Existence of optimal controls on hybrid time domains

Hybrid control systems are considered, combining continuous-time dynamics and discrete-time dynamics, and modeled by differential equations or inclusions, by difference equations or inclusions, and by constraints on the resulting dynamics. Solutions are defined on hybrid time domains. Finite-horizon and infinite-horizon optimal control problems for such control systems are considered. Existence of optimal open-loop controls is shown. The assumptions used include, essentially, the existence for the (non-hybrid) continuous-time case; the existence for the (non-hybrid) discrete-time case; mild conditions on the endpoint penalties; and closedness and boundedness, in the finite-horizon case, of the set of admissible hybrid time domains. Examples involving switching systems and hybrid automata are included.     

Elliptic quadratic forms,focal points,and a generalized theory of oscillation

The purpose of this paper is to generalize the theory, methods, and results for oscillation of second-order normal ordinary differential equations. This purpose is obtained by use of a theory of quadratic forms on Hilbert spaces given by Hestenes and the author.In particular, the ideas of this paper may be applied to second-order abnormal problems of differential equations, higher-order control problems, integral and partial differential equations, abstract approximation problems, and to finite dimensional approximations which lead to meaningful computer algorithms.For expository purposes some examples are included. Finally we show that specific existence and comparison theorems for the second-order case may be generalized to the 2nth-order case.     

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.     

Period two implies chaos for a class of ODEs

We extend a result of J. Andres and K. Pastor concerning scalar time-periodic first order ordinary differential equations without uniqueness, by proving that the existence of just one subharmonic implies the existence of large sets of subharmonics of all given orders. Since these periodic solutions must coexist with complicated dynamics, we might paraphrase T. Y. Li and J. A. Yorke by loosely saying that in this setting even period two implies chaos. Similar results are obtained for a class of differential inclusions.

     

A Comparison between Two Theories for Multi-Valued Semiflows and Their Asymptotic Behaviour

This paper presents a comparison between two abstract frameworks in which one can treat multi-valued semiflows and their asymptotic behaviour. We compare the theory developed by Ball (1997) to treat equations whose solutions may not be unique, and that due to Melnik and Valero (1998) tailored more for differential inclusions. Although they deal with different problems, the main ideas seem quite similar. We study their relationship in detail and point out some essential technical problems in trying to apply Ball's theory to differential inclusions.     

On four-point boundary value problems for differential inclusions and differential equations with and without multivalued moving constraints

We deal with the problems of four boundary points conditions for both differential inclusions and differential equations with and without moving constraints. Using a very recent result we prove existence of generalized solutions for some differential inclusions and some differential equations with moving constraints. The results obtained improve the recent results obtained by Papageorgiou and Ibrahim-Gomaa. Also by means of a rather different approach based on an existence theorem due to O. N. Ricceri and B. Ricceri we prove existence results improving earlier theorems by Gupta and Marano.     

Problems of Thin Inclusions in a Two-Dimensional Viscoelastic Body

Under study are the equilibrium problems for a two-dimensional viscoelastic body with delaminated thin inclusions in the cases of elastic and rigid inclusions. Both variational and differential formulations of the problems with nonlinear boundary conditions are presented; their unique solvability is substantiated. For the case of a thin elastic inclusion modelled as a Bernoulli–Euler beam, we consider the passage to the limit as the rigidity parameter of the inclusion tends to infinity. In the limit it is the problem about a thin rigid inclusion. Relationship is established between the problems about thin rigid inclusions and the previously considered problems about volume rigid inclusions. The corresponding passage to the limit is justified in the case of inclusions without delamination.     

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.     

ON THE APPLICATIONS OF LERAY-SCHAUDER CONTINUATION THEOREM TO BOUNDARY VALUE PROBLEMS OF SEMILINEAR DIFFERENTIAL EQUATIONS

1IntroductionLeray-Schauderdegree,anditsvariantssuchascoinidencedegree113]andtopologicaltransversality[4],isanefficienttooltostudythesolvabilityofboundaryvalueproblemsfornonlineardifferentialequations,especiallyf'Orsemilineardifferentialequations.Bytherec…     

