Asymptotic behavior of extremal solutions and structure of extremal norms of linear differential inclusions of order three

Asymptotic properties of extremal solutions of linear inclusions of order three with zero Lyapunov exponent are investigated. Under certain conditions it is shown that all extremal solutions of such inclusions tend to the same (up to a multiplicative factor) solution, which is central symmetric. The structure of the convex set of extremal norm is studied. A number of extremal points of this set are described.     

Optimal solutions to differential inclusions

We treat a control problem given in terms of a differential inclusion $$\dot x(t) \in E(t,x(t))$$ and develop necessary conditions for a minimum in the problem. These conditions are given in terms of certain normals to arbitrary closed sets, and require no smoothness or convexity in the problem. The results subsume related works that incorporate convexity assumptions.     

Semiconcave solutions of partial differential inclusions

We obtain new results on the propagation of singularities for semiconcave solutions of partial differential inclusions. These results will be used to study the behavior of singularities of the value function for a reflected control problem.     

Approximation of slow solutions to differential inclusions

To approach a viable solution of a differential inclusion, i.e., staying at any time in a closed convexK, a sufficient condition is given implying the convergence of an approximation sequence defined from the Euler or Runge-Kutta methods applied to a selection process which corresponds to the slowsolution concept. WhenK is smooth, the convergence condition is satisfied. This proves that the method is implementable on a computer for solving, for instance, differentiable equations with a noncontinuous right-hand side. Since the usual best approximation operator is difficult to implement, we introduce a class of quasi-projectors much more suitable for computation.     

On local solutions of degenerate differential inclusions

We study the existence and properties of local solution sets for differential inclusions of the form (Ax)′ ∈ F(t, x), where A is a closed linear surjective operator with nontrivial null space and F is a compact set-valued mapping.     

On generalized-periodic solutions of nonautonomous differential inclusions

We introduce the notion of a generalized-periodic solution of a classical nonautonomous differential inclusion with periodic right-hand side. We show that the existence of a bounded solution implies the existence of a generalized-periodic solution, and vice versa.     

Periodic solutions of linear impulsive differential inclusions

We establish sufficient conditions for the existence of periodic R-solutions of linear differential inclusions with impulses at fixed times. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1287–1296, September, 2008.     

Bifurcation of periodic solutions in differential inclusions

Ordinary differential inclusions depending on small parameters are considered such that the unperturbed inclusions are ordinary differential equations possessing manifolds of periodic solutions. Sufficient conditions are determined for the persistence of some of these periodic solutions after multivalued perturbations. Applications are given to dry friction problems.     

Some remarks on extremal solutions of multivalued differential equations

In this paper, we consider extremal solutions of multivalued differential equations, i.e., solutions that steer to the boundary of the attainable set. Multivalued differential equations arise in a natural way from control systems governed by ordinary differential equations that have a variable control-constraint set. Extremal solutions of multi-valued differential equations are important in the study of the optimal control of such systems. We give conditions under which extremality of a solution at a certain time implies extremality of the solution at all previous times where it is defined. Necessary conditions for extremality are also obtained. We treat both the time-dependent case and the time-independent case.     

On the existence of solutions of differential inclusions

We consider differential inclusions corresponding to accretive operators in a Banach space X for the case in which X is the one-dimensional Euclidean space. We prove existence theorems and an asymptotic stability theorem. We also introduce the notion of a generalizednonincreasing (nondecreasing) multivalued function and establish a relationship between nondecreasing multivalued functions and accretive operators in the one-dimensional Euclidean space.     

Brownian motion and exposed solutions of differential inclusions

We present a research program designed by A. Bressan and some partial results related to it. First, we construct a probability measure supported on the space of solutions to a planar differential inclusion, where the right-hand side is a Lipschitz continuous segment. Such measure assigns probability one to solutions having derivatives a.e. equal to one of the endpoints of the segment. Second, for a class of planar differential inclusions with Hölder continuous right-hand side F, we prove existence of solutions whose derivatives are exposed points of F. Finally, we complete the research program if the right-hand side of the differential inclusion does not depend on the state and prove a result on the Lipschitz continuity of an auxiliary map. The proofs rely on basic properties of Brownian motion.     

Discrete approximations and periodic solutions of differential inclusions

We consider two numerical methods for solving a periodic boundary value problem for a system of differential inclusions, the Galerkin method and the polygon method. To the original problem, we assign a sequence of its discretizations. Conditions under which the existence of solutions of the periodic boundary value problem implies the solvability of its discrete versions are presented. The convergence of the sequence of approximate solutions is analyzed.     

Remarks on extremal solutions of differential equations with advanced argument

We show the existence of absolutely continuous extremal solutions to the problemx′(t)=f(t, x)h(t)))+g(t)),x(0)=x ₀, whereh is an arbitrary continuous deviated argument. Conditions for the uniqueness of solutions are given. Research partialy supported by grant UG BW 5100 - 5 - 0143 - 4     

Periodic solutions to impulsive differential inclusions with constraints

The existence of a periodic solution to an impulsive differential inclusion being invariant with respect to a non-convex set of state constraints is established by the use a Lefschetz type fixed-point theorem for set-valued maps.     

On the stability of solutions of stochastic differential inclusions

We study the stability in probability of a solution of a stochastic differential inclusion in a finite-dimensional space with nonrandom coefficients and a maximally monotone operator in the drift coefficient.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 551–554, April, 1995.     

On the approximation of solutions of stochastic differential inclusions

A sequence of approximating equations is constructed for stochastic differential inclusions, and the properties of the measures corresponding to solutions of the approximating equations are studied for the class of stochastic differential inclusions.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 43–48, 1986.