Relaxation Theorem Research Articles

Control problems governed by time-dependent maximal monotone operators

The paper concerns on an infinite dimensional Hilbert space, the existence and uniqueness of absolutely continuous solutions, for Lipschitz single-valued perturbations of evolution problems involving maximal-monotone operators. This result allows us to extend to optimal control problems associated with such equations, the relaxation theorems with Young measures proved recently in [S. Saidi, L. Thibault and M.F. Yarou, Numer. Funct. Anal. Optim. 34 (2013) 1156–1186].

Periodic solutions for time-dependent subdifferential evolution inclusions

We consider evolution inclusions driven by a time dependent subdifferential plus a multivalued perturbation. We look for periodic solutions. We prove existence results for the convex problem (convex valued perturbation), for the nonconvex problem (nonconvex valued perturbation) and for extremal trajectories (solutions passing from the extreme points of the multivalued perturbation). We also prove a strong relaxation theorem showing that each solution of the convex problem can be approximated in the supremum norm by extremal solutions. Finally we present some examples illustrating these results.

Subdifferential inclusions with unbounded perturbation: Existence and relaxation theorems

The paper studies an evolution inclusion in a separable Hilbert space whose right-hand side contains the subdifferential of a proper convex lower semicontinuous function of time and a set-valued perturbation. Together with this inclusion, an inclusion with convexified perturbation values is considered. The existence and density of the solution set of the initial inclusion in the closure of the solution set of the inclusion with convexified perturbation are proved. This property is usually called relaxation. Traditional assumptions for relaxation theorems are the compactness property of the convex function and the boundedness of the perturbation. In the present paper, such assumptions are not made. Assumptions for subdifferential inclusions described by polyhedral sweeping processes and variational inequalities with time-dependent obstacles and constraints are specified.

Solutions of evolution inclusions generated by a difference of subdifferentials

An evolution inclusion with the right-hand side containing the difference of subdifferentials of proper convex lower semicontinuous functions and a multivalued perturbation whose values are nonconvex closed sets is considered in a separable Hilbert space. In addition to the original inclusion, we consider an inclusion with convexified perturbation and a perturbation whose values are extremal points of the convexified perturbation that also belong to the values of the original perturbation. Questions of the existence of solutions under various perturbations are studied and relations between solutions are established. The primary focus is on the weakening of assumptions on the perturbation as compared to the known assumptions under which existence and relaxation theorems are valid. All our assumptions, in contrast to the known assumptions, concern the convexified rather than original perturbation.

Dissipation in monotonic and non-monotonic relaxation to equilibrium.

Using molecular dynamics simulations, we study field free relaxation from a non-uniform initial density, monitored using both density distributions and the dissipation function. When this density gradient is applied to colour labelled particles, the density distribution decays to a sine curve of fundamental wavelength, which then decays conformally towards a uniform distribution. For conformal relaxation, the dissipation function is found to decay towards equilibrium monotonically, consistent with the predictions of the relaxation theorem. When the system is initiated with a more dramatic density gradient, applied to all particles, non-conformal relaxation is seen in both the dissipation function and the Fourier components of the density distribution. At times, the system appears to be moving away from a uniform density distribution. In both cases, the dissipation function satisfies the modified second law inequality, and the dissipation theorem is demonstrated.

Generalized Solutions of Semilinear Evolution Inclusions

In this paper we study different types of (generalized) solutions for semilinear evolution inclusions in general Banach spaces, called limit and weak solutions, which are extensions of the weak solutions studied by T. Donchev [Nonlinear Anal., 16 (1991), pp. 533--542] and the directional solutions studied by J. Tabor [Set-Valued Anal., 14 (2006), pp. 121--148]. Under appropriate assumptions, we show that the set of the limit solutions is compact $R_\delta$. When the right-hand side satisfies the one-sided Perron condition, a variant of the well-known lemma of Filippov--Plis, as well as a relaxation theorem, are proved.

A Derivation of the Gibbs Equation and the Determination of Change in Gibbs Entropy from Calorimetry

In this paper, we give a succinct derivation of the fundamental equation of classical equilibrium thermodynamics, namely the Gibbs equation. This derivation builds on our equilibrium relaxation theorem for systems in contact with a heat reservoir. We reinforce the comments made over a century ago, pointing out that Clausius’ strict inequality for a system of interest is within Clausius’ set of definitions, logically undefined. Using a specific definition of temperature that we have recently introduced and which is valid for both reversible and irreversible processes, we can define a property that we call the change in calorimetric entropy for these processes. We then demonstrate the instantaneous equivalence of the change in calorimetric entropy, which is defined using heat transfer and our definition of temperature, and the change in Gibbs entropy, which is defined in terms of the full N-particle phase space distribution function. The result shows that the change in Gibbs entropy can be expressed in terms of physical quantities.

Differential inclusions with unbounded right-hand side: Existence and relaxation theorems

A differential inclusion in which the values of the right-hand side are nonconvex closed possibly unbounded sets is considered in a finite-dimensional space. Existence theorems for solutions and a relaxation theorem are proved. Relaxation theorems for a differential inclusion with bounded right-hand side, as a rule, are proved under the Lipschitz condition. In our paper, in the proof of the relaxation theorem for the differential inclusion, we use the notion of ρ - H Lipschitzness instead of the Lipschitzness of a multivalued mapping.

Structure of the solution set for a partial differential inclusion

In this paper, we consider the biharmonic problem of a partial differential inclusion with Dirichlet boundary conditions. We prove existence theorems for related partial differential inclusions with convex and nonconvex multivalued perturbations, and obtain an existence theorem on extremal solutions, and a strong relaxation theorem. Also we prove that the solution set is compact $R_{\delta}$ if the perturbation term of the related partial differential inclusion is convex, and its solution set is path-connected if the perturbation term is nonconvex.

