Sufficient Geometric Conditions Research Articles

Reduced-order dynamic observer error linearisation: a natural extension of observer error linearisation

This paper introduces the concept of reduced-order dynamic observer error linearisation (RDOEL) for multi-output systems, which is the problem of transforming a nonlinear system into a nonlinear observer canonical form with the aid of auxiliary dynamics. The proposed RDOEL framework is not only a modified version of dynamic observer error linearisation (DOEL) but also a natural extension of observer error linearisation (OEL). We provide three necessary conditions for the RDOEL problem, and then derive a necessary and sufficient condition described in terms of Lie algebras of vector fields. Furthermore, from the result, we also give a geometric necessary and sufficient condition for the OEL problem, which has not yet been completely established in the case where a diffeomorphism on the output of general form is considered.

A general regularity theory for stable codimension 1 integral varifolds

We give a necessary and sufficient geometric structural condition, which we call the a -Structural Hypothesis, for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities. The a -Structural Hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded C 1,a hypersurfaces-with-boundary meeting (only) along their common boundary. We establish that whenever a stable integral n -varifold on a smooth (n+1) -dimensional Riemannian manifold satisfies the a -Structural Hypothesis for some a?(0,1/2) , its singular set is empty if n=6 , discrete if n=7 and has Hausdorff dimension =n-7 if n=8 ; in view of well-known examples, this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses. We also establish compactness of mass-bounded subsets of the class of stable codimension 1 integral varifolds satisfying the a -Structural Hypothesis for some a?(0,1/2) . The a -Structural Hypothesis on an n -varifold for any a?(0,1/2) is readily implied by either of the following two hypotheses: (i) the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary, (ii) the singular set of the varifold has vanishing (n-1) -dimensional Hausdorff measure. Thus, our theory subsumes the well-known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions. An optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems

Extended output depending normal form

This paper concerns the design of a nonlinear observer through a transformation of a nonlinear system into an observer form that supports a high gain observer. Sufficient geometrical condition is deduced to guarantee the existence of change of coordinates allowing the transformation of a nonlinear system into the proposed normal form. The result is highlighted by studying the Susceptible Exposed Infected and Recovered (SEIR) model.

Linearization of time-delay systems by input–output injection and output transformation

In the present paper necessary and sufficient geometric conditions for the equivalence of a single input single output nonlinear time-delay system, under bicausal change of coordinates and regular output transformation, to a linear weakly observable time-delay system up to input–output injection are given. The obtained results are then applied to the observer design problem.

Mixed gravitational field equations on globally hyperbolic spacetimes

For every globally hyperbolic spacetime M, we derive new mixed gravitational field equations embodying the smooth Geroch infinitesimal splitting of M, as exhibited by Bernal and Sánchez (2005 Commun. Math. Phys. 257 43–50). We give sufficient geometric conditions (e.g. is isoparametric and is totally umbilical) for the existence of exact solutions to mixed field equations in free space. We linearize and solve the mixed field equations for empty space, where is the mixed scalar curvature of foliated spacetime (due to Rovenski (2010 arXiv:1010.2986 v1[math.DG])). If gϵ = g0 + ϵγ is a solution to the linearized field equations, then each leaf of is totally geodesic in to order O(ϵ). We derive the equations of motion of a material particle in the gravitational field gμν governed by the mixed field equations . In the weak field (ϵ ≪ 1) and low velocity (‖v‖/c ≪ 1) limit, the motion equations are d2r/dt2 = ∇ϕ + F, where ϕ = (ϵ/2)c2γ00.

Nonlinear observer normal form with output injection and extended dynamic

This paper presents a new extended output depending nonlinear observer normal form. A sufficient geometrical conditions that guarantee a change of coordinates allowing the transformation of a given nonlinear dynamical system into the proposed observer form are given. Throughout this work, it will be showed that, unlike to the existing observer normal forms, this new form enables to design an observer for the Susceptible, Exposed, Infected, and Recovered (SEIR) model of population under an infectious disease.

