Hyperbolic Differential Inclusions Research Articles

Duality in the problems of optimal control described by Darboux-type differential inclusions

AbstractThis paper is devoted to the optimization of the Mayer problem with hyperbolic differential inclusions of the Darboux type and duality. We use the discrete approximation method to get sufficient conditions of optimality for the convex problem given by Darboux differential inclusions and the polyhedral problem for a hyperbolic differential inclusion with state constraint. We formulate the adjoint inclusions in the Euler-Lagrange inclusion and Hamiltonian forms. Then, we construct the dual problem to optimal control problem given by Darboux differential inclusions with state constraint and prove so-called duality results. Moreover, we show that each pair of primal and dual problem solutions satisfy duality relations and that the optimal values in the primal convex and dual concave problems are equal. Finally, we establish the dual problem to the polyhedral Darboux problem and provide an example to demonstrate the main constructions of our approach.

Existence of Solutions of Impulsive Partial Hyperbolic Differential Inclusion of Fractional Order

This paper is concerned with the existence of solutions of a class of Cauchy problems for hyperbolic partial fractional differential inclusions (HPFD) involving the Caputo fractional derivative with an impulse whose right hand side is convex and non-convex valued. Our results are achieved within the framework of the nonlinear alternative of Leray-Schauder type and contraction multivalued maps. A detailed example was provided to support the theorem.

Optimal control of hyperbolic type discrete and differential inclusions described by the Laplace operator

The paper is devoted to the optimization of a first mixed initial-boundary value problem for hyperbolic differential inclusions (DFIs) with Laplace operator. For this, an auxiliary problem with a hyperbolic discrete inclusion is defined and, using locally conjugate mappings, necessary and sufficient optimality conditions for hyperbolic discrete inclusions are proved. Then, using the method of discretization of hyperbolic DFIs and the already obtained optimality conditions for discrete inclusions, the optimality conditions for the discrete approximate problem are formulated in the form of the Euler-Lagrange type inclusion. Thus, using specially proved equivalence theorems, which are the only tool for constructing Euler-Lagrangian inclusions, we establish sufficient optimality conditions for hyperbolic DFIs. Further, the way of extending the obtained results to the multidimensional case is indicated. To demonstrate the above approach, some linear problems and polyhedral optimization with hyperbolic DFIs are investigated.

Fractional order differential inclusions on an unbounded domain with infinite delay

We provide sufficient conditions for the existence of solutions to initial value problems, for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with infinite delay by applying the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces.

Viability in a non-local population model structured by size and spatial position

Motivated conceptually by multiscale models of cancer cell migration, the focus is on a broad class of set-valued population models structured both physiologically and spatially. For time-dependent sets of constraints given, sufficient conditions are provided such that every permitted density function initializes at least one admissible weak solution. This result about viability a.k.a. weak invariance concerns non-local hyperbolic differential inclusions of first order with memory and with one boundary condition for the minimal value of the scalar structure parameter.

Darboux problem for perturbed hyperbolic differential inclusions with fractional order

Darboux problem for perturbed hyperbolic differential inclusions with fractional order

Set-valued $$(\Psi ,\Phi )$$ ( Ψ , Φ ) - $$\Theta $$ Θ ordered contractions with applications in differential inclusions

In this article we prove some results for coincidence points of a new type of set-valued $$(\Psi ,\Phi )$$ - $$\Theta $$ contractions in the settings of ordered metric spaces. Our results generalize many results in the literature. We also provide some nontrivial examples to support our main theorem. For application of our main result, we provide an existence theorem for solutions of hyperbolic type differential inclusions.

АППРОКСИМАЦИЯ ГИПЕРБОЛИЧЕСКИХ ДИФФЕРЕНЦИАЛЬНЫХ ВКЛЮЧЕНИЙ ДРОБНОГО ПОРЯДКА С ИМПУЛЬСНЫМИ ВОЗДЕЙСТВИЯМИ

In this paper there are considered hyperbolic differential inclusions of fractional order with impulses. Here we represent the concept of approximate solution (δ-solution) for a hyperbolic differential inclusion of fractional order with impulses. The asymptotic properties of solutions sets to approximating differential inclusions of fractional order with external disturbance are derived.