An oscillation theory for second-order integral differential equations

The primary purpose of this paper is to give an oscillation theory for second-order integral differential equations. It is shown that this theory follows in a natural way as “a corollary” from the more abstract approximation theory of quadratic forms given previously by the author. Thus, our ideas are primarily constructive and quantitative as opposed to the usual qualitative methods. We also note that the usual oscillation theory for second-order differential equations follows directly by our methods. Furthermore, our methods provide a unified theory for eigenvalue problems, optimization problems, and numerical approximation problems within this setting.In Section 1 we give the preliminaries for the remainder of the paper. In Section 2 we define the basic quadratic form and integral differential equation and give the relationships between them. These relationships are used (in Section 3) to give a theory of oscillation in our setting and some basic oscillation results. Finally, in Section 4 we give some deeper oscillation results.To emphasize the unifying methods of our ideas, this paper is presented as a companion paper to “A Numerical Approximation Theory for Second Order Integral Differential Equations.”     

Global existence of solutions for perturbed differential equations

In this paper we consider sufficient conditions for the continuability of solutions for perturbed differential equations. We obtain also some results for the global existence of solutions for differential inclusions and for stochastic differential equations of McShane and Ito type. We give an application to the global inversion of local diffeomorphisms.     

Phase spaces and periodic solutions of set functional dynamic equations with infinite delay

Very recently, a new theory known as set dynamic equations on time scales has been built. In this paper, a phase space is built for set functional dynamic equations with infinite delay on time scales and sufficient criteria are established for the existence of periodic solutions of such equations, which generalize and incorporate as special cases some known results for set differential equations and for set difference equations when the time scale is the real number set or the integer set, respectively, moreover, for differential inclusions and difference inclusions if the variable under consideration is a single valued mapping. Our results show that one can unify the study of some continuous or discrete problems in the sense of (set) dynamic equations on general time scales.     

Asymptotic boundary value problems in Banach spaces

A continuation principle is given for solving boundary value problems on arbitrary (possibly infinite) intervals to Carathéodory differential inclusions in Banach spaces. For this aim, the appropriate fixed point index is defined to condensing decomposable multivalued operators in Fréchet spaces. This index extends and unifies the one for compact maps in Andres et al. [Trans. Amer. Math. Soc. 351 (1999) 4861-4903] as well as the one for operators in Banach spaces in Bader [Ph.D. Thesis, University of Munich, 1995]. As an application, we prove the existence of an entirely bounded solution of a semilinear evolution inclusion.     

Some questions of qualitative theory in dynamics of systems with the variable dissipation

In this work, we consider some problems of the qualitative theory of ordinary differential equations; the study of dissipative systems, as well as variable dissipation system considered below, which, in particular, arise in the dynamics of a rigid body interacting with a medium and in the oscillation theory, depends on solutions of these problems. We consider such problems as existence and uniqueness problems for trajectories having infinitely remote points as limit sets for systems on the plane, elements of qualitative theory of monotone vector fields, and also existence problems for families of long-period and Poisson stable trajectories. In conclusion, we study the possibility of extending the Poincar´e two-dimensional topographical system and the comparison system to the many-dimensional case.     

Nontrivial application of Nielsen theory to differential systems

In reply to a problem of Jean Leray concerning application of the Nielsen theory to differential systems for obtaining multiplicity results, we present a nontrivial example of such an application. The emphasis is on the parameter space in order to ensure that no subdomain becomes subinvariant under the related Hammerstein solution operator. To achieve this goal, we develop a general method applicable also for ordinary differential equations with or without uniqueness as well as for upper-Carathéodory differential inclusions. We are not aware that any alternative approach can be employed, even in the single-valued case.     

Boundary value problems for second order nonlinear differential equations on infinite intervals

In this paper, we consider boundary value problems for nonlinear differential equations on the semi-axis (0,∞) and also on the whole axis (−∞,∞), under the assumption that the left-hand side being a second order linear differential expression belongs to the Weyl limit-circle case. The boundary value problems are considered in the Hilbert spaces L²(0,∞) and L²(−∞,∞), and include boundary conditions at infinity. The existence and uniqueness results for solutions of the considered boundary value problems are established.     

Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions

In this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations and the topological structure of the solutions set are considered too.     

A Survey on Existence Results for Boundary Value Problems of Nonlinear Fractional Differential Equations and Inclusions

In this survey paper, we shall establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative. The both cases of convex and nonconvex valued right hand side are considered. The topological structure of the set of solutions is also considered.