On the Existence of Mixed Strategy Nash Equilibria

The focus of this paper is on developing geometric sufficient conditions for the existence of a mixed strategy Nash equilibrium for both diagonally transfer continuous and better-reply secure games. First, we show that employing the concept of diagonal transfer continuity in place of better-reply security might be advantageous when the existence of a mixed strategy Nash equilibrium is concerned. Then, we study equilibrium existence in better-reply secure games possessing a payoff secure mixed extension. With the aid of an example we show that such games need not have mixed strategy Nash equilibria. We provide some easily verifiable conditions for the mixed extension of a two-person game that is reciprocally upper semicontinuous and uniformly payoff secure to be better-reply secure.

On restricting Cauchy–Pexider functional equations to submanifolds

Sufficient geometric conditions are given which determine when the Cauchy–Pexider functional equation f(x)g(y) = h(x + y) restricted to x, y lying on a hypersurface in $${\mathbb{R}^d}$$ has only solutions which extend uniquely to exponential affine functions $${\mathbb{R}^d \to \mathbb{C}}$$ (when f, g, h are assumed to be measurable and non-trivial). The Cauchy–Pexider-type functional equations $${\prod_{j=0}^df_j(x_j)=F(\sum_{j=0}^dx_j)}$$ for $${x_0, \ldots,x_d}$$ lying on a curve and $${f_1(x_1)f_2(x_2)f_3(x_3)=F(x_1+x_2+x_3)}$$ for x 1, x 2, x 3 lying on a hypersurface are also considered.

The Spinc Dirac operator on hypersurfaces and applications

We extend to the eigenvalues of the hypersurface Spinc Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Furthermore, we prove a correspondence between the existence of a Spinc Killing spinor on homogeneous 3-dimensional manifolds E⁎(κ,τ) with 4-dimensional isometry group and isometric immersions of E⁎(κ,τ) into the complex space form M4(c) of constant holomorphic sectional curvature 4c, for some c∈R⁎. As applications, we show the non-existence of totally umbilic surfaces in E⁎(κ,τ) and we give necessary and sufficient geometric conditions to immerse a 3-dimensional Sasaki manifold into M4(c).

A new reduced-order observer normal form for nonlinear discrete time systems

This paper presents a new observability normal form for discrete-time nonlinear systems. This form enables us to design a reduced-order observer. Necessary and sufficient geometrical conditions for the existence of a coordinate change to transform a discrete-time nonlinear system into such normal form are given. An illustrative example is given to show the effectiveness of our approach.

Walking control for a planar biped robot using 0-flat normal form

This paper deals with the use of 0-flat normal form to control a 7 d.o.f-biped robot to follow a specified trajectory. Sufficient geometrical conditions are given to transform the studied nonlinear systems into a 0-flat normal form and determine the flat outputs. On the other hand, a controller design strategy is proposed to control the walking robot. Simulations are carried out using Matlab. The results obtained are very convincing and show the usefulness of such a method in studying highly non-linear systems and designing control laws to drive them.

Stabilisation of second-order LTI switched positive systems

The stabilisation problem of second-order switched positive systems consisting of two unstable subsystems is considered in this article. By considering the vector fields and geometric characteristics, a necessary and sufficient condition for the stabilisability of second-order switched positive systems with two unstable subsystems is provided. Furthermore, it is shown via this condition that neither second-order switched positive systems consisting of two subsystems with unstable nodes nor second-order switched positive systems consisting of one subsystem with unstable nodes and the other with a saddle point can be stabilised via any switching law.

Some results on surjectivity of augmented differential operators

Considering a problem of Bonet and Domański (2006) [1, Problem 9.1], we prove that for a polynomial P on R 2 surjectivity of the differential operator P ( D ) on D ′ ( X ) implies surjectivity of the augmented operator P + ( D ) on D ′ ( X × R ) , where P + ( x 1 , x 2 , x 3 ) : = P ( x 1 , x 2 ) . Moreover we give a sufficient geometrical condition on an open subset X of R d such that an analogous implication is true for arbitrary dimension d in case of P being homogeneous, semi-elliptic, or of principal type.

