Lipschitz Continuous Research Articles

Linearization of holomorphic Lipschitz functions

AbstractLet and be complex Banach spaces with denoting the open unit ball of . This paper studies various aspects of the holomorphic Lipschitz space , endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets of Lipschitz mappings and of bounded holomorphic mappings, from to . Thanks to the Dixmier–Ng theorem, is indeed a dual space, whose predual shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of . We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that contains a 1‐complemented subspace isometric to and that has the (metric) approximation property whenever has it. We also analyze when is a subspace of , and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.

On the Coincidence between Campanato Functions and Lipschitz Functions: A New Approach via Elliptic PDES

Abstract Let $({\mathcal{M}},d,\mu)$ be the metric measure space with a Dirichlet form $\mathscr{E}$. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on ${\mathcal{M}}$ to its Poisson integral Ptu on ${\mathcal{M}}\times\mathbb{R}_+$. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math., 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971].

Investigating integrodifferential equations associated with fractal–fractional differential operators

This study focuses on integrodifferential equations involving fractal–fractional differential operators characterized by exponential decay, power law, and generalized Mittag–Leffler kernels. Utilizing linear growth and Lipschitz conditions, we investigate the existence and uniqueness of solutions, as well as the Hyers–Ulam stability of the proposed equations. For every instance, a numerical method is utilized to derive a numerical solution for the specified equation. The paper includes illustrations of fractal–fractional integrodifferential equations, with their precise solutions determined and subsequently compared with the numerical outcomes. This methodology can be applied to demonstrate convergence, and graphical presentations are included in relevant examples to illustrate our proposed approach.

Unconditionally optimal error estimates of linearized Crank-Nicolson virtual element methods for quasilinear parabolic problems on general polygonal meshes

In this paper, we construct, analyze, and numerically validate a linearized Crank-Nicolson virtual element method (VEM) for solving quasilinear parabolic problems on general polygonal meshes. In particular, we consider the more general nonlinear term a(x, u), which does not require Lipschitz continuity or uniform ellipticity conditions. To ensure that the fully discrete solution remains bounded in L∞-norm, we construct two novel elliptic projections and apply a new error splitting technique. With the help of the boundedness of numerical solution and delicate analysis of the nonlinear term, we derive the optimal error estimates for any k-order VEMs without any time-step restrictions. Numerical experiments on various polygonal meshes validate the accuracy of theoretical analysis and the unconditional convergence of the proposed scheme.

On Some Problems with Multivalued Mappings

We consider some problems with a set-valued mapping, which can be reduced to minimization of a homogeneous Lipschitz function on the unit sphere. Latter problem can be solved in some cases with a first order algorithm—the gradient projection method. As one of the examples, the case when set-valued mapping is the reachable set of a linear autonomous controlled system is considered. In several settings, the linear convergence is proven. The methods used in proofs follow those introduced by B.T. Polyak for the case where Lezanski– Polyak–Lojasiewicz condition holds. Unlike algorithms that use approximation of the reachable set, the proposed algorithms depend far less on dimension and other parameters of the problem. Efficient error estimation is possible. Numerical experiments confirm the effectiveness of the considered approach. This approach can also be applied to various set-theoretical problems with general set-valued mappings.

A higher dimensional Marcinkiewicz exponent and the Riemann boundary value problems for polymonogenic functions on fractals domains

We use a high-dimensional version of the Marcinkiewicz exponent, a metric characteristic for non-rectifiable plane curves, to present a direct application to the solution of some kind of Riemann boundary value problems on fractal domains of Euclidean space Rn+1,n≥2 for Clifford algebra-valued polymonogenic functions with boundary data in classes of higher order Lipschitz functions. Sufficient conditions to guarantee the existence and uniqueness of solution to the problems are proved. To illustrate the delicate nature of this theory we described a class of hypersurfaces where the results are more refined than those that exist in literature.

A note on 𝐶^{1,𝛼}-smooth approximation of Lipschitz functions

We show that on super-reflexive spaces a Moreau-Yosida type of regularisation by infimal convolution together with a known insertion-type theorem (a variant of Ilmanen’s lemma) easily give an approximation of a Lipschitz function by a C 1 , α C^{1,\alpha } -smooth Lipschitz function with the same Lipschitz constant. This is a generalisation of the well-known theorem of J.-M. Lasry and P.-L. Lions from Hilbert spaces. It also gives a new self-contained and probably simpler proof of the Lasry-Lions theorem.

