Relaxation Theorem Research Articles

A relaxation theorem for a first-order set differential inclusion in a metric space

Abstract The aim of this article is to present, in a specific metric space, a density theorem. Then, as an application, we give a relaxation theorem for a first-order set-differential inclusion, through an abstract convexity notion.

On the Filippov-Ważewski relaxation theorem for a certain class of fractional differential inclusions

The purpose of this text is to propose an attempt of an extension of the Filippov-Ważewski Relaxation Theorem for a certain class of fractional differential inclusions. The classical result devoted to ordinary differential inclusions is a part of the qualitative theory: a description of the relationship between the solutions to the differential inclusion and the convexified differential inclusion was given. There exist several generalizations of that result and in this note a method is proposed so that the range of some parameters is extended. An example of a possible application may arise in control theory and the question is whether it is possible to have the same reachable set economizing the set of controls.

Relaxation Theorem for Stieltjes Differential Inclusions on Infinite Intervals

For a very general first-order differential problem on an infinite-time horizon involving the Stieltjes derivative with respect to a left-continuous non-decreasing function g:[0,infty )rightarrow mathbb {R}1xg′(t)∈F(t,x(t)),t∈[0,∞)x(0)=x0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} x'_g(t) \\in F(t,x(t)),\\; t\\in [0,\\infty )\\\\ x(0)=x_0, \\end{array} \\right. \\end{aligned}$$\\end{document}we study the possibility to approximate the solutions of the convexified inclusion by the solutions of the non-convexified problem. Via a generalization to this framework of a classical result concerning continuous selection of trajectory, we thus present a relaxation theorem which states that, similarly to the setting of usual differential inclusions, the approximation can be achieved once we allow to the initial value to differ (but remaining close to) from the initial value of the considered solution of the relaxed inclusion.

Relaxation result for differential inclusions with Stieltjes derivative

The aim of this paper is to provide a Filippov-Ważewski Relaxation Theorem for the very general setting of Stieltjes differential inclusions. New relaxation results can be deduced for generalized differential problems, for impulsive differential inclusions with multivalued impulsive maps and possibly countable impulsive moments and also for dynamic inclusions on time scales.

How to tighten the control set?

We examine the problem of reducing the control set in a dynamical system so that the solution set and the attainable sets remain essentially unchanged. We cite some classical results to exhibit the problem and promote a set-valued approach. A necessary condition is formulated by using a concept of set-valued derivative, which can be regarded as an extension of the classical Relaxation theorem.

Second-order problems involving time-dependent subdifferential operators and application to control

<p style='text-indent:20px;'>The paper provides a new result concerning the existence of solutions for second-order evolution problems associated with time-dependent subdifferential operators involving both single-valued and mixed semi-continuous set-valued perturbations. Optimal control problems corresponding to such differential inclusions using relaxation theorems with Young measures are investigated. The existence of solutions for a coupled system governed by a second-order differential equation with an evolution problem is also addressed.</p>

Nonlocal semilinear evolution inclusions with time lag

In this paper, nonautonomous semilinear evolution inclusions with time lag and nonlocal initial conditions are studied. We define the so-called limit and weak solutions. We prove the existence of solutions when the multivalued part of the right-hand side and the initial functions are Lipschitz under natural conditions on the Lipschitz constants. Afterwards, under the same conditions, a variant of the important in optimal control relaxation theorem is proved.

Integral representation and relaxation of local functionals on Cheeger–Sobolev spaces

We prove an integral representation theorem for local functionals with polynomial growth defined on Cheeger–Sobolev spaces. More precisely, we give a version of the well-known Buttazzo–Dal maso’s integral representation theorem in the framework of Cheeger–Sobolev spaces. The integral representation theorem is used to prove a relaxation theorem.

Existence and Relaxation Results for Second Order Multivalued Systems

We consider nonlinear systems driven by a general nonhomogeneous differential operator with various types of boundary conditions and with a reaction in which we have the combined effects of a maximal monotone term A(x) and of a multivalued perturbation F(t,x,y) which can be convex or nonconvex valued. We consider the cases where D(A)neq mathbb{R}^{N} and D(A)= mathbb{R}^{N} and prove existence and relaxation theorems. Applications to differential variational inequalities and control systems are discussed.

Existence and relaxation of solutions for evolution differential inclusions with maximal monotone operators

In this paper, we consider evolution problems involving time-dependent maximal monotone operators in Hilbert spaces. Existence and relaxation theorems are proved.

Differential inclusions in Wasserstein spaces: The Cauchy-Lipschitz framework

In this article, we propose a general framework for the study of differential inclusions in the Wasserstein space of probability measures. Based on earlier geometric insights on the structure of continuity equations, we define solutions of differential inclusions as absolutely continuous curves whose driving velocity fields are measurable selections of multifunction taking their values in the space of vector fields. In this general setting, we prove three of the founding results of the theory of differential inclusions: Filippov's theorem, the Relaxation theorem, and the compactness of the solution sets. These contributions – which are based on novel estimates on solutions of continuity equations – are then applied to derive a new existence result for fully non-linear mean-field optimal control problems with closed-loop controls.

