On the solvability of implicit differential inclusions

In the paper, the set-valued covering mappings are studied. The statements on solvability, solution estimates, and well-posedness of inclusions with conditionally covering mappings are proved. The results obtained are applied to the investigation of differential inclusions unsolved for the unknown function. The statements on solvability, solution estimates, and well-posedness of these inclusions are derived.

INDEX OF AN IMPLICIT DIFFERENTIAL EQUATION

In this paper we introduce the concept of the index of an implicit differential equation F(x,y,p) = 0, where F is a smooth function, p = dy/dx, F(p) = 0 and F(pp) = 0 at an isolated singular point. We also apply the results to study the geometry of surfaces in R(5).

Implicit differential equations with impasse singularities and singular perturbation problems

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Documentation for DFASUB--a program for the solution of simultaneous implicit differential and nonlinear equations,

Mode of access: Internet.

Invariants of binary differential equations

In this paper, we study binary differential equations a(x, y)dy (2) + 2b(x, y) dx dy + c(x, y)dx (2) = 0, where a, b, and c are real analytic functions. Following the geometric approach of Bruce and Tari in their work on multiplicity of implicit differential equations, we introduce a definition of the index for this class of equations that coincides with the classical Hopf`s definition for positive binary differential equations. Our results also apply to implicit differential equations F(x, y, p) = 0, where F is an analytic function, p = dy/dx, F (p) = 0, and F (pp) not equal aEuro parts per thousand 0 at the singular point. For these equations, we relate the index of the equation at the singular point with the index of the gradient of F and index of the 1-form omega = dy -aEuro parts per thousand pdx defined on the singular surface F = 0.

Necessary conditions of optimality for state constrained infinite horizon differential inclusions

This article presents and discusses necessary conditions of optimality for infinite horizon dynamic optimization problems with inequality state constraints and set inclusion constraints at both endpoints of the trajectory. The cost functional depends on the state variable at the final time, and the dynamics are given by a differential inclusion. Moreover, the optimization is carried out over asymptotically convergent state trajectories. The novelty of the proposed optimality conditions for this class of problems is that the boundary condition of the adjoint variable is given as a weak directional inclusion at infinity. This improves on the currently available necessary conditions of optimality for infinite horizon problems. © 2011 IEEE.

Description of the attainable sets of one-dimensional differential inclusions

The role played by the attainable set of a differential inclusion, in the study of dynamic control systems and fuzzy differential equations, is widely acknowledged. A procedure for estimating the attainable set is rather complicated compared to the numerical methods for differential equations. This article addresses an alternative approach, based on an optimal control tool, to obtain a description of the attainable sets of differential inclusions. In particular, we obtain an exact delineation of the attainable set for a large class of nonlinear differential inclusions.

A new approach for modeling and control of nonlinear systems via norm-bounded linear differential inclusions

A systematic approach to model nonlinear systems using norm-bounded linear differential inclusions (NLDIs) is proposed in this paper. The resulting NLDI model is suitable for the application of linear control design techniques and, therefore, it is possible to fulfill certain specifications for the underlying nonlinear system, within an operating region of interest in the state-space, using a linear controller designed for this NLDI model. Hence, a procedure to design a dynamic output feedback controller for the NLDI model is also proposed in this paper. One of the main contributions of the proposed modeling and control approach is the use of the mean-value theorem to represent the nonlinear system by a linear parameter-varying model, which is then mapped into a polytopic linear differential inclusion (PLDI) within the region of interest. To avoid the combinatorial problem that is inherent of polytopic models for medium- and large-sized systems, the PLDI is transformed into an NLDI, and the whole process is carried out ensuring that all trajectories of the underlying nonlinear system are also trajectories of the resulting NLDI within the operating region of interest. Furthermore, it is also possible to choose a particular structure for the NLDI parameters to reduce the conservatism in the representation of the nonlinear system by the NLDI model, and this feature is also one important contribution of this paper. Once the NLDI representation of the nonlinear system is obtained, the paper proposes the application of a linear control design method to this representation. The design is based on quadratic Lyapunov functions and formulated as search problem over a set of bilinear matrix inequalities (BMIs), which is solved using a two-step separation procedure that maps the BMIs into a set of corresponding linear matrix inequalities. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

Method of Averaging for Impulsive Differential Inclusions

AMS subject classification: Primary 49N25, Secondary 49J24, 49J25.