Dirichlet forms and stochastic completeness of graphs and subgraphs

We study Laplacians on graphs and networks via regular Dirichlet forms. We give a sufficient geometric condition for essential selfadjointness and explicitly determine the generators of the associated semigroups on all ℓ p , 1 ≦ p < ∞, in this case. We characterize stochastic completeness thereby generalizing all earlier corresponding results for graph Laplacians. Finally, we study how stochastic completeness of a subgraph is related to stochastic completeness of the whole graph.

Geometric Sufficient Conditions for Compactness of the Complex Green Operator

We establish compactness estimates for \(\overline{\partial}_{M}\) on a compact pseudoconvex CR-submanifold M of ℂn of hypersurface type that satisfies the (analogue of the) geometric sufficient conditions for compactness of the \(\overline{\partial}\)-Neumann operator given in (Straube in Ann. Inst. Fourier, 54(3):699–710, 2004; Munasinghe and Straube in Pac. J. Math., 232(2):343–354 2007). These conditions are formulated in terms of certain short time flows in complex tangential directions.

A triangular canonical form for a class of 0-flat nonlinear systems

This article proposes a triangular canonical form for a class of 0-flat nonlinear systems. Necessary and sufficient geometrical conditions are given in order to guarantee the existence of a local diffeomorphism to transform the studied nonlinear systems into the proposed 0-flat canonical form, which enables us to compute the flat output as well.

On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form

This article deals with the transformation of a class of nonlinear systems into an extended nonlinear observable canonical form (ENOCF) by dynamic extension. The dynamic extension is obtained by adding auxiliary dynamics and virtual outputs to the original system, thus giving rise to a higher dimensional system in an extended state space. Sufficient geometrical conditions to guarantee the existence of a local diffeomorphism which allow the transformation of the extended system into the ENOCF are given. In particular, an algorithm that permits us to compute such a diffeomorphism is derived.

A canonical discrete-time nonlinear form for reduced order observers design

AbstractThis paper deals with a normal form of a class of nonlinear dynamical systems. This form enables us to design a reduced-order observer for a class of nonlinear dynamical system. Necessary and sufficient geometrical conditions for the existence of a coordinate change to transform a given singular dynamical system into such normal form are also given.

State and feedback linearizations of single-input control systems

In this paper we address the problem of state (resp. feedback) linearization of nonlinear single-input control systems using state (resp. feedback) coordinate transformations. Although necessary and sufficient geometric conditions have been provided in the early eighties, the problems of finding the state (resp. feedback) linearizing coordinates are subject to solving systems of partial differential equations. We will provide here a solution to the two problems by defining algorithms allowing to compute explicitly the linearizing state (resp. feedback) coordinates for any nonlinear control system that is indeed linearizable (resp. feedback linearizable). Each algorithm is performed using a maximum of n − 1 steps ( n being the dimension of the system) and they are made possible by explicitly solving the Flow-box or straightening theorem. We illustrate with several examples borrowed from the literature.

Geometry Behind Chordal Loewner Chains

Loewner Theory is a deep technique in Complex Analysis affording a basis for many further important developments such as the proof of famous Bieberbach’s conjecture and well-celebrated Schramm’s stochastic Loewner evolution. It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the radial and the chordal Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there has been introduced a new approach (Bracci F et al. in Evolution families and the Loewner equation I: the unit disk. Preprint 2008. Available on ArXiv 0807.1594; Bracci F et al. in Math Ann 344:947–962, 2009; Contreras MD et al. in Loewner chains in the unit disk. To appear in Revista Matemática Iberoamericana; preprint available at arXiv:0902.3116v1 [math.CV]) bringing together, and containing as quite special cases, radial and chordal variants of Loewner Theory. In the framework of this approach we address the question what kind of systems of simply connected domains can be described by means of Loewner chains of chordal type. As an answer to this question we establish a necessary and sufficient condition for a set of simply connected domains to be the range of a generalized Loewner chain of chordal type. We also provide an easy-to-check geometric sufficient condition for that. In addition, we obtain analogous results for the less general case of chordal Loewner evolution considered in (Aleksandrov IA et al. in Complex Analysis. PWN, Warsaw, pp 7–32, 1979; Bauer RO in J Math Anal Appl 302: 484–501, 2005; Goryainov VV and Ba I in Ukrainian Math J 44:1209–1217, 1992).