The best approximation of a given function in L 2-norm by Lipschitz functions with gradient constraint

Abstract The starting point of this paper is the study of the asymptotic behavior, as p → ∞ {p\to\infty} , of the following minimization problem: min ⁡ { 1 p ⁢ ∫ Ω | ∇ ⁡ v | p + 1 2 ⁢ ∫ Ω ( v - f ) 2 , v ∈ W 1 , p ⁢ ( Ω ) } . \min\biggl{\{}\frac{1}{p}\int_{\Omega}|\nabla v|^{p}+\frac{1}{2}\int_{\Omega}(% v-f)^{2},\quad v\in W^{1,p}(\Omega)\biggr{\}}. We show that the limit problem provides the best approximation, in the L 2 {L^{2}} -norm, of the datum f among all Lipschitz functions with Lipschitz constant less or equal than one. Moreover, such an approximation verifies a suitable PDE in the viscosity sense. After the analysis of the model problem above, we consider the asymptotic behavior of a related family of nonvariational equations and, finally, we also deal with some functionals involving the ( N - 1 ) {(N-1)} -Hausdorff measure of the jump set of the function.

Exponential stability of non-conformable fractional-order systems

Abstract Recently, the authors Guzman et al. (2018) introduced a new simple well-behaved definition of the fractional derivative called non-conformable fractional derivative. In this paper we study the exponential stability of non-conformable fractional-order systems by using the Lyapunov function and Gronwall inequality. These inequalities can be used as handy tools to research stability problems of nonlinear systems. Sufficient conditions for exponential stability are given using the Lyapunov theory. Further, deals with the state feedback stabilization problems for a family of nonlinear systems satisfying a Lipschitz continuity condition. The stability of the controller is proved by means of the new Lyapunov stability theorem given in this paper. A numerical example is given to illustrate the efficiency of the obtained result.

Lipschitz continuity of the solutions to the Dirichlet problems for the invariant Laplacians

This short note is motivated by an attempt to understand the distinction between the Laplace operator and the hyperbolic Laplacian on the unit ball of Rn, regarding the Lipschitz continuity of the solutions to the corresponding Dirichlet problems.We investigate the Dirichlet problem{Δϑu=0, in Bn,u=ϕ, on Sn−1, whereΔϑ:=(1−|x|2){1−|x|24Δ+ϑ∑j=1nxj∂∂xj+ϑ(n2−1−ϑ)I}.We show that: (i) when ϑ>0, the Lipschitz continuity of boundary data always implies the Lipschitz continuity of the solutions; (ii) when ϑ=0, there exists a Lipschitz continuous function ϕ:Sn−1→R such that the solution is not Lipschitz continuous; (iii) when ϑ<0, the solution is definitely not Lipschitz continuous unless ϕ≡0.

Restrictive Lipschitz continuity, basis property of a real sequence, and fixed-point principle in metrically convex spaces

