Continuous Data Dependence Research Articles

Some characteristic properties of the solutions in the three-phase-lag heat conduction

In this paper we consider the three-phase-lag model of heat conduction that involves second-order effects in phase lag of the heat flux vector. This model leads to a fourth-order in time equation of Moore–Gibson–Thompson type. We use the thermodynamic restrictions derived from the compatibility of the constitutive equation with the Second Law of Thermodynamics to study the properties of the solutions of the initial boundary value problems associated with the model in concern. In this connection we establish a series of well-posedness results concerning the related solutions like: uniqueness, continuous data dependence, exponentially stability or domain of influence. Furthermore, based on the thermodynamic restrictions, we show that the thermal model in question admits damped in time propagating waves as well as exponentially decaying standing modes. We also show that when the thermodynamic restrictions are not fulfilled, then wave solutions appear that cause the energy blows up as time goes to infinity.

Contracting differential equations in weighted Banach spaces

Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite in the Riemannian metric. In the infinite-dimensional setting, however, such analysis is generally restricted to norm-contracting systems. We develop a generalization of geodesic contraction rates to Banach spaces using a smoothly-weighted semi-inner product structure on tangent spaces. We show that negative contraction rates in bijectively weighted spaces imply asymptotic norm-contraction, and apply recent results on asymptotic contractions in Banach spaces to establish the existence of fixed points. We show that contraction in surjectively weighted spaces verifies non-equilibrium asymptotic properties, such as convergence to finite- and infinite-dimensional subspaces, submanifolds, limit cycles, and phase-locking phenomena. We use contraction rates in weighted Sobolev spaces to establish existence and continuous data dependence in nonlinear PDEs, and pose a method for constructing weak solutions using vanishing one-sided Lipschitz approximations. We discuss applications to control and order reduction of PDEs.

Generalized Contractions and Fixed Point Results in Spaces with Altering Metrics

In this paper, we have provided some fixed point results for self-mappings fulfilling generalized contractive conditions on altered metric spaces. In addition, some applications of the main results to continuous data dependence of the fixed points of operators defined on these spaces were shown.

Some Fixed Point Results in Spaces with Perturbed Metrics

"n this paper, the concept of perturbed metric was introduced within the metric spaces and some fixed point results were established for self-mappings satisfying such contractive conditions, using Picard operators technique and generalized contractions. Moreover, some applications of the main result to continuous data dependence of the fixed points of Picard operators defined on these spaces were presented. Also, the main result of this paper was applied to study the existence and uniqueness of the solution for an integral equation which models an epidemiological problem"

A contact problem for a piezoelectric actuator on an elasto-plastic obstacle

A problem of motion of a piezoelectric actuator in contact with an elasto-plastic obstacle is reformulated as a PDE in one spatial dimension with hysteresis in the bulk and on the contact boundary. The model is shown to dissipate energy in agreement with the principles of thermodynamics. The main result includes existence, uniqueness, and continuous data dependence of solutions.

Stability results for the functional differential equations associated to water hammer in hydraulics

There is considered a system of two sets of partial differential equations describing the water hammer in a hydroelectric power plant containing the dynamics of the tunnel, turbine penstock, surge tank and hydraulic turbine. Under standard simplifying assumptions (negligible Darcy–Weisbach losses and dynamic head variations), a system of functional differential equations of neutral type, with two delays, can be associated to the aforementioned partial differential equations and existence, uniqueness and continuous data dependence can be established. Stability is then discussed using a Lyapunov functional deduced from the energy identity. The Lyapunov functional is "weak" i.e. its derivative function is only non-positive definite. Therefore only Lyapunov stability is obtained while for asymptotic stability application of the Barbashin–Krasovskii–LaSalle invariance principle is required. A necessary condition for its validity is the asymptotic stability of the difference operator associated to the neutral system. However, its properties in the given case make the asymptotic stability non-robust (fragile) in function of some arithmetic properties of the delay ratio.