The averaging of fuzzy hyperbolic differential inclusions

In this paper the substantiation of a possibility of application of partial averaging method for hyperbolic differential inclusions with the fuzzy right-hand side with the small parameters is considered.

On a system of hyperbolic partial fractional differential inclusions

On a system of hyperbolic partial fractional differential inclusions

Some fixed point theorems of multivalued operators in partially ordered metric spaces and applications to hyperbolic differential inclusions

In this paper, we introduce the concept of generalized contraction for multivalued operators defined on ordered complete metric spaces. We analyze the existence of fixed points for generalized multivalued operators. Moreover, as an application of our main theorem, we give an existence theorem for the solution of a hyperbolic differential inclusion problem.

Existence and general decay for nondissipative hyperbolic differential inclusions with acoustic/memory boundary conditions

In this paper, we prove the existence and general energy decay rate of global solution to the mixed problem for nondissipative multi‐valued hyperbolic differential inclusions urn:x-wiley:0025584X:media:mana201400343:mana201400343-math-0001with memory boundary conditions on a portion of the boundary and acoustic boundary conditions on the rest of it. For the existence of solutions, we prove the global existence of weak solution by using Galerkin's method and compactness arguments. For the energy decay rates, we first consider the general nonlinear case of h satisfying a smallness condition, and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h: and prove the general decay estimates of equivalent energy.

Energy decay rates for a hyperbolic differential inclusion with viscoelastic boundary conditions

In this paper, we consider a hyperbolic differential inclusion with viscoelastic boundary conditions localized on a part of the boundary. For a wider class of relaxation functions, we establish a more general decay result, from which the usual exponential and polynomial decay rates are only special cases. Copyright © 2013 John Wiley & Sons, Ltd.

Existence and uniqueness of solution of nonconvex impulsive hyperbolic differential inclusions

Existence and uniqueness of solution of nonconvex impulsive hyperbolic differential inclusions

Approximation and optimization of Darboux type differential inclusions with set-valued boundary conditions

The present paper is devoted to an optimal control problem given by hyperbolic discrete (PD) and differential inclusions (PC) of generalized Darboux type and ordinary discrete inclusions. The results are extended to non-convex problems. An approach concerning necessary and sufficient conditions for optimality is proposed. In order to formulate sufficient conditions of optimality for problem (PC) the approximation method is used. Formulation of these conditions is based on locally adjoint mappings. Moreover for construction of adjoint partial differential inclusions the equivalence theorems of locally adjoint mappings are proved. One example with homogeneous boundary conditions is considered.

Weak solutions for hyperbolic partial fractional differential inclusions in Banach spaces

In this paper, we investigate the existence of solutions under the Pettis integrability assumption for an initial value problem for a hyperbolic differential inclusion of Caputo fractional order. We apply the measure of weak noncompactness technique.

A thermoviscoelastic problem with damage

This work studies, from the numerical point of view, a problem which models the dynamic evolution of damage in a thermoviscoelastic body. Material damage which results from tension or compression is taken into account in the constitutive law, and its evolution is described by a hyperbolic partial differential inclusion. The variational problem is formulated as a coupled system of nonlinear evolutionary equations, for which the existence of a unique weak solution is recalled. A fully discrete numerical scheme is introduced, by using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity assumptions, the linear convergence of the numerical scheme is obtained. Finally, the results of simulations of three two-dimensional examples are presented.

Fractional order impulsive partial hyperbolic differential inclusions with variable times

Fractional order impulsive partial hyperbolic differential inclusions with variable times

Fixed points of multivalued operators in ordered metric spaces with applications

Abstract The purpose of this paper is to present some new (hybrid) fixed point theorems involving multivalued operators which satisfy weakly generalized contractive conditions in an ordered complete metric space. An example of the existence of solutions for a perturbed impulsive hyperbolic differential inclusion with variable times is given to illustrate the usability of our results.

Impulsive partial hyperbolic differential inclusions of fractional order

AbstractIn this paper we investigate the existence of solutions of a class of partial impulsive hyperbolic differential inclusions involving the Caputo fractional derivative. Our main tools are appropriate fixed point theorems from multivalued analysis.