These results are new and can be considered as an extension of many known ones in the literature for the classical nonlinear case. Our approach is based on tools from variational analysis, where the metric regularity concept plays an important role in our analysis. Nonsmooth Lyapunov pairs for differential inclusions goverened by operators with nonempty interior domain, with A. Hantoute and M.
This new contribution focuses on the natural case when the maximally monotone operator governing the given inclusion has a domain with nonempty interior. This setting permits to have nonincreasing Lyapunov functions on the whole trajectory of the solution to the given differential inclusion. It also allows some more explicit criteria for Lyapunov's pairs. Our analysis makes use of standard tools from convex and variational analysis. Trang and Vu N. Based on the Lyapunov functional method and a generalized Jensen integral inequality, novel delay-dependent conditions for the existence of output feedback controllers are established in terms of linear matrix inequalities LMIs.
The proposed conditions allow us to design the output feedback controllers which robustly stabilize the closed-loop system in the finite-time sense. A numerical example is given to illustrate the efficiency of the proposed method. Trang and V. By introducing a set of improved Lyapunov-Krasovskii functionals, a novel delay-dependent condition for output feedback guaranteed cost control with guar- anteed exponential stability is derived in terms of linear matrix inequalities LMIs.
Then, a design method of robust guaranteed cost control via output feedback controller is applied for uncertain linear systems. The design of output feedback controllers can be carried out in a systematic and computationally efficient manner via the use of LMI-based algorithms. Numerical examples are included to illustrate the effectiveness of the obtained result.
First online: 04 July Abstract Abstract: In this paper, we study the well-posedness and stability analysis of set-valued Lur'e dynamical systems in infinite-dimensional Hilbert spaces. The existence and uniqueness results are established under the so-called passivity condition.
Our approach uses a regularization procedure for the term involving the maximal monotone operator. The Lyapunov stability as well as the invariance properties are considered in detail. In addition, we give some sufficient conditions ensuring the robust stability of the system in finite-dimensional spaces.
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The theoretical developments are illustrated by means of some examples dealing with nonregular electrical circuits. Our methodology is based on tools from set-valued and variational analysis. Newton's method for solving inclusions using set-valued approximations, with R. Cibulka and H. As an application, we study super- linear convergence of the Newton-type iterative process for solving generalized equations.
The possibility to choose set-valued approximations allows us to describe several iterative schemes in a unified way such as inexact Newton method, non-smooth Newton method for semi-smooth functions, inexact proximal point algorithm, etc. Moreover, it also covers a forward-backward splitting algorithm for finding a common zero of the sum of two multivalued not necessarily monotone operators.
Finally, a globalization of the Newton's method is discussed. Rammal Journal of Optimization Theory and Applications, , no.
Theory and Applications
In this paper, we extend EiCP to problem where the nonnegative orthant, i. We reformulate such problem to find the roots of a semismooth function. Surprisngly, this kind of subject has never been studied before due to the difficulty of this problem in the sense that the Lorentz spectrum is not always finite. On one-sided Lipschitz stability of set-valued contractions, with A. Dontchev and M. Lim [On fixed-point stability for set-valued contractive mappings with applications to generalized differential equations, J.
A global version of the Lyusternik-Graves theorem is a corollary of this estimate as well.
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We apply the generalization of Lim's result to derive one-sided Lipschitz properties of the solution mapping of a differential inclusion with a parameter. Quantitative stability of a generalized equation. Application to non-regular electrical circuits, with R. The theoretical results are applied in the theory of non-regular electrical circuits involving electronic devices like ideal diode, practical diode, and DIACs DIode Alternating Current.
Thibault , Mathematical Programming, , no.
Napsu Karmitsa - Nonsmooth Optimization (NSO)
B, pages: 5—47 Abstract Abstract: In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by J. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation.
The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like the other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by the Moreau's seminal work. Le Applicable Analysis, 93 , no.
Many recent researches deal with the case when the set-valued part is the subdifferential of some proper, convex, lower semicontinuous function in order to use the nice properties of maximally monotone operators.