Non-convex sweeping processes in the space of regulated functions

<p style='text-indent:20px;'>The aim of this paper is to study a wide class of non-convex sweeping processes with moving constraint whose translation and deformation are represented by regulated functions, i. e., functions of not necessarily bounded variation admitting one-sided limits at every point. Assuming that the time-dependent constraint is uniformly prox-regular and has uniformly non-empty interior, we prove existence and uniqueness of solutions, as well as continuous data dependence with respect to the sup-norm.</p>

On the quasi-static approximation in the initial boundary value problem of linearised elastodynamics

Continuous data dependence estimates are employed to rigorously derive conditions that validate the quasi-static approximation for the initial homogeneous boundary value problem in the theory of small elastic deformations superposed upon large elastic deformations. This theory imposes no sign-definite assumptions on the linearised elastic moduli and in consequence the requisite estimates are established using methods principally motivated by known Lagrange identity arguments.

Modeling triple porosity under local thermal non-equilibrium

In this paper, we present a model describing the evolutionary behavior of materials with triple porosity under local thermal non-equilibrium. The constitutive equations are presented for anisotropic and inhomogeneous materials and so they take into account the terms representing connectivity between the pressures and temperatures in the various pore scales. Further, the basic initial boundary value problems are investigated by using the Lagrange identity and the logarithmic convexity methods. We introduce a functional representing an appropriate measure of the solution to the initial boundary value problem for the model in concern and by means of the Lagrange identity method we are able to establish a uniqueness result. We also establish some estimates describing the continuous data dependence of solution with respect to the initial data and with respect to the various supply terms. Finally, we show how the above functional can be used in combination with the logarithmic convexity method in order to prove again that the solution is unique.

Control of a Time Delay System Arising From Linearized Conservation Laws

A structure of energy cogeneration with distributed parameters is considered, modeled by an initial boundary value problem for hyperbolic conservation laws with a nonlinear and nonstandard boundary condition. Under a standard simplifying assumption leading to linear partial differential equations, the model is studied by associating a system of functional differential and difference equations of neutral type: considered are basic theory (existence, uniqueness, and continuous data dependence), invariant sets (positiveness of some state variables), equilibria, and their inherent stability (without control). These properties are illustrated by simulation results. Furthermore, a control Lyapunov functional is constructed, and feedback stabilizing structure is designed. This paper ends with a conclusion section, where open problems, such as stability by the first approximation/robustness and stability preservation under singular perturbations, are pointed out.

Analysis of an optimization problem for a piezoelectric energy harvester

The problem of optimal energy harvesting for a piezoelectric element driven by mechanical vibrations is stated in terms of an ODE system with hysteresis under the time derivative coupling a mechanical oscillator with an electric circuit with or without inductance. In the piezoelectric constitutive law, both the self-similar piezoelectric butterfly character of the hysteresis curves and feedback effects are taken into account in a thermodynamically consistent way. The physical parameters of the harvester are chosen to be the control variable, and the goal is to maximize the harvested energy for a given mechanical load and a given time interval. If hysteresis is modeled by the Preisach operator, the system is shown to be well-posed with continuous data dependence. For the special case of the play operator, we derive first-order necessary optimality conditions and an explicit form of the gradient of the total harvested energy functional in terms of solutions to the adjoint system.

Neutral functional differential equations and systems of conservation laws

A natural way of introducing neutral functional differential equations is integration along the characteristics of the Riemann invariants of the hyperbolic partial differential equations. Up to now the problem has been discussed by A.D. Myshkis, K.L. Cooke and their followers in the linear or quasilinear case. The conservation laws are nonlinear hyperbolic partial differential equations whose study goes back to the fifties of the previous century, being due to the pioneering papers of O.A. Oleinik and P.D. Lax. These equations describe important phenomena in Physics and Engineering, being also subject to control issues. The present paper is an attempt to extend the method of integrating along the characteristics to the systems of conservation laws. The main purpose of this attempt is to emphasize the occurring nonlinear neutral functional differential equations and to construct the augmented validation (existence, uniqueness, continuous data dependence, Stability Postulate) at least at the level of the classical solutions.

