Semicontinuous Right-hand Sides Research Articles

A subgradient-based neurodynamic algorithm to constrained nonsmooth nonconvex interval-valued optimization

In this paper, a subgradient-based neurodynamic algorithm is presented to solve the nonsmooth nonconvex interval-valued optimization problem with both partial order and linear equality constraints, where the interval-valued objective function is nonconvex, and interval-valued partial order constraint functions are convex. The designed neurodynamic system is constructed by a differential inclusion with upper semicontinuous right-hand side, whose calculation load is reduced by relieving penalty parameters estimation and complex matrix inversion. Based on nonsmooth analysis and the extension theorem of the solution of differential inclusion, it is obtained that the global existence and boundedness of state solution of neurodynamic system, as well as the asymptotic convergence of state solution to the feasible region and the set of LU-critical points of interval-valued nonconvex optimization problem. Several numerical experiments and the applications to emergency supplies distribution and nondeterministic fractional continuous static games are solved to illustrate the applicability of the proposed neurodynamic algorithm.

A mean field type differential inclusion with upper semicontinuous right-hand side

Mean field type differential inclusions appear within the theory of mean field type control through the convexification of a right-hand side. We study the case when the right-hand side of a differential inclusion depends on the state of an agent and the distribution of agents in an upper semicontinuous way. The main result of the paper is the existence and the stability of the solution of a mean field type differential inclusion. Furthermore, we show that the value function of the mean field type optimal control problem depends on an initial state and a parameter semicontinuously.

INTEGRAL OPERATORS, GUIDING POTENTIALS AND PERIODIC SOLUTIONS OF DIFFERENTIAL INCLUSIONS ON NON-COMPACT LIE GROUPS

We obtain the existence of a periodic solution for differential inclusions with lower semi-continuous right-hand sides on non-compact Lie groups. The construction is based on the combination of the method of guiding functions and the technique of integral operators with parallel translation.

Differential inclusions with mean derivatives having extreme right-hand sides and optimal control

First we prove the existence of solutions of some special stochastic differential inclusion with mean derivatives having lower semi-continuous right-hand sides that may not be convex. Then we show that among those solutions there is a solution that minimizes a certain cost criterion. After that this result is applied to investigation of controlled stochastic differential equations with feed back, whose right-hand sides take values in extreme sets of Hausdorff continuous set-valued vector field with bounded convex images. By reducing the equation to the inclusion of above-mentioned sort we prove that there exists a control that realizes the optimal solution of the inclusion as an optimal solution of the equation (an analogue of Filippov's theorem).

Smooth Lyapunov Functions for Multistable Differential Inclusions

We provide a converse Lyapunov theorem for differential inclusions with upper semicontinuous right-hand side, admitting a finite number of compact, globally attractive, weakly invariant sets, and evolving on Riemannian manifolds. Such properties entail multistable behavior in differential inclusions and may gather interest in a number of applications where uncertainty and discontinuities of the vector field play a major role.

Global attractors of impulsive parabolic inclusions

In this work we consider an impulsive multi-valued dynamical system generated by a parabolic inclusion with upper semicontinuous right-hand side \begin{document}$\varepsilon F(y)$\end{document} and with impulsive multi-valued perturbations. Moments of impulses are not fixed and defined by moments of intersection of solutions with some subset of the phase space. We prove that for sufficiently small value of the parameter \begin{document}$\varepsilon>0$\end{document} this system has a global attractor.

Discrete Dirichlet boundary value problems with upper semicontinuous right-hand sides

This paper deals with the relationship between solutions of Dirichlet boundary value problems (BVPs) for second order systems of differential inclusions with upper semicontinuous right-hand sides and associated numerical discrete Dirichlet BVPs of second order difference inclusions. First, the existence and estimate of solutions to the discrete BVP is discussed uniformly with respect to the discrete step size. Then convergence of solutions of the numerical discrete BVP and the corresponding semicontinous BVP is studied. Related results are also mentioned which motivated our study of this problem.

Continuous ε-selection theorems for almost lower semicontinuous multifunctions and applications

Continuous ε-selection theorems for almost lower semicontinuous multifunctions with decomposable values and some of their applications are presented. In particular, the existence of approximation solutions of differential inclusions with almost lower semicontinuous right-hand sides is obtained.

Morse equation of attractors for nonsmooth dynamical systems

This paper is concerned with a Morse theory of attractors for finite-dimensional nonsmooth dynamical systems described by differential inclusions with upper semi-continuous righthand sides. We first show that all open attractor neighborhoods of an attractor share the same homotopy type. Then based on this basic fact we introduce the concept of homology index for Morse sets and establish Morse inequalities and Morse equation by using smooth Morse–Lyapunov functions.

Morse Decompositions for Periodic General Dynamical Systems and Differential Inclusions

We first establish the Morse decomposition theory of periodic invariant sets for non-autonomous periodic general dynamical systems (set-valued dynamical systems). Then we discuss the stability of Morse decompositions of periodic uniform forward attractors. We also apply the abstract results to non-autonomous periodic differential inclusions with only upper semi-continuous right-hand side. We show that Morse decompositions are robust with respect to both internal and external perturbations (upper semi-continuity of Morse sets). Finally as an application we study the effect of small time delays to asymptotic behavior of control systems from the point of view of Morse decompositions.

