Parabolic Differential Inclusion Research Articles

Polyhedral optimization of discrete and partial differential inclusions of parabolic type

ABSTRACT The purpose of this paper is to look into the optimization of the first mixed boundary value problems for partial differential inclusions of the parabolic type. More specifically, we discuss a constructive approach to the study and solution of optimization problems for partial differential inclusions based on the discrete-approximate method. We formulate necessary and sufficient conditions of optimality by reducing the original first mixed boundary value problems to finite-dimensional problems in mathematical programming. All of our conditions are presented through the notion of Euler-Lagrange inclusions and transversality conditions. The effectiveness of our approach and its superiority over well-known methods are demonstrated by the example of the linear optimal control problem. Then, by passing formally to the limit as the discrete steps approach zero, we derive sufficient conditions of optimality for parabolic differential inclusions in the discrete-approximate problem.

Analysis and Approximation of Hemivariational Inequality for a Frictional Thermo-electro-visco-elastic Contact Problem with Damage

The aim of this paper is to investigate a contact problem involving thermo-electro-visco-elastic body with damage and a rigid foundation. The friction is modelled with a subgradient of a locally Lipschitz mapping, and the contact is described by the Signorini's unilateral contact condition. A parabolic differential inclusion for the damage function is used to include the damaging effect in the model. We establish the model's variational formulation using four systems of three hemivariational inequalities and a parabolic equation, we prove an existence and uniqueness result of this problem. The proof is based on a fixed point argument and a recent finding from hemivariational inequality theory. Finally, by employing the finite element approach, we investigate a fully discrete approximation of the model and we derive error estimates on the approximate solution.

Optimal control of parabolic differential inclusions in one space dimension

The paper is devoted to the optimisation of parabolic differential inclusions (DFIs) in a bounded rectangular region. For this, a problem with a parabolic discrete inclusion is defined, which is the main inevitable auxiliary problem. With the help of locally adjoint mappings, necessary and sufficient conditions for the optimality of parabolic discrete inclusions are proved. Then, using the method of discretization of parabolic DFIs and the already obtained optimality conditions for discrete inclusions, the necessary and sufficient conditions for the discrete-approximate problem are formulated in the form of the Euler–Lagrange type inclusion. Thus, using a specially proved equivalence theorem, we establish sufficient optimality conditions for a parabolic DFI. To demonstrate the above approach, some linear problems and polyhedral optimisation with inclusions of parabolic type are investigated. In addition, numerical results are also presented.

Numerical studies of a hemivariational inequality for a viscoelastic contact problem with damage

This paper is devoted to the study of a hemivariational inequality modeling the quasistatic bilateral frictional contact between a viscoelastic body and a rigid foundation. The damage effect is built into the model through a parabolic differential inclusion for the damage function. A solution existence and uniqueness result is commented. A fully discrete scheme is introduced with the time derivative of the damage function approximated by the backward finite difference and the spatial derivatives approximated by finite elements. An optimal order error estimate is derived for the fully discrete scheme when linear elements are used for the velocity and displacement variables, and piecewise constants are used for the damage function. Simulation results on numerical examples are reported illustrating the performance of the fully discrete scheme and the theoretically predicted convergence orders.

Sliding mode control of the Hodgkin–Huxley mathematical model

In this paper we deal with a feedback control design for the action potential of a neuronal membrane in relation with the non-linear dynamics of the Hodgkin-Huxley mathematical model. More exactly, by using an external current as a control expressed by a relay graph in the equation of the potential, we aim at forcing it to reach a certain manifold in finite time and to slide on it after that. From the mathematical point of view we solve a system involving a parabolic differential inclusion and three nonlinear differential equations via an approximating technique and a fixed point result. The existence of the sliding mode and the determination of the time at which the potential reaches the prescribed manifold is proved by a maximum principle argument. Numerical simulations are presented.

