolic heat equation, which are solved at each iteration step using the boundary element method. The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for a typical benchmark test example, which has the input measured data perturbed by increasing amounts of random noise. The numerical results show that the proposed procedure gives accurate numerical approximations in relatively few iterations.     

Estimation of terrestrial heat flow from temperature measurements in bottom sediments

Under study is the problem of estimation of the terrestrial heat flow from the temperature measurements in the bottom sediments. The problem is divided into the two subproblems: first, we solve the one-dimensional inverse problem of estimating the heat conductivity λ and, second, compute the heat flow value by solving the direct stationary problem using the just-found value of λ. We develop a sweep method for solving the direct problem which differs from the standard. An optimization approach is used for solving the inverse problem, and the explicit formulas are obtained for computing the gradient of the error functional. We analyze the factors that cause errors in estimating the heat flow. We show that the main contribution to the errors is given by the presence of harmonics with the periods exceeding the temperature monitoring time interval. We show that if the parameters of the harmonics are known then we can calculate some corrections for the obtained value of the heat flow. The results were applied to the data of temperature measurements carried out at the bottom of Lake Teletskoye from June of 2008 to September of 2010. For finding the long-period harmonics, we use the meteorological data about the bottom water temperature from 1968 to 2011. This allowed us to estimate the heat flow through the bottom of Lake Teletskoye as well as the thermal diffusivity in the upper layer of the sediments.     

On the null-controllability of the heat equation in unbounded domains

We make two remarks about the null-controllability of the heat equation with Dirichlet condition in unbounded domains. Firstly, we give a geometric necessary condition (for interior null-controllability in the Euclidean setting) which implies that one cannot go infinitely far away from the control region without tending to the boundary (if any), but also applies when the distance to the control region is bounded. The proof builds on heat kernel estimates. Secondly, we describe a class of null-controllable heat equations on unbounded product domains. Elementary examples include an infinite strip in the plane controlled from one boundary and an infinite rod controlled from an internal infinite rod. The proof combines earlier results on compact manifolds with a new lemma saying that the null-controllability of an abstract control system and its null-controllability cost are not changed by taking its tensor product with a system generated by a non-positive self-adjoint operator.     

On the complete damping of oscillations of an inhomogeneous rod by means of a boundary control at one end

The process of damping oscillations of a rod consisting of two dissimilar segments and fixed at one end is considered. The damping is carried out by means of a boundary control at the other end of the rod.     

Single-axis equilibrium orientations to the attracting center of a symmetrical prolate orbital gyrostat with an elastic rod

Under study in the restricted formulation is the motion of a symmetrical prolate stationary gyrostat along a Keplerian circular orbit in a central Newtonian field of forces. An elastic homogeneous rod, rectilinear in the undeformed state, is rigidly clamped by one end in the body of gyrostat along its axis of symmetry. There is a point mass at the free end of the rod. The inextensible elastic rod, for simplicity of constant circular cross-section, performs infinitesimal space oscillations in the process of system motion. In this case, we neglect the terms in the system’s tensor of inertia which are nonlinear with respect to displacements of the points of the rod.We consider the following (so-called semi-inverse) problem: Under what kinetic momentumof the flywheel, among the relative equilibriums of the system (the states of rest relative to the orbital coordinate system) does there exist an equilibrium such that the axis, arbitrarily chosen in the coordinate system associated with the gyrostat, is collinear with the local vertical? In the discretization of the problem, we present the values of the Lagrange coordinates that define the deformation of the rod for these equilibria and the value of gyrostatic moment providing the presence of the equilibrium in question.     

Biegeschwingungen verwundener,einseitig eingespannter und am andern Ende gelenkig gelagerter Stäbe

Summary The lateral and torsional vibrations of twisted rods can be treated separately if we consider as usual only first order terms. The eigen-frequencies of the lateral vibrations can be calculated exactly if we restrict ourselves to isotropic homogeneous rods with constant mass and twist per unit length and constant principal flexural rigidities. In this paper the eigen-frequencies for a rod built in at one end and supported at the other are given for 3 different cross-sections.     

Small penalty control of the end temperature in a long rod

The problem considered is that of maintaining the end temperature of a long rod near a prescribed level over a fixed time interval. Control is achieved via the heat flux at the near end, and it is optimal in the sense that it minimizes a given performance index of quadratic form. The performance index contains a penalty parameter associated with the magnitude of the control. Particular attention is given to the determination of the optimal control when the penalty parameter is small (i.e., cheap control). This gives rise to a singularly perturbed integral equation, which is solved asymptotically by a methodology which has recently been developed for a related class of problems.The work of the first author was supported by the Applied Mathematical Sciences Subprogram, Office of Energy Research, US Department of Energy under Contract W-7405-ENG-82. The work of the second author was supported by NSF under Grant DMS-87-00962.     

Problems of maximizing the compliance of curvilinear rods

A number of problems of finding the shape of a thin curvilinear rod (the support element of an artificial lens) of constant cross-section and specified length with its ends at specified points and under specified loading conditions with maximum compliance for characteristic types of end restraint and loading are considered. It is shown that the boundary-value problem arising for the non-linear Euler equation may have a set (possibly denumerable) of solutions, one of which gives the absolute maximum compliance, and the others the local maxima. The problem is analysed in detail, analytical solutions are obtained and the corresponding shapes are constructed in a number of important cases.     

Thermal regimes of high burn-up nuclear fuel rod

The temperature distribution in the nuclear fuel rods for high burn-up is studied. We use the numerical and analytical approaches. It is shown that the time taken to have the stationary thermal regime of nuclear fuel rod is less than one minute. We can make the inference that the behavior of the nuclear fuel rod can be considered as a stationary task. Exact solutions of the temperature distribution in the fuel rods in the stationary case are found. Thermal regimes of high burn-up the nuclear fuel rods are analyzed.     

The dual nature of the transverse vibrations of an elastic rod

The transverse vibrations of an elastic rod, to one of which displacements are applied while the other end is free, are investigated. It is assumed that the propagation velocity of the perturbations in the rod is finite. The unperturbed part performed rotational motion around the centre line. The angle of rotation is expressed by the angle of curvature of the centre line of the perturbed part of the rod. Two types of elastic vibrations are obtained: (1) the rod vibrates elastically due to displacements applied at the end, and (2) when performing rotational motion elastic vibrations and additional forces occur in the rod due to elasticity [1].