On Differential Inclusions with Unbounded Right-Hand Side

2000 Mathematics Subject Classification: 58C06, 47H10, 34A60.

Fixed point results for multivalued contractions on graphs and their applications

Nous présentons dans cette thèse des théorèmes de point fixe pour des contractions multivoques définies sur des espaces métriques, et, sur des espaces de jauges munis d’un graphe. Nous illustrons également les applications de ces résultats à des inclusions intégrales et à la théorie des fractales. Cette thèse est composée de quatre articles qui sont présentés dans quatre chapitres. Dans le chapitre 1, nous établissons des résultats de point fixe pour des fonctions multivoques, appelées G-contractions faibles. Celles-ci envoient des points connexes dans des points connexes et contractent la longueur des chemins. Les ensembles de points fixes sont étudiés. La propriété d’invariance homotopique d’existence d’un point fixe est également établie pour une famille de Gcontractions multivoques faibles. Dans le chapitre 2, nous établissons l’existence de solutions pour des systèmes d’inclusions intégrales de Hammerstein sous des conditions de type de monotonie mixte. L’existence de solutions pour des systèmes d’inclusions différentielles avec conditions initiales ou conditions aux limites périodiques est également obtenue. Nos résultats s’appuient sur nos théorèmes de point fixe pour des G-contractions multivoques faibles établis au chapitre 1. Dans le chapitre 3, nous appliquons ces mêmes résultats de point fixe aux systèmes de fonctions itérées assujettis à un graphe orienté. Plus précisément, nous construisons un espace métrique muni d’un graphe G et une G-contraction appropriés. En utilisant les points fixes de cette G-contraction, nous obtenons plus d’information sur les attracteurs de ces systèmes de fonctions itérées. Dans le chapitre 4, nous considérons des contractions multivoques définies sur un espace de jauges muni d’un graphe. Nous prouvons un résultat de point fixe pour des fonctions multivoques qui envoient des points connexes dans des points connexes et qui satisfont une condition de contraction généralisée. Ensuite, nous étudions des systèmes infinis de fonctions itérées assujettis à un graphe orienté (H-IIFS). Nous donnons des conditions assurant l’existence d’un attracteur unique à un H-IIFS. Enfin, nous appliquons notre résultat de point fixe pour des contractions multivoques définies sur un espace de jauges muni d’un graphe pour obtenir plus d’information sur l’attracteur d’un H-IIFS. Plus précisément, nous construisons un espace de jauges muni d’un graphe G et une G-contraction appropriés tels que ses points fixes sont des sous-attracteurs du H-IIFS.

Stability for impulsive control systems

In this paper we extend the notion of the control Lyapounov pair of functions and derive a stability theory for impulsive control systems. The control system is a measure driven differential inclusion that is partly absolutely continuous and partly singular. Some examples illustrating the features of Lyapounov stability are provided.

Necessary conditions of optimality for vector-valued impulsive control problems

A vector-valued impulsive control problem is considered whose dynamics, defined by a differential inclusion, are such that the vector fields associated with the singular term do not satisfy the so-called Frobenius condition. A concept of robust solution based on a new reparametrization procedure is adopted in order to derive necessary conditions of optimality. These conditions are obtained by taking a limit of those for an appropriate sequence of auxiliary standard optimal control problems approximating the original one. An example to illustrate the nature of the new optimality conditions is provided. © 2000 Elsevier Science B.V. All rights reserved.

Aeroelastic stability analysis using linear matrix inequalities

The present work describes an alternative methodology for identification of aeroelastic stability in a range of varying parameters. Analysis is performed in time domain based on Lyapunov stability and solved by convex optimization algorithms. The theory is outlined and simulations are carried out on a benchmark system to illustrate the method. The classical methodology with the analysis of the system's eigenvalues is presented for comparing the results and validating the approach. The aeroelastic model is represented in state space format and the unsteady aerodynamic forces are written in time domain using rational function approximation. The problem is formulated as a polytopic differential inclusion system and the conceptual idea can be used in two different applications. In the first application the method verifies the aeroelastic stability in a range of air density (or its equivalent altitude range). In the second one, the stability is verified for a rage of velocities. These analyses are in contrast to the classical discrete analysis performed at fixed air density/velocity values. It is shown that this method is efficient to identify stability regions in the flight envelope and it offers promise for robust flutter identification.

Espaços vetoriais e topológicos de intervalos generalizados com alguns conceitos de cálculo e otimização intervalar

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)