Tilting instability of magnetically confined spheromaks

We consider the tilting instability of a magnetically confined spheromak using three-dimensional magnetohydrodynamic and relativistic particle-in-cell calculations with an application to astrophysical plasmas, specifically those occurring in magnetar magnetospheres. The instability is driven by the counter-alignment of the spheromak's intrinsic magnetic dipole with the external magnetic field. Initially, the spheromak rotates – tilts – trying to lower its magnetic potential energy. As a result, a current sheet forms between the internal magnetic field of a spheromak and the confining field. Magnetic reconnection sets in; this leads to the annihilation of the newly counter-aligned magnetic flux of the spheromak. This occurs on a few Alfvén time scales. In the case of a higher-order (second-order) spheromak, the internal core is first pushed out of the envelope, resulting in formation of two nearly independent tilting spheromaks. Thus, the magnetically twisted outer shell cannot stabilize the inner core. During dissipation, helicity of the initial spheromak is carried away by torsional Alfvén waves, violating the assumptions of the Taylor relaxation theorem. In applications to magnetar giant flares, fast development of tilting instabilities and no stabilization of the higher-order spheromaks make it unlikely that trapped spheromaks are responsible for the tail emission lasting hundreds of seconds.

Local existence conditions for sweeping process solutions

A sufficient condition for the existence of an absolutely continuous solution for a sweeping process is given by the absolute continuity, in a definite sense, of the multivalued mapping which generates the process. This property is described in terms of the Hausdorff distance between values of the multivalued mapping. However, there exist multivalued mappings for which the Hausdorff distance between those values is infinite; for instance, mappings which take hyperplanes as values. For such mappings absolute continuity cannot be described in terms of the Hausdorff distance. In this paper we study conditions which provide local absolute continuity of a multivalued mapping. By using these conditions we prove an existence theorem for an absolutely continuous solution of a sweeping process. We apply the results obtained to the study of sweeping processes with nonconvex and with convexified perturbations. For such sweeping processes we prove an existence theorem for solutions and a relaxation theorem. Bibliography: 13 titles.

Nonlinear vector Duffing inclusions with no growth restriction on the orientor field

We consider nonlinear multivalued Dirichlet Duffing systems. We do not impose any growth condition on the multivalued perturbation. Using tools from the theory of nonlinear operators of monotone type, we prove existence theorems for the convex and the nonconvex problems. Also we show the existence of extremal trajectories and show that such solutions are $C_0^1(T,\mathbb{R}^N)$-dense in the solution set of the convex problem (strong relaxation theorem).

Relaxation of functionals in the space of vector-valued functions of bounded Hessian

In this paper it is shown that if $$\varOmega \subset \mathbb {R}^N$$ is an open, bounded Lipschitz set, and if $$f: \varOmega \times \mathbb {R}^{d \times N \times N} \rightarrow [0, \infty )$$ is a continuous function with $$f(x, \cdot )$$ of linear growth for all $$x \in \varOmega $$ , then the relaxed functional in the space of functions of Bounded Hessian of the energy $$\begin{aligned} F[u] = \int _{\varOmega } f(x, \nabla ^2u(x)) dx \end{aligned}$$ for bounded sequences in $$W^{2,1}$$ is given by $$\begin{aligned} {{\mathcal {F}}}[u] = \int _\varOmega {{\mathcal {Q}}}_2f(x, \nabla ^2u) dx + \int _\varOmega ({{\mathcal {Q}}}_2f)^{\infty }\bigg (x, \frac{d D_s(\nabla u)}{d |D_s(\nabla u)|} \bigg ) d |D_s(\nabla u) |. \end{aligned}$$ This result is obtained using blow-up techniques and establishes a second order version of the BV relaxation theorems of Ambrosio and Dal Maso (J Funct Anal 109:76–97, 1992) and Fonseca and Muller (Arch Ration Mech Anal 123:1–49, 1993). The use of the blow-up method is intended to facilitate future study of integrands which include lower order terms and applications in the field of second order structured deformations.

Nonlinear multivalued Duffing systems

We consider a multivalued nonlinear Duffing system driven by a nonlinear nonhomogeneous differential operator. We prove existence theorems for both the convex and nonconvex problems (according to whether the multivalued perturbation is convex valued or not). Also, we show that the solutions of the nonconvex problem are dense in those of the convex (relaxation theorem). Our work extends the recent one by Kalita–Kowalski [7].

New relaxation theorems with applications to strong materials

Recently, Sychev showed that conditions both necessary and sufficient for lower semicontinuity of integral functionals with p-coercive extended-valued integrands are the W1,p-quasi-convexity and the validity of a so-called matching condition (M). Condition (M) is so general that we conjecture whether it always holds in the case of continuous integrands. In this paper we develop the relaxation theory under the validity of condition (M). It turns out that a better relaxation theory is available in this case. This motivates our research since it is an important old open problem to develop the relaxation theory in the case of extended-value integrands. Then we discuss applications of the general relaxation theory to some concrete cases, in particular to the theory of strong materials.

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

Abstract We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.

Value function, relaxation, and transversality conditions in infinite horizon optimal control

We investigate the value function V:R+×Rn→R+∪{+∞} of the infinite horizon problem in optimal control for a general—not necessarily discounted—running cost and provide sufficient conditions for its lower semicontinuity, continuity, and local Lipschitz regularity. Then we use the continuity of V(t,⋅) to prove a relaxation theorem and to write the first order necessary optimality conditions in the form of a, possibly abnormal, maximum principle whose transversality condition uses limiting/horizontal supergradients of V(0,⋅) at the initial point. When V(0,⋅) is merely lower semicontinuous, then for a dense subset of initial conditions we obtain a normal maximum principle augmented by sensitivity relations involving the Fréchet subdifferentials of V(t,⋅). Finally, when V is locally Lipschitz, we prove a normal maximum principle together with sensitivity relations involving generalized gradients of V for arbitrary initial conditions. Such relations simplify drastically the investigation of the limiting behavior at infinity of the adjoint state.

A relaxation theorem for a differential inclusion with “maxima”

Abstract We consider a Cauchy problem associated to a nonconvex differential inclusion with “maxima” and we prove a Filippov type existence result. This result allows to obtain a relaxation theorem for the problem considered.