Order Maximum Principle Research Articles

A MAXIMUM PRINCIPLE FOR COMPLETE HYPERSURFACES IN LOCALLY SYMMETRIC RIEMANNIAN MANIFOLD

In this article, we apply the weak maximum principle in order to obtain a suitable characterization of the complete linearWeingarten hypersurfaces immersed in locally symmetric Riemannian manifold <TEX>$N^{n+1}$</TEX>. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or hypersurface is an isoparametric hypersurface with two distinct principal curvatures one of which is simple.

A new characterization of complete linear Weingarten hypersurfaces in real space forms

We apply the Hopf’s strong maximum principle in order to obtain a suitable characterization of the complete linear Weingarten hypersurfaces immersed in a real space form Q nC1 c of constant sectional curvature c. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a Clifford torus, if cD 1, a circular cylinder, if cD 0, or a hyperbolic cylinder, if cD 1.

On the geometry of linear Weingarten hypersurfaces in the hyperbolic space

In this paper, we apply some forms of generalized maximum principles in order to study the geometry of complete linear Weingarten hypersurfaces with nonnegative sectional curvature immersed in the hyperbolic space. In this setting, under the assumption that the mean curvature attains its maximum, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder.

On the Unicity of Complete Hypersurfaces Immersed in a Semi-Riemannian Warped Product

Our purpose is to apply appropriate generalized maximum principles in order to study the unicity of complete hypersurfaces immersed in a semi-Riemannian warped product, which is supposed to obey a suitable convergence condition. In this setting, by assuming a natural comparison inequality between the r-th mean curvatures of the hypersurface and the ones of the slices of the slab where the hypersurface is contained, we establish sufficient conditions to guarantee that such a hypersurface must be a slice.

Presymplectic high order maximum principle

Pontryagin's Maximum Principle is an outstanding result for solving optimal control problems by means of optimizing a specific function on some particular variables, the so called controls. However, this is not always enough for solving all these problems. A high order maximum principle (Krener, 1977) must be used in order to obtain more necessary conditions for optimality. These new conditions determine candidates to be optimal controls for a wider range of optimal control problems. Here, we focus on control-affine systems. Krener's high order perturbations are redefined following the notions introduced in Aguilar-Lewis (2008). A weaker version of Krener's high order maximum principle is stated in the framework of presymplectic geometry. As a result, the presymplectic constraint algorithm in the sense of Gotay-Nester-Hinds (1979) can be used. We establish the connections between the presymplectic constraint algorithm and the candidates to be optimal curves obtained from the necessary conditions in Krener's high order maximum principle. In this paper we obtain weaker geometric necessary conditions for optimality of abnormal solutions than the ones in Krener (1977) and the ones in the weak high order maximum principle. These new necessary conditions are more useful, computationally speaking, for finding curves candidate to be optimal. The theory is supported by describing specifically some of the above-mentioned conditions for some mechanical control systems.

Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes

In Zhang and Shu (J. Comput. Phys. 229:3091---3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and Shu (J. Comput. Phys. 229:8918---8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.

Behavior of the solution of a Stefan problem by changing thermal coefficients of the substance

We consider a one-dimensional one-phase Stefan problem for a semi-infinite substance. We suppose that there is a transient heat flux at the fixed face and the thermal coefficients are constant. The goal of this paper is to determine the behavior of the free boundary and the temperature by changing the thermal coefficients. We use the maximum principle in order to obtain properties of monotony with respect to the latent heat of fusion, the specific heat and the mass density. We compute approximate solutions through the quasi-stationary, the Goodman’s heat-balance integral and the Biot’s variational methods and a numerical solution through a finite difference scheme. We show that the solution is not monotone with respect to the thermal conductivity. The results obtained are important in technological applications.