Well-posedness of an extended model for water-ice phase transitions

We propose an improvedmodel explaining the occurrence of high stresses due to thedifference in specific volumes during phase transitions betweenwater and ice. The unknowns of the resulting evolution problem are the absolute temperature,the volume increment, and the liquid fraction. The main novelty here consistsin including the dependence of the specific heat and of the speed of soundupon the phase. These additional nonlinearities bringnew mathematical difficulties which require new estimation techniques based onMoser iteration. We establish the existence of a global solution tothe corresponding initial-boundary value problem, as well as lower andupper bounds for the absolute temperature. Assuming constant heat conductivity,we also prove uniqueness and continuous data dependence of the solution.

Several Results on Uniqueness and Continuous Data Dependence in Thermoelasticity of Type III

In the present paper we consider the linear theory of thermoelasticity of type III for anisotropic and inhomogeneous materials as developed by Green and Naghdi (1992, 1995). We consider the general initial-boundary value problems associated with the theory in concern and then we identify some mild constitutive hypotheses that lead to appropriate energetic measures of solutions. On this basis we are able to prove some new uniqueness and continuous data dependence theorems. When the boundary surface is maintained at zero thermal displacement, we are able to obtain uniqueness and continuous data dependence results by using only the positiveness of the density mass, the positive definiteness of the elasticity tensor and the positive definiteness of the heat conductivity tensor, without any sign defined assumption upon the specific heat. These results are established by means of new techniques which are essentially different from those previously used for the classical theory of thermoelasticity.

Regularity and uniqueness in quasilinear parabolic systems

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.

A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity

In this paper, we prove the existence and global boundedness from above for a solution to an integro-differential model for nonisothermal multi-phase transitions under nonhomogeneous third type boundary conditions. The system couples a quasilinear internal energy balance ruling the evolution of the absolute temperature with a vectorial integro-differential inclusion governing the (vectorial) phase-parameter dynamics. The specific heat and the heat conductivity k are allowed to depend both on the order parameter χ and on the absolute temperature θ of the system, and the convex component of the free energy may or may not be singular. Uniqueness and continuous data dependence are also proved under additional assumptions.

Lipschitz continuous data dependence of sweeping processes in BV spaces

For a rate independent sweeping process with a time dependent smooth convex constraint, we prove that the Kurzweil solution for possibly discontinuous inputs depends locally Lipschitz continuously on the data in terms of the $BV$-norm.

Modeling Multivariate Count Data Using Copulas

Multivariate count data occur in several different disciplines. However, existing models do not offer great flexibility for dependence modeling. Models based on copulas nowadays are widely used for continuous data dependence modeling. Modeling count data via copulas is still in its infancy; see the recent article of Genest and Nešlehová (2007). A series of different copula models providing various residual dependence structures are considered for vectors of count response variables whose marginal distributions depend on covariates through negative binomial regressions. A real data application related to the number of purchases of different products is provided.

Rate independent Kurzweil processes

The Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in BV spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree one.

Some uniqueness and stability results in the theory of micropolar solid–fluid mixture

This paper is devoted to the study of a binary homogeneous mixture of an isotropic micropolar linear elastic solid with an incompressible micropolar viscous fluid. Assuming that the internal energy of the solid and the dissipation energy are positive definite forms, some uniqueness and continuous data dependence results are presented. Also, some estimates which describe the time behaviour of solution are established, provided the above two energies are positive definite forms. These estimates are used to prove that for the linearized equations, in the absence of body loads and for null boundary conditions, the solution is asymptotically stable. Then a uniqueness result under mild assumptions on the constitutive constants is given using the Lagrange–Brun identities method. These mathematical results prove that the approach of a micropolar solid–fluid mixture is in agreement with physical expectations.