Semilinear parabolic differential inclusions with one-sided Lipschitz nonlinearities

We present an existence result for a partial differential inclusion with linear parabolic principal part and relaxed one-sided Lipschitz multivalued nonlinearity in the framework of Gelfand triples. Our study uses discretizations of the differential inclusion by a Galerkin scheme, which is compatible with a conforming finite element method, and we analyze convergence properties of the discrete solution sets.

The Krasnosel'skii formula for parabolic differential inclusions with state constraints

We consider a constrained semilinear evolution inclusion of parabolic type involving an $m$-dissipative linear operator and a source term of multivalued type in a Banach space and topological properties of the solution map. We establish the $R_δ$-description of the set of solutions surviving in the constraining area and show a relation between the fixed point index of the Krasnosel'skii-Poincaré operator of translation along trajectories associated with the problem and the appropriately defined constrained degree of the right-hand side in the equation. This provides topological tools appropriate to obtain results on the existence of periodic solutions to studied differential problems.

Pullback attractors of reaction-diffusion inclusions with space-dependent delay

Inspired by biological phenomena with effects of switching off (maybe just for a while), we investigate non-autonomous reaction-diffusion inclusions whose multi-valued reaction term may depend on the essential supremum over a time interval in the recent past (but) pointwise in space. The focus is on sufficient conditions for the existence of pullback attractors. If the multi-valued reaction term satisfies a form of inclusion principle standard tools for non-autonomous dynamical systems in metric spaces can be applied and provide new results (even) for infinite time intervals of delay. More challenging is the case without assuming such a monotonicity assumption. Then we consider the parabolic differential inclusion with the time interval of delay depending on space and extend the approaches of norm-to-weak semigroups to a purely metric setting. This provides completely new tools for proving pullback attractors of non-autonomous dynamical systems in metric spaces.

On the differential variational inequalities of parabolic-elliptic type

We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd.

Variational-Hemivariational Approach to a Quasistatic Viscoelastic Problem with Normal Compliance, Friction and Material Damage

This work studies a model for quasistatic frictional contact between a viscoelastic body and a reactive foundation. The constitutive law is assumed to be nonlinear and contains damage effects modeled by a parabolic differential inclusion. Contact is described by the normal compliance condition and a subdifferential frictional condition. A variational-hemivariational formulation of the problem is provided and the existence and uniqueness of its solution is proved. The proof is based on a surjectivity result for pseudomonotone coercive operators and a fixed point argument.

Reaction-diffusion equations with a switched--off reaction zone

Reaction-diffusion equations are considered on a bounded domain $\Omega$ in $\mathbb{R}^d$ with a reaction term that is switched off at a point in space when the solutionfirst exceeds a specified threshold and thereafter remains switched off at that point, which leads to a discontinuous reaction term with delay.This problem is formulated as a parabolic partial differential inclusion with delay. The reaction-free region forms what could be called dead corein a biological sense rather than that used elsewhere in the literature for parabolic PDEs. The existence of solutions in $L^2(\Omega)$ is establishedfirstly for initial data in $L^{\infty}(\Omega)$ and in $W_0^{1,2}(\Omega)$ by different methods, with $d$ $=$ $2$ or $3$ in the first case and $d$ $\geq$ $2$ in the second. Solutions here are interpreted in the sense of integral or strong solutions of nonhomogeneous linear parabolic equations in $L^2(\Omega)$ that are generalised to selectors of the corresponding nonhomogeneous linear parabolic differential inclusions and are shown to be equivalent under the assumptions used in the paper.

An existence and uniqueness result for a strain-adaptive bone remodeling problem

In this paper we prove an existence and uniqueness result for a strain-adaptive bone remodeling model that couples the displacements and the apparent density (the porosity) of the bone. The rate of this density at a particular location is described as an objective function, which depends on a particular stimulus at that location. Then, a new version of this problem is considered, based on a regularization on the bone remodeling law by using convolution operators. The existence and uniqueness result is proved by applying classical results for nonlinear parabolic differential inclusions, Gronwall’s inequality and the Banach fixed point theorem.