But in practice, particularly in electronics, there are some devices such as diac, silicon controller rectifier SCR This fact motivates us to write the paper which is organized as follows: firstly the existence and uniqueness of solutions are proved by using Filippov's method and local hypo-monotonicity; then the stability analysis and generalized LaSalle's invariance principle are presented. The theoretical results are supported by numerical simulations for some examples in electronics. Our methology is based on nonsmooth and variational analysis.
Numerical Functional Analysis and Optimization, 35 , no. Under appropriate assumptions, for such a problem we give necessary and sufficient optimality conditions via reverse convex problems. SpringerBriefs in Mathematics. Available on February About this book About this book: This brief examines mathematical models in nonsmooth mechanics and nonregular electrical circuits, including evolution variational inequalities, complementarity systems, differential inclusions, second-order dynamics, Lur'e systems and Moreau's sweeping process. The field of nonsmooth dynamics is of great interest to mathematicians, mechanicians, automatic controllers and engineers.
Full Professor in Mathematics
The present volume acknowledges this transversality and provides a multidisciplinary view as it outlines fundamental results in nonsmooth dynamics and explains how to use them to study various problems in engineering. In particular, the author explores the question of how to redefine the notion of dynamical systems in light of modern variational and nonsmooth analysis.
With the aim of bridging between the communities of applied mathematicians, engineers and researchers in control theory and nonlinear systems, this brief outlines both relevant mathematical proofs and models in unilateral mechanics and electronics. Submitted Papers Well-Posedness of nonconvex degenerate sweeping process via unconstrained evolution problems, with T. Refereed Publications A coderivative approach to the robust stability of composite parametric variational systems. First the existence and uniqueness of solutions are analyzed, then the Lyapunov stability, the Krasovskii-LaSalle invariance principle and finite-time convergence properties are studied.
Rammal , Computational Optimization and Applications 55, No 3, pp Abstract Abstract: In this paper, we introduce a new method, called the Lattice Projection Method LPM , for solving eigenvalue complementarity problems. The original problem is reformulated to find the roots of a nonsmooth function. A semismooth Newton type method is then applied to approximate the eigenvalues and eigenvectors of the complementarity problems.
The performance measures, used to analyze the three solvers on a set of matrices mostly taken from the Matrix Market, are computing time, number of iterations, number of failures and maximum number of solutions found by each solver. The numerical experiments highlight the efficiency of the LPM and show that it is a promising method for solving eigenvalue complementarity problems. Finally, Pareto bi-eigenvalue complementarity problems were solved numerically as an application to confirm the efficiency of our method.
Variational Analysis and Generalized Equations in Electronics.
Stability and Simulation Issues, with R. Massias , Set-Valued and Variational Analysis 21 , no. The variational and non-smooth analysis is applied in the theory of non-regular electrical circuits involving electronic devices like ideal diodes, practical diodes, DIACs, silicon controlled rectifiers SCR , and transistors.
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We also discuss the relationship of our results to the ones using classical techniques from smooth analysis and provide a simulation for several simple electrical circuits which are chosen in order to cover the most common non-smooth elements in electronics. The simulations of the electrical circuits discussed in this paper are performed by using Xcos a component of Scilab.
On some dynamic thermal non clamped contact problems, with O. Chau , Mathematical Programming serie B, No , pp Abstract Abstract: We study a class of dynamic thermal sub-differential contact problems with friction, for long memory visco-elastic materials, without the clamped condition, which can be put into a general model of system defined by a second order evolution inequality, coupled with a first order evolution equation. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods.
Finally a fully discrete scheme for numerical approximations is provided, and corresponding various numerical computations in dimension two will be given. Application in Electronics, with J. Outrata , Journal of Convex Analysis 20, 1 Abstract Abstract: The main concern of this paper is to investigate some stability properties namely Aubin property and isolated calmness of a special non-monotone variational inclusion. We provide a characterization of these properties in terms of the problem data and show their importance for the design of electrical circuits involving nonsmooth and non-monotone electronic devices like DIAC DIode Alternating Current.
Regularity Concepts in Nonsmooth Analysis
Circuits with other devices like SCR Silicon Controlled Rectifiers , Zener diodes, thyristors, varactors and transistors can be analyzed in the same way. Goeleven and B. We prove the existence of trajectories for the model.