Singular Trajectories and Their Role in Control Theory

Linear Systems and the Time Optimal Control Problem. - Exercises. - Optimal Control for Nonlinear Systems. - Exercises. - Geometric Optimal Control. - Exercises. - Singular Trajectories and Feedback Classification. - Exercises. - Controllability, Higher Order Maximum Principle, Legendre-Clebsch and Goh Necessary Optimality Conditions. - Exercises. - The Concept of Conjugate Points in the Time Minimal Control Problem for Singular Trajectories, C0-Optimality. - Time Minimal Control of Chemical Batch Reactors and Singular Trajectories. - Generic Properties of Singular Trajectories. - Exercises. - Singular Trajectories in Sub-Riemannian Geometry. - Exercises. - Micro-Local Resolution of the Singularity near a Singular Trajectory, Lagrangian Manifolds and Symplectic Stratifications. - Exercises. - Numerical Computations. - Conclusion and Perspectives. - Exercises. - References. - Index

The Immersed Interface/Multigrid Methods for Interface Problems

New multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic PDEs that involve interfaces and discontinuities. For elliptic interface problems, the multigrid solver developed in this paper works while some other multigrid solvers do not. For linear parabolic equations, we have developed the second order maximum principle preserving finite difference scheme in this paper. We use the Crank--Nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives. Numerical examples are also presented.

Explicit decay bounds in some quasilinear one-dimensional parabolic problems

In the first part of this paper we study a thermal diffusion process described by a semilinear parabolic problem and we introduce a new maximum principle in order to obtain explicit decay bounds for the temperature and its gradient. In the second part we find analogous bounds for the so-called ground water equation in a more general form.

Again on Tangent Cones and High Order Maximum Principles

The paper concerns the definition of a cone of tangent vectors of the reachable set of a control system which is a regular tangent cone. This property implies that the Lagrange multiplier rule applies so that an high order Maximum Principle for Mayer optimization control problem with constraints on the final point can be stated. The tangent vectors considered in the paper are generated by control variations concentrated in “small” intervals, i.e. needle like variations; they are more general than the ones of the same type which can be found in the literature. Therefore the paper gives an insight to the properties one has to look for in order that a needle like control variation produces a “good” tangent vector.

Semigroups in the simply connected covering of SL(2)

Based on the parametrization proposed in [HH85], K.-H. Neeb investigated in [Ne91] the globality of a certain one-parameter family of Lorentzian cones in the universal covering group $$\widetilde{Sl}(2)$$ of SI(2). In this article, we use the Pontrjagin Maximum Principle in order to obtain an explicit description of the semigroups generated by these cones, resp., to find conal curves which return to the identity. Furthermore we improve the description of the exponential function of $$\widetilde{Sl}(2)$$ . An adequate general framework should be that of J. Hilgert in [Hi92] who investigated the globality resp. controllability of pointed generating conesC which are invariant under the adjoint action of a compact subgroupK that comes from an Iwasawa decompositionG=NAK.

Reactivity Effect of Non-Uniformly Distributed Fuel in Fuel Solution Systems.

A numerical method to determine the optimal fuel distribution for minimum critical mass, or maximum k-effective, is developed using the Maximum Principle in order to evaluate the maximum effect of non-uniformly distributed fuel on reactivity. This algorithm maximizes the Hamiltonian directly by an iterative method under a certain constraint-the maintenance of criticality or total fuel mass. It ultimately reaches the same optimal state of a flattened fuel importance distribution as another algorithm by Dam based on perturbation theory. This method was applied to two kinds of spherical cores with water reflector in the simu-lating reprocessing facility. In the slightly-enriched uranyl nitrate solution core, the minimum critical mass decreased by less than 1% at the optimal moderation state. In the plutonium nitrate solution core, the k-effective increment amounted up to 4.3%Δk within the range of present study.

Weakly trapped submanifolds in standard static spacetimes

We study weakly trapped submanifolds of codimension two in a standard static spacetime. In this setting, we apply some generalized maximum principles in order to investigate the geometry of these trapped submanifolds. For instance, we establish sufficient conditions to guarantee that such a spacelike submanifold must be a hypersurface of the Riemannian base of the ambient spacetime. As a consequence, we prove that there do not exist $n$-dimensional compact (without boundary) trapped submanifolds immersed in an $(n+2)$-dimensional standard static spacetime. Such a nonexistence result was originally obtained for stationary spacetimes by Mars and Senovilla [20]. Furthermore, we also investigate parabolic weakly trapped submanifolds immersed in a standard static spacetime.