Numerical analysis and simulations of a dynamic frictionless contact problem with damage

The process of dynamic frictionless contact between a viscoelastic body and a reactive foundation, which includes material damage, is modelled, numerically analyzed, and simulated. Contact is modelled with the normal compliance condition. The damage of the material, resulting from tension or compression, is described by the damage field which measures the pointwise fractional decrease in the load carrying capacity of the material. The evolution of the damage is governed by a parabolic differential inclusion. A fully discrete scheme for the problem is proposed, and error estimates on the numerical solutions derived. The scheme has been implemented and numerical simulations of the evolution of the mechanical state of the system and its material damage are presented.

An energy balance climate model with hysteresis

Energy balance climate models of Budyko type lead to reaction–diffusion equations with slow diffusion and memory on the 2-sphere. The reaction part exhibits a jump discontinuity (at the snow line). Here we introduce a Babuška–Duhem hysteresis in order to account for a frequent repetition of sudden and fast warming followed by much slower cooling as observed from paleoclimate proxy data. Existence of global solutions and of a trajectory attractor will be established for the resulting system of a parabolic differential inclusion and an ode.

Extremal solutions of quasilinear parabolic subdifferential inclusions

In this paper we consider initial boundary value problems for quasilinear parabolic differential inclusions governed by general operators of Leray-Lions type and continuously perturbed subdifferentials. Our main goal is to show the existence of extremal solutions within a sector formed by appropriately defined upper and lower solutions. One of the main difficulties that arises in the treatment of the parabolic problems considered here is due to the fact that the underlying solution space does not possess lattice structure. Furthermore, the variational techniques that can be used for the corresponding stationary problems are not applicable here. The main tools used in the proof of our result are abstract results on nonlinear evolution equations, regularization, comparison and truncation techniques as well as special test function techniques.

A Tikhonov type Theorem for abstract parabolic differential inclusions in Banach spaces

We consider a class of singularly perturbed systems of semilinear parabolic differential inclusions in infinite dimensional spaces. For such a class we prove a Tikhonov-type theorem for a suitably defined subset of the set of all solutions for e ≥ 0, where e is the perturbation parameter. Specifically, assuming the existence of a Lipschitz selector of the involved multivalued maps we can define a nonempty subset ZL(e) of the solution set of the singularly perturbed system. This subset is the set of the Holder continuous solutions defined in [0, d], d > 0 †Research supported by the project “Qualitative analysis and control of dynamical systems” at the University of Siena and FRBR grant 99-01-00333. 208 A. Gudovich, M. Kamenski and P. Nistri with prescribed exponent and constant L. We show that ZL(e) is uppersemicontinuous at e = 0 in the C[0, d]× C[δ, d] topology for any δ ∈ (0, d].

Existence and comparison results for quasilinear parrabolic inclusions woth state-dependent subdifferentials*

In this paper we consider initial boundary value problems for quasilinear parabolic differential inclusions governed by nonmonotone operators of Leray-Lions type and state-dependent subdifferentials. Our main goal is to prove the existence of solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof are abstract results on nonlinear evolution equations, regularization, comparison and truncation techniques as well as special test function techniques

Asymptotic analysis of a phase field model with memory for vanishing time relaxation

A phase-field model accounting for memory effects is considered. This model consists of a hyperbolic integrodifferential equation coupled with a parabolic differential inclusion. The latter relation rules the evolution of the phase field and contains a time relaxation parameter which happens to be very small in the appli- cations. A well-posed initial and boundary value problem for the evolution system is introduced and the asymptotic behavior of its solution as the time relaxation goes to zero is analyzed rigorously. Convergence results and error estimates are obtained under suitable assumptions ensuring that the limit problem has a unique solution.