Abstract This work deals with boundary value problems for second‐order nonlinear elliptic equations in divergence form, which emerge as Euler–Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. Integrands with nonpolynomial growth are included in our discussion. The ‐regularity of the stress‐field associated with solutions, namely, the nonlinear expression of the gradient subject to the divergence operator, is established under the weakest possible assumption that the datum on the right‐hand side of the equation is a merely ‐function. Global regularity estimates are offered in domains enjoying minimal assumptions on the boundary. They depend on the weak curvatures of the boundary via either their degree of integrability or an isocapacitary inequality. By contrast, none of these assumptions is needed in the case of convex domains. An explicit estimate for the constants appearing in the relevant estimates is exhibited in terms of the Lipschitz characteristic of the domains, when their boundary is endowed with Hölder continuous curvatures.

On a Theorem of S.L. Sobolev Generated by Anisotropic Norms of Partial Integral Operators

We consider a discrete-time-invariant system with multiplicative noise with implementation in the state space. The exogenous disturbance is chosen from the class of time-invariant ergodic sequences of nonzero colorness. We consider the level of mean anisotropy of the exogenous disturbance to be bounded by a known value. Conditions for the anisotropic norm to be bounded by a given number are obtained in terms of solving a matrix system of inequalities with a convex constraint of a special type. It is demonstrated how, on the basis of the obtained conditions, to construct a static state control that ensures the minimum value of the anisotropic norm of the system enclosed by this control.

Discrete linear polytopic systems affected by random correlated stationary disturbances are considered. New numerical methods for estimating of the anisotropic norm of a polytopic system using linear matrix inequalities are proposed.

Bounded Real Lemma for the Anisotropic Norm of Time-invariant Systems with Multiplicative Noises

Zero divisors of Cayley–Dickson algebras over an arbitrary field 𝔽, char 𝔽 ≠= 2, are studied. It is shown that the zero divisors whose components alternate strongly pairwise and have nonzero norm form hexagonal structures in the zero-divisor graph of a Cayley–Dickson algebra. Properties of the doubly alternative zero divisors at least one of whose components has nonzero norm are established, and explicit forms of their annihilators, orthogonalizers, and centralizers are obtained. Properties of the zero divisors in Cayley–Dickson algebras with anisotropic norm are described, and it is shown that in this case, directed hexagons in the zero-divisor graph can be extended to undirected double hexagons in the orthogonality graph. A criterion of C-equivalence for elements of Cayley–Dickson algebras with anisotropic norm is obtained. Possible values of dimension for the annihilators of elements in Cayley–Dickson algebras are considered.

We study a lattice point-counting problem for spheres arising from the Heisenberg groups. In particular, we prove an upper bound on the number of points on and near large dilates of the unit spheres generated by the anisotropic norms $$\Vert (z,t)\Vert _\alpha = ( \left| z\right| ^\alpha + \left| t\right| ^{\alpha /2})^{1/\alpha }$$ for $$\alpha \ge 2$$ . As a first step, we reduce our counting problem to one of bounding an energy integral. The primary new challenges that arise are the presence of vanishing curvature and nonisotropic dilations. In the process, we establish bounds on the Fourier transform of the surface measures arising from these norms. Further, we utilize the techniques developed here to estimate the number of lattice points in the intersection of two such surfaces.

The sensor network estimation with dropouts: Anisotropy-based approach

The paper considers the problem of setting up a communication scheme associated with the adjacency matrix between separate non-ideal sensors and known probability of their failsafe operation. As the evaluation object, a linear discrete non-stationary model in the state space was chosen, which was affected by external perturbations with the inaccurately specified stochastic characteristics. For external perturbations, upper limit of the anisotropy of the extended vector consisting of all the perturbing sequence elements was determined. Sensors were combined into a common network, where each separate node was able to use not only the own measurements to build an estimate of the desired output, but also the measurements received from the adjacent sensors. The model took into account the failure of specific sensors, where failures had the Bernoulli distribution. A failure should be understood as the random readings of a measurement device containing no useful information. The criterion is anisotropic norm of the system in the estimation errors from the perturbing action to the estimated output error. The problem was in selecting such adjacency matrix coefficients, where the anisotropic norm value in the estimation errors was not exceeding a certain threshold value. Solution to the problem was reduced to a numerical procedure of solving a special system of matrix inequalities ensuring boundedness of the system anisotropic norm in the estimation errors

We study the set of possible traces of anisotropic least gradient functions. We show that even on the unit disk it changes with the anisotropic norm: for two sufficiently regular strictly convex norms the trace spaces coincide if and only if the norms coincide. The example of a function in exactly one of the trace spaces is given by a characteristic function of a suitably chosen Cantor set.