On the asymptotic stability of discontinuous systems analysed via the averaging method

The averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions. A large number of practical problems lead to differential equations with discontinuous right-hand sides. In a rigorous theory of such systems, developed by Filippov, solutions of a differential equation with discontinuous right-hand side are regarded as being solutions to a special differential inclusion with upper semi-continuous right-hand side. The averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion. In this paper we prove natural extensions of Bogolyubov’s first theorem and the Samoilenko–Stanzhitskii theorem to differential inclusions with an upper semi-continuous right-hand side. We prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one. The extension of Bogolyubov’s theorem concerns finite time intervals, while the extension of the Samoilenko–Stanzhitskii theorem deals with solutions defined on the infinite interval. The averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required.

On the Existence of Solutions to Differential Inclusions with Nonconvex Right-Hand Sides

We study the existence of solutions of differential inclusions with upper semicontinuous right-hand sides. The investigation was prompted by the well-known Filippov examples. We define a new concept, “colliding on a set.” In the case when the admissible velocities do not “collide” on the set of discontinuities of the right-hand side, we expect that at least one trajectory emanates from every point. If the velocities do “collide” on the set of discontinuities of the right-hand side, the existence of solutions is not guaranteed, as is seen from one of Filippov's examples. In this case we impose an additional condition in order to prove the existence of a solution starting at a point of the discontinuity set. For the right-hand sides under consideration, we assume the following: whenever the velocities “collide” on a set S there exist tangent velocities (belonging to the Clarke tangent cone to S) on a dense subset of S. Then we prove the existence of an $\varepsilon$-solution for every $\varepsilon >0$. Under additional assumptions we can pass to the limit as $\varepsilon \to 0$ and obtain a solution of the considered differential inclusion.

On the dynamics of nonautonomous periodic general dynamical systems and differential inclusions

We establish the existence and stability results for periodic nonautonomous uniform forward attractors of periodic general dynamical systems (set-valued dynamical systems). We also investigate the dynamical behavior of nonautonomous periodic differential inclusion x ′ ( t ) ∈ f ( t , x ( t ) ) on R m with only upper semi-continuous right-hand side by applying the abstract results. Firstly, we show that if the system has a compact uniformly attracting set, then it has a periodic nonautonomous uniform forward attractor A . Secondly, we prove that A is robust with respect to both internal and external perturbations. Finally, we apply the robustness result to discuss the effects of small time delays to asymptotic stability of the system.

Bogolyubov's theorem under constraints generated by a lower semicontinuous differential inclusion

An analogue of the classical theorem of Bogolyubov with non-convex constraint is proved. The constraint is the solution set of a differential inclusion with non-convex lower semicontinuous right-hand side. As an application we study the interrelation between the solutions of the problem of minimizing an integral functional with non-convex integrand on the solutions of the original inclusion and the solutions of the relaxation problem.

Existence results for functional and neutral functional integrodifferential inclusions with lower semicontinuous right-hand side

In this paper, Schaefer's fixed point theorem combined with a selection theorem due to Bressan and Colombo is used to investigate the existence of solutions for the first- and second-order functional and neutral functional integrodifferential inclusions with lower semicontinuous and nonconvex-valued right-hand side.

Mixed Semicontinuous Mappings and Their Applications to Differential Inclusions

A class of multivalued mappings, called by us mixed semicontinuous mappings, which are upper semicontinuous in some points and lower semicontinuous in remaining points is considered. A general selection theorem for mixed semicontinuous maps is proved. Then applications to differential inclusions with mixed semicontinuous right-hand sides are presented. Note that our applications generalize earlier existence results obtained both on the whole Euclidean space R n and on a compact proximate retract of R n .

Lower semicontinuous differential inclusions. One-sided Lipschitz approach

Some properties of differential inclusions with lower semicontinuous right-hand side are considered. Our essential assumption is the one-sided Lipschitz condition introduced in [4]. Using the main idea of [10], we extend the well known relaxation theorem,

Implicit runge-kutta methods for differential inclusions

This paper is concerned with the application of implicit Runge-Kutta methods suitable for stiff initial value problems to initial value problems for differential inclusions with upper semicontinuous right-hand sides satisfying a uniform one-sided Lipschitz condition and a growth condition. The problems could stem from differential equations with state discontinuous right-hand sides. It is shown that there exist methods with higher order of convergence on intervals where the solution is smooth enough. Globally we get at least the order one.

On a class of implicit differential inclusions

Existence of solutions is established for a class of implicit differential inclusions equivalent to explicit relations with nonconvex-valued, yet only upper semicontinuous right-hand side. Moreover, the set of stationary points is shown to be asymptotically stable if both of the set-valued maps involved are monotone and one of them is a subdifferential. Essentially, the primitive of the latter can be used as a natural Lyapunov function.