Periodic solutions for nonlinear differential inclusions with multivalued perturbations

In this paper, the periodic solutions for nonlinear differential inclusion governed by convex subdifferential and different perturbations are studied. It is firstly proved that the differential inclusion has unique periodic solution, if the perturbation function is a single-valued function. Then, by Schauder's fixed point theorem and Kakutani's fixed point theorem, we prove that the differential inclusion has at least a periodic solution, when the perturbation function is an upper semicontinuous (or lower semicontinuous) multifunction. Moreover, the existence of the extremal solution for the differential inclusion is also studied. Finally, based on one-sided Lipschitz (OSL) assumption, we prove the related relaxation theorem.

A local proof for the characterization of Young measures generated by sequences in BV

This work presents a new proof of the recent characterization theorem for generalized Young measures generated by sequences in BV by Kristensen and Rindler (2010) [14]. The present argument is based on a localization technique together with a local Hahn–Banach argument in novel function spaces combined with an application of Alberti's Rank-One Theorem. This strategy avoids employing a relaxation theorem as in the previously known proof, and the new tools introduced in its course should prove useful in other contexts as well. In particular, we introduce “homogeneous” Young measures, separately at regular and singular points, which exhibit rather different behavior than the classical homogeneous Young measures. As an application, we show how for BV-Young measures with an “atomic” part one can find a generating sequence respecting this structure.

A LYAPUNOV CHARACTERIZATION OF ASYMPTOTIC CONTROLLABILITY FOR NONLINEAR SWITCHED SYSTEMS

In this paper, we show that general nonlinear switched systems are asymptotically controllable if and only if there exist control-Lyapunov functions for their relaxation systems. If the switching signal is dependent on the time, then the control-Lyapunov functions are continuous. And if the switching signal is dependent on the state, then the control-Lyapunov functions are <TEX>$C^1$</TEX>-smooth. We obtain the results from the viewpoint of control system theory. Our approach is based on the relaxation theorems of differential inclusions and the classic Lyapunov characterization.

Differential inclusions with measurable-pseudo-Lipschitz right-hand side

We obtain existence theorems and Filippov-Wazewski type relaxation theorems for differential inclusions in Banach spaces with measurable-pseudo-Lipschitz right-hand side. For the solution sets of these differential inclusions, we also describe some properties that extend classical theorems on continuous dependence and on differentiation of solutions with respect to initial data.

Existence of Solutions for a Class of Nonlinear Evolution Inclusions with Nonlocal Conditions

This paper deals with the nonlocal problems for a class of nonlinear first-order evolution inclusions. Some existence results are established for the cases of a convex and of a nonconvex valued perturbation terms. Also, the existence of extremal solutions and a strong relaxation theorem are obtained. Subsequently a nonlinear hyperbolic optimal control problem is considered and the existence theorems based on the proven results are obtained. Then the nonlinear version of "bang---bang" principle for control systems is given as well by utilizing the relaxation theorem.

Relaxation of Optimal Control Problems Involving Time Dependent Subdifferential Operators

This article provides in the setting of infinite dimensional Hilbert space, a result concerning the existence and uniqueness of solutions for Lipschitz single-valued perturbations of evolution problems associated with time-dependent subdifferential operators. The result is used to extend to optimal control problems associated with such equations the relaxation theorems with Young measures established in Casting et al. [7, 11] and Edmond and Thibault [11].

Properties of the solutions set for a class of nonlinear evolution inclusions with nonlocal conditions

In this paper, we consider the nonlocal problems for nonlinear first-order evolution inclusions in an evolution triple of spaces. Using techniques from multivalued analysis and fixed point theorems, we prove existence theorems of solutions for the cases of a convex and of a nonconvex valued perturbation term with nonlocal conditions. Also, we prove the existence of extremal solutions and a strong relaxation theorem. Some examples are presented to illustrate the results. MSC:34B15, 34B16, 37J40.

One-sided Perron Differential Inclusions

The main qualitative properties of the solution set of almost lower (upper) semicontinuous one-sided Perron differential inclusion with state constraints in finite dimensional spaces are studied. Using the technique introduced by Veliov (Nonlinear Anal 23:1027–1038, 1994) we give sufficient conditions for the solution map of the above state constrained differential inclusion to be continuous in the sense of Hausdorff metric. An application on the propagation of the continuity of the state constrained minimum time function associated with the nonautonomous differential inclusion and the target zero is given. Some relaxation theorems are proved, which are used afterward to derive necessary and sufficient conditions for invariance.

Localization principle and relaxation

Relaxation theorems for multiple integrals on , where , are proved under general conditions on the integrand which is Borel measurable and not necessarily finite. We involve a localization principle that we previously used to prove a general lower semicontinuity result. We apply these general results to the relaxation of nonconvex integrals with exponential-growth.

A RELAXATION RESULT FOR AN INHOMOGENEOUS FUNCTIONAL PRESERVING POINT-LIKE AND CURVE-LIKE SINGULARITIES IN IMAGE PROCESSING

In this paper we address a relaxation theorem for a new integral functional of the calculus of variations defined on the space of functions in [Formula: see text] whose gradient is an Lp-vector field with distributional divergence given by a Radon measure. The result holds for integrand of type f(x, Δu) without any coerciveness condition, with respect to the second variable, and C1-continuity assumptions with respect to the spatial variable x.

A Proof of the Relaxation Theorem for Differential Inclusions Based on Euler Approximations

In this note, an alternative proof of the relaxation theorem is presented that is based on a two-level Euler approximation.