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In this paper, we study the behaviors of fixed points sets of non necessarily pseudo-contractive multifunctions. Rather than comparing the images of the involved multifunctions, we make use of some conditions on the fixed points sets to establish general re­sults on their stability and continuous dependence. We illustrate our results by applications to differential inclusions and give stability results of fixed points sets of non necessarily pseudo-contractive multifunctions with respect to the bounded proximal convergence.
This paper deals with stability analysis of hybrid systems. Various stability concepts related to hybrid systems are introduced. The paper advocates a local analysis. It involves the equivalence relation generated by reset maps of a hybrid system. To establish a tangible method for stability analysis, we introduce the notion of a chart, which locally reduces the complexity of the hybrid system. In a chart, a hybrid system is particularly simple and can be analyzed with the use of methods borrowed from the theory of differential inclusions. Thus, the main contribution of this paper is to show how stability of a hybrid system can be reduced to a specialization of the well established stability theory of differential inclusions. A number of examples illustrate the concepts introduced in the paper.
Content available remote On a boundary value problem for a third differential inclusion
We consider a boundary value problem for third order nonconvex differential inclusion and we obtain some existence results by using the set-valued contraction principle.
In this paper we study the global existence of positive integrable solution for the nonlinear integral inclusion of fractional order..[formula] As an application the global existence of the solution for the initial-value problem of the arbitrary (fractional) orders differential inclusion..[formula] will be studied.
Content available remote Second-order viability problem: a baire category approach
The paper deals with the existence of viable solutions to the differential inclusion x(t) ∈ ƒ(t, x(t)) + ext F(t, x(t)), where ƒ is a single-valued map and ext F(t, x) stands for the extreme points of a continuous, convex and noncompact set-valued mapping F with nonempty interior.
Content available remote The dynamic response of an elastic background under undetermined variable loads
The problem of Lagrange stability of an elastic background excited by undetermined variable loads is considered. A discrete model of the background with bounded excitations is studied. Applying optimal Lyapunov functions, upper bounds on the dynamic response of the background are estimated and interpreted in terms of rough sets. The described approach is useful to estimate vibration transmission in ground as well as to analyze vibration of a road surface or rail-track loaded by moving vehicles.
Content available remote Filippov Lemma for certain differential inclusion of third order
We propose a version of the Filippov Lemma for differential inclusions of the type y'" + k2y' is an element of F(x,y) defined on [—1,1] with boundary conditions y(-1)=y(1)=y'(1)=0.
Content available remote Viability and generalized differential quotients
Necessary and sufficient conditions for a set-valued map K : R → Rn to be GDQ-differentiable are given. It is shown that K is GDQ differentiate at to if and only if it has a local multiselection that is Cellina continuously approximable and Lipschitz at to. It is also shown that any minimal GDQ of K at (to,yo) is a subset of the contingent derivative of K at (to,yo), evaluated at 1. Then this fact is used to prove a viability theorem that asserts existence of a solution to the initial value problem y(t) ∈ F(t, y(t)), with y(to) =yo, where F : Gr(K) → Rn is an orientor field (i.e. multivalued vector field) defined only on the graph of K and K : T → Rn is a time-varying constraint multifunction. One of the assumptions is GDQ differentiability of K.
Content available remote Stability of an elastic column subjected to non-stationary compressive loads
The problem of dynamic stability of an elastic column with pinned ends subjected to nonstationary compressive axial loads is considered. The method of optical Lyapunov functions for differential inclusions is applied to obtain sufficient conditions of stability of the column in the case of bounded loads. The obtained results, improving and generalising the classical solutions to the dynamic Euler problem, may be useful in designing civil engineering structures and mechanical systems consisting of compressed columns. The possibility of optimisation of the column characteristics with respect to its stability properties ( e. g. stability margins in the space of parameters) is pointed out.
Content available remote Optimal synthesis via superdifferentials of value function
We derive a differential inclusion governing the evolution of optimal trajectories to the Mayor problem. The value function is allowed to be discontinuous. This inclusion has convex compact right-hand sides.
Content available remote On necessary conditions in variable end-time optimal control problems
The paper contains a new necessary optimality condition for optimal control problems with free terminal time and discontinuous time dependence, which, actually is a family of conditions, each corresponding to a continuation of optimal trajectory beyond the optimal time for an arbitrary small interval.
Content available remote Inversion of multifunctions and differential inclusions
We present a new inverse mapping theorem for correspondences. It uses a notion of differentiability for multifunctions which seems to be new. We compare it with previous versions. We provide an application to differential inclusions.
Content available remote Relaxing constrained control systems
In this paper we provide a relaxation result for control systems under both equality and inequality constraints involving the state and the control. In particular we show that the Mangasarian-Fromowitz constraint qualification allows to rewrite constrained systems as differential inclusions with locally Lipschitz right-hand side. Then Filippov-Ważewski relaxation theorem may be applied to show that ordinary solutions are dense in the set of relaxed solutions. If, besided agreeing with the above constraints, the state has to remain in a control-independent set K, then we provide a condition on the feasible velocities on the boundary of K to get a relaxation theorem.
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