Approximation of the viability kernel

We study recursive inclusionsx ⁿ⁺¹ G(x ⁿ ). For instance, such systems appear for discrete finite-difference inclusionsx ⁿ⁺¹ G (x ⁿ) whereG :=1+F. The discrete viability kernel ofG , i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withx ⁿ⁺¹ (xⁿ) where (x) =x + F(x) + (ML/2) ². Secondly, we show that it can be approached by finite viability kernels associated withx _h ⁿ⁺¹ ( (x _h ⁿ⁺¹ ) +(h) X _h .     

Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions

The main objective of this paper is to provide new explicit criteria to characterize weak lower semicontinuous Lyapunov pairs or functions associated to first-order differential inclusions in Hilbert spaces. These inclusions are governed by a Lipschitzian perturbation of a maximally monotone operator. The dual criteria we give are expressed by means of the proximal and basic subdifferentials of the nominal functions while primal conditions are described in terms of the contingent directional derivative. We also propose a unifying review of many other criteria given in the literature. Our approach is based on advanced tools of variational analysis and generalized differentiation.     

On nonoscillatory solutions of differential inclusions

This paper introduces a nonoscillatory theory for differential inclusions based on fixed point theory for multivalued maps.

     

Elliptic differential inclusions on non-compact Riemannian manifolds

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace–Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions for the studied differential inclusion. The proofs are based on nonsmooth variational analysis as well as isometric actions and fine eigenvalue properties on Riemannian manifolds. The results are also new in the smooth setting.     

On the existence of periodic solutions for differential inclusions

In this paper we study the existence of periodic solutions for differential inclusions. We prove existence theorems under various sets of hypotheses for both the nonconvex and convex problems. Also we show the existence of extreme solutions. Some feedback control systems are also considered.     

Generic properties of multifunctions: Application to differential inclusions

We prove that almost all in Baire sense Caratheodory multifunctions in finite dimensional space are Kamke continuous. Further the main properties of differential inclusions with Kamke and one sided Kamke right-hand sides are studied.As a corollary we prove that for almost all optimal control problems, the relaxation and relaxation stability properties hold.     

On solutions of differential inclusions in homogeneous spaces of functions

In this paper we study the solvability of some classes of differential inclusions with multivalued linear operators in homogeneous spaces of functions. These spaces include a large number of functional spaces like periodic functions and Bohr and Stepanov almost periodic functions. As an application, we consider some existence results for feedback control systems governed by degenerate differential equations of Sobolev type in a Banach space.     

On singular perturbations for differential inclusions on the infinite interval

We consider a differential inclusion subject to a singular perturbation, i.e., part of the derivatives are multiplied by a small parameter >0. We show that under some stability and structural assumptions, every solution of the singularly perturbed inclusion comes close to a solution of the degenerate inclusion (obtained for =0) when tends to 0. The goal of the present paper is to provide a new result of Tikhonov type on the time interval [0,+∞[.     

On integro‐differential inclusions with operator‐valued kernels

We study integro‐differential inclusions in Hilbert spaces with operator‐valued kernels and give sufficient conditions for the well‐posedness. We show that several types of integro‐differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well‐posedness within this framework. As an example, we apply our results to the equations of visco‐elasticity and to a class of nonlinear integro‐differential inclusions describing phase transition phenomena in materials with memory. Copyright © 2014 John Wiley & Sons, Ltd.     

On the controllability of a class of differential inclusions depending on a parameter

In this paper, we deal with the following stability problem: given a differential inclusion of the formx'F(t,x,), where is a parameter varying in a topological space , find conditions under which the set of all , such that the differential inclusion is controllable, is open in . Applying Theorem 3.1 of Ref. 3, we get a result in this direction, assuming, as leader hypotheses, thatF(t,·,) is a convex process, from into itself, and thatF(t,x,·) is lower semicontinuous.     

A generalized Nielsen number and multiplicity results for differential inclusions

The Nielsen number is defined for a rather general class of multivalued maps on compact connected ANRs, including, e.g., admissible maps (in the sense of Górniewicz (1976); compare also Górniewicz (1995)) on tori. Since the Poincaré maps generated by the Marchaud vector fields are of this type (see (Andres, 1997)), we can obtain in such a way multiplicity results for differential inclusions. More precisely, the nontrivial Nielsen number gives a lower estimate of coincidence points (in particular, fixed points) corresponding to the desired solutions.     

Solution bundle for a class of impulsive differential inclusions on Banach spaces

In this paper a class of impulsive differential inclusions is investigated. The existence of solution bundle is proved. And we also construct a nonlinear semigroup of operators on cb(E) (closed-bounded subset of E) to describe the set of attainable states.     

Homogeneous feedback design of differential inclusions based on control Lyapunov functions

This paper is concerned with the stabilization of differential inclusions. By using control Lyapunov functions, a design method of homogeneous controllers for differential equation systems is first addressed. Then, the design method is developed to two classes of differential inclusions without uncertainties: homogeneous differential inclusions and nonhomogeneous ones. By means of homogeneous domination theory, a homogeneous controller for differential inclusions with uncertainties is constructed. Comparing to the existing results in the literature, an improved formula of homogeneous controllers is proposed, and the difficulty of the controller design for uncertain differential inclusions is reduced. Finally, two simulation examples are given to verify the preset design.     

On boundary-value problems for second order perturbed differential inclusions

This article studies the existence of solutions to boundary-value problems for second order multi-valued perturbed differential inclusions under the mixed Lipschitz and Carathéodory conditions. The existence of extremal solutions is also obtained under certain monotonicity conditions and the weaker nonconvexity conditions for multi-valued functions.     

Systems of p-Laplacian differential inclusions with large diffusion

In this paper we consider coupled systems of p-Laplacian differential inclusions and we prove, under suitable conditions, that a homogenization process occurs when diffusion parameters become arbitrarily large. In fact we obtain that the attractors are continuous at infinity on L²(Ω)×L²(Ω) topology, with respect to the diffusion coefficients, and the limit set is the attractor of an ordinary differential problem.     

Near viability for fully nonlinear differential inclusions

We consider the nonlinear differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A is an m-dissipative operator on a separable Banach space X and F is a multi-function. We establish a viability result under Lipschitz hypothesis on F, that consists in proving the existence of solutions of the differential inclusion above, starting from a given set, which remain arbitrarily close to that set, if a tangency condition holds. To this end, we establish a kind of set-valued Gronwall’s lemma and a compactness theorem, which are extensions to the nonlinear case of similar results for semilinear differential inclusions. As an application, we give an approximate null controllability result.     

Optimal solutions to differential inclusions in presence of state constraints

Necessary conditions in terms of the Hamiltonian are given for optimal solutions to the differential inclusion problem when state constraints are present. This result extends a result of Clarke for the unconstrained problem. The data are nonsmooth, nonlinear, nonconvex. The method incorporates the state constraint in the cost functional as a penalty term for a sequence of unconstrained problems that approximate our problem. An application of Ekeland's variational principle, the known necessary conditions for the auxiliary problems, and a limiting process provide the necessary conditions.