A mapping T of a metric space X,d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( X,d\\right) $$\\end{document} into a metric space Y,ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( Y,\\rho \\right) $$\\end{document} is called restrictive Lipschitz if there exist: a positive decreasing to zero sequence tn:n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( t_{n}:n\\in \\mathbb {N} \\right) $$\\end{document} and a nonnegative sequence Ln:n∈N,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( L_{n}:n\\in \\mathbb {N}\\right) ,$$\\end{document} with L:=lim infn→∞Ln<∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L:=\\liminf _{n\\rightarrow \\infty }L_{n}<\\infty ,$$\\end{document} such that for all x,y∈X,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ x,y\\in X,$$\\end{document}n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\in \\mathbb {N}$$\\end{document}dx,y=tn⟹ρTx,Ty≤Lntn.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} d\\left( x,y\\right) =t_{n}\\Longrightarrow \\rho \\left( Tx,Ty\\right) \\le L_{n}t_{n}\ ext {.} \\end{aligned}$$\\end{document}Using a basis property of the sequence tn:n∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( t_{n}:n\\in \\mathbb {N}\\right) $$\\end{document} (Lemma 1), we prove that if T is a continuous and restrictive Lipschitz mapping of a complete metrically convex space X,d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( X,d\\right) $$\\end{document} into a metric space Y,ρ,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( Y,\\rho \\right) ,$$\\end{document} then T is Lipschitz continuous with the constant L, that is ρTx,Ty≤Ldx,y,x,y∈X,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\rho \\left( Tx,Ty\\right) \\le Ld\\left( x,y\\right) , \\quad x,y\\in X, \\end{aligned}$$\\end{document}and, in the case when the set n∈N:Ln<L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left\\{ n\\in \\mathbb {N}:L_{n}<L\\right\\} $$\\end{document} is infinite, even essentially more, namely ρTx,Ty≤Lαdx,y,x,y∈X,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\rho \\left( Tx,Ty\\right) \\le L\\alpha \\left( d\\left( x,y\\right) \\right) , \\quad x,y\\in X, \\end{aligned}$$\\end{document}where the function α:0,∞→0,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha :\\left[ 0,\\infty \\right) \\rightarrow \\left[ 0,\\infty \\right) $$\\end{document} is continuous, increasing, concave (so subadditive) and such that αt<t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\left( t\\right) <t$$\\end{document} for all t>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t>0$$\\end{document}. This result leads to the following fixed-point principle: Every continuous selfmapping T of a nonempty metrically convex complete metric space X,d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( X,d\\right) $$\\end{document} that is restrictive Lipschitz with a sequence Ln:n∈N,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( L_{n}:n\\in \\mathbb {N}\\right) ,$$\\end{document} such that0≤Ln<1(n∈N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ 0\\,\\le L_{n}<1\\ (n\\in \\mathbb {N) }$$\\end{document} and lim infn→∞Ln≤1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\liminf _{n\\rightarrow \\infty }L_{n}\\le 1,$$\\end{document} has a unique fixed point, and either it is a Banach contraction, or there is an increasing concave function α:0,∞→0,∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha :\\left[ 0,\\infty \\right) \\rightarrow \\left[ 0,\\infty \\right) $$\\end{document}, such that αt<t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\left( t\\right) <t$$\\end{document} for t>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t>0$$\\end{document} and dTx,Ty≤αdx,y,x,y∈X.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} d\\left( Tx,Ty\\right) \\le \\alpha \\left( d\\left( x,y\\right) \\right) , \\quad x,y\\in X. \\end{aligned}$$\\end{document}Some applications of these results to the theory of iterative functional equations are proposed.

Functional Solutions of Stochastic Differential Equations

We present an integration condition ensuring that a stochastic differential equation dXt=μ(t,Xt)dt+σ(t,Xt)dBt, where μ and σ are sufficiently regular, has a solution of the form Xt=Z(t,Bt). By generalizing the integration condition we obtain a class of stochastic differential equations that again have a functional solution, now of the form Xt=Z(t,Yt), with Yt an Ito process. These integration conditions, which seem to be new, provide an a priori test for the existence of functional solutions. Then path-independence holds for the trajectories of the process. By Green’s Theorem, it holds also when integrating along any piece-wise differentiable path in the plane. To determine Z at any point (t,x), we may start at the initial condition and follow a path that is first horizontal and then vertical. Then the value of Z can be determined by successively solving two ordinary differential equations. Due to a Lipschitz condition, this value is unique. The differential equations relate to an earlier path-dependent approach by H. Doss, which enables the expression of a stochastic integral in terms of a differential process.

A proximal neurodynamic model for a system of non-linear inverse mixed variational inequalities

In this article, we introduce a system of non-linear inverse mixed variational inequalities (SNIMVIs). We propose a proximal neurodynamic model (PNDM) for solving SNIMVIs, leveraging proximal mappings. The uniqueness of the continuous solution for the PNDM is proved by assuming Lipschitz continuity. Moreover, we establish the global asymptotic stability of equilibrium points of the PNDM, contingent upon Lipschitz continuity and strong monotonicity. Additionally, an iterative algorithm involving proximal mappings for solving the SNIMVIs is presented. Finally, we provide illustrative examples to support our main findings. Furthermore, we provide an example where the SNIMVIs violate the strong monotonicity condition and exhibit the divergence nature of the trajectories of the corresponding PNDM.