<p style='text-indent:20px;'>Recently, the authors proved [<xref ref-type="bibr" rid="b2">2</xref>] that the Maxwell-Stefan system with an incompressibility-like condition on the total flux can be rigorously derived from the multi-species Boltzmann equation. Similar cross-diffusion models have been widely investigated, but the particular case of a perturbative incompressible setting around a non constant equilibrium state of the mixture (needed in [<xref ref-type="bibr" rid="b2">2</xref>]) seems absent of the literature. We thus establish a quantitative perturbative Cauchy theory in Sobolev spaces for it. More precisely, by reducing the analysis of the Maxwell-Stefan system to the study of a quasilinear parabolic equation on the sole concentrations and with the use of a suitable anisotropic norm, we prove global existence and uniqueness of strong solutions and their exponential trend to equilibrium in a perturbative regime around any macroscopic equilibrium state of the mixture. As a by-product, we show that the equimolar diffusion condition naturally appears from this perturbative incompressible setting.</p>

In this paper, the anisotropy-based analysis problem for the linear discrete time-invariant system with state-multiplicative noises is considered. The mean anisotropy of stationary input sequence is assumed to be bounded by given nonnegative value. For this setting, the boundedness condition for anisotropic norm of the system is obtained in terms of Riccati equations.

Lemma on Boundedness of Anisotropic Norm for Systems with Multiplicative Noises under a Noncentered Disturbance

The synthesis problem of static output feedback controllers within the anisotropic-norm setup is revisited. A tractable synthesis approach involving iterations over a convex optimisation problem is suggested, similarly to existing results for the H∞-norm minimisation case. The results are formulated by a couple of Linear Matrix Inequalities coupled via a bilinear equality, revealing, as in the H∞ case the duality of between the control-type and filtering type LMIs and allowing a tractable iterative method to cope with practical static output feedback synthesis problems. The resulting optimisation scheme is then applied to a flight control problem, where the merit of the anisotropic norm setup is shown to provide a useful trade-off between closed loop response and feedback gains

The synthesis problem of static output feedback controllers within the anisotropic-norm setup is revisited. A tractable synthesis approach involving iterations over a convex optimisation problem is suggested, similarly to existing results for the H∞-norm minimisation case. The results are formulated by a couple of Linear Matrix Inequalities coupled via a bilinear equality, revealing, as in the H∞ case the duality of between the control-type and filtering type LMIs and allowing a tractable iterative method to cope with practical static output feedback synthesis problems. The resulting optimisation scheme is then applied to a flight control problem, where the merit of the anisotropic norm setup is shown to provide a useful trade-off between closed loop response and feedback gains

Least gradient problem with respect to a non-strictly convex norm

In this paper, linear discrete-time systems with polytopic uncertainties affected by random external disturbances are under consideration. The input disturbance is supposed to be a stochastic signal with known mean anisotropy, which stands for a spectral color of the signal. The anisotropic norm of the system indicates its stochastic gain from the input disturbance with the same mean anisotropy level to the output of the system. The problem is to find state-feedback control law that robustly stabilizes the uncertain system and guarantees desired performance index subject to input disturbance for all possible uncertainties. In order to solve this problem, Lyapunov functions technique and matrix inequalities approach are used to obtain the numerically effective procedure. To illustrate the efficiency of the proposed conditions, a numerical example is considered.

Finite-horizon Anisotropy-based Estimation with Packet Dropouts

The design of static output feedback controllers in an anisotropic norm setup is considered. The aim is to determine a stabilizing static output feedback for a given four block system such that the resulting closed loop system has the a-anisotropic norm less than a given γ > 0. The solvability conditions are expressed in terms of the solution of a rank minimization problem with linear matrix inequalities constraints. Based on the specific form of these constraints it is shown that a solution of this problem may be obtained solving a semidefinite programming problem.

This paper considers the problem of synthesis of a proportional-integral-derivative control law (PID controller) for a linear discrete time-invariant system with scalar control input and measured output operating under influence of the stochastic disturbances with uncertainty described in terms of the mean anisotropy. The closed-loop system abilities to attenuate the disturbances are quantitatively characterized by the anisotropic norm. Sufficient existence conditions for the anisotropic suboptimal controller that stabilizes the closed-loop system and guarantees that its anisotropic norm is strictly bounded by a given threshold value are derived.