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Complex calculus of variations

Michel Gondran, Rita Hoblos Saade (2003)

Kybernetika

In this article, we present a detailed study of the complex calculus of variations introduced in [M. Gondran: Calcul des variations complexe et solutions explicites d’équations d’Hamilton–Jacobi complexes. C.R. Acad. Sci., Paris 2001, t. 332, série I]. This calculus is analogous to the conventional calculus of variations, but is applied here to 𝐂 n functions in 𝐂 . It is based on new concepts involving the minimum and convexity of a complex function. Such an approach allows us to propose explicit solutions...

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

Yuri L. Sachkov (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.

Continuous extension of order-preserving homogeneous maps

Andrew D. Burbanks, Colin T. Sparrow, Roger D. Nussbaum (2003)

Kybernetika

Maps f defined on the interior of the standard non-negative cone K in ℝ N which are both homogeneous of degree 1 and order-preserving arise naturally in the study of certain classes of Discrete Event Systems. Such maps are non-expanding in Thompson’s part metric and continuous on the interior of the cone. It follows from more general results presented here that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that the extension must have at least...

Control affine systems on solvable three-dimensional Lie groups, I

Rory Biggs, Claudiu C. Remsing (2013)

Archivum Mathematicum

We seek to classify the full-rank left-invariant control affine systems evolving on solvable three-dimensional Lie groups. In this paper we consider only the cases corresponding to the solvable Lie algebras of types II, IV, and V in the Bianchi-Behr classification.

Control structures

Robert Bryant, Robert Gardner (1995)

Banach Center Publications

We define an extension of the classical notion of a control system which we call a control structure. This is a geometric structure which can be defined on manifolds whose underlying topology is more complicated than that of a domain in ℝ n . Every control structure turns out to be locally representable as a classical control system, but our extension has the advantage that it has various naturality properties which the (classical) coordinate formulation does not, including the existence of so-called...

Control Systems on the Orthogonal Group SO(4)

Ross M. Adams, Rory Biggs, Claudiu C. Remsing (2013)

Communications in Mathematics

We classify the left-invariant control affine systems evolving on the orthogonal group S O ( 4 ) . The equivalence relation under consideration is detached feedback equivalence. Each possible number of inputs is considered; both the homogeneous and inhomogeneous systems are covered. A complete list of class representatives is identified and controllability of each representative system is determined.

Controllability of 3D low Reynolds number swimmers

Jérôme Lohéac, Alexandre Munnier (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done...

Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane*

Yuri L. Sachkov (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained....

Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane*

Yuri L. Sachkov (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is considered. In the previous works [Moiseev and Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009004; Sachkov, ESAIM: COCV, DOI: 10.1051/cocv/2009031], extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained....

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