Relaxed Inertial Method for Solving Split Monotone Variational Inclusion Problem with Multiple Output Sets Without Co-coerciveness and Lipschitz Continuity

Relaxed Inertial Method for Solving Split Monotone Variational Inclusion Problem with Multiple Output Sets Without Co-coerciveness and Lipschitz Continuity

Designing stable neural networks using convex analysis and ODEs

Motivated by classical work on the numerical integration of ordinary differential equations we present a ResNet-styled neural network architecture that encodes non-expansive (1-Lipschitz) operators, as long as the spectral norms of the weights are appropriately constrained. This is to be contrasted with the ordinary ResNet architecture which, even if the spectral norms of the weights are constrained, has a Lipschitz constant that, in the worst case, grows exponentially with the depth of the network. Further analysis of the proposed architecture shows that the spectral norms of the weights can be further constrained to ensure that the network is an averaged operator, making it a natural candidate for a learned denoiser in Plug-and-Play algorithms. Using a novel adaptive way of enforcing the spectral norm constraints, we show that, even with these constraints, it is possible to train performant networks. The proposed architecture is applied to the problem of adversarially robust image classification, to image denoising, and finally to the inverse problem of deblurring.

A geometric representative for the fundamental class in KK-duality of Smale spaces

A fundamental ingredient in the noncommutative geometry program is the notion of KK-duality, often called K-theoretic Poincaré duality, that generalises Spanier-Whitehead duality. In this paper we construct a θ-summable Fredholm module that represents the fundamental class in KK-duality between the stable and unstable Ruelle algebras of a Smale space. To find such a representative, we construct dynamical partitions of unity on the Smale space with highly controlled Lipschitz constants. This requires a generalisation of Bowen's Markov partitions. Along with an aperiodic point-sampling technique we produce a noncommutative analogue of Whitney's embedding theorem, leading to the Fredholm module.

Existence Result for a Stochastic Functional Differential System Driven by G-Brownian Motion with Infinite Delay

Abstract In this article, we are interested in the study of a class of stochastic functional differential systems driven by G-Brownian motion with infinite delay. We prove the existence and uniqueness of the solutions when two basic conditions are met: the linear growth condition and the Lipschitz condition.

A comparison principle for semilinear Hamilton–Jacobi–Bellman equations in the Wasserstein space

The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution.

Passivity-based control of underactuated mechanical systems with Coulomb friction: Application to earthquake prevention

Passivity property gives a sense of energy balance. The classical definitions and theorems of passivity in dynamical systems require time invariance and locally Lipschitz functions. However, these conditions are not met in many systems. Characteristic examples are nonautonomous and discontinuous systems due to the presence of Coulomb friction. This paper presents an extended result for the negative feedback connection of two passive nonautonomous systems with set-valued right-hand side based on an invariance-like principle. Such extension is the base of a structural passivity-based control synthesis for underactuated mechanical systems with Coulomb friction. The first step consists of designing the control able to restore the passivity in the considered friction law, achieving stabilization of the system trajectories to a domain with zero velocities. Then, an integral action is included to improve the latter result and perform a tracking over a constant reference (regulation). Finally, the control is designed considering dynamics in the actuation. These control objectives are obtained using fewer control inputs than degrees of freedom, as a result of the underactuated nature of the plant. The presented control strategy is implemented in an earthquake prevention scenario, where a mature seismogenic fault represents the considered frictional underactuated mechanical system. Simulations are performed to show how the seismic energy can be slowly dissipated by tracking a slow reference, thanks to fluid injection far from the fault, accounting also for the slow dynamics of the fluid’s diffusion.

A comprehensive study on Milne-type inequalities with tempered fractional integrals

AbstractIn the framework of tempered fractional integrals, we obtain a fundamental identity for differentiable convex functions. By employing this identity, we derive several modifications of fractional Milne inequalities, providing novel extensions to the domain of tempered fractional integrals. The research comprehensively examines significant functional classes, including convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation.