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Mathematical Control & Related Fields

2013 , Volume 3 , Issue 3

Special Issue dedicated to Professor George Leitmann on the occasion of his 88th birthday (Part II)

Special Issue Papers: 389-489; Regular Papers: 491-599

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Preface
J.-B. Caillau, Monique Chyba, Dominique Sugny and Emmanuel Trélat
2013, 3(3): i-ii doi: 10.3934/mcrf.2013.3.3i +[Abstract](230) +[PDF](85.7KB)
Abstract:
Life is singular in many ways, and Prof. Bernard Bonnard is aware of this better than anybody else. Throughout his work and career, he has shown us that following a singular path is not only exciting and interesting but most importantly that it is often the most efficient way to accomplish a goal!

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Estimates on trajectories in a closed set with corners for $(t,x)$ dependent data
Piernicola Bettiol and Richard Vinter
2013, 3(3): 245-267 doi: 10.3934/mcrf.2013.3.245 +[Abstract](300) +[PDF](474.5KB)
Abstract:
Estimates on the distance of a given process from the set of processes that satisfy a specified state constraint in terms of the state constraint violation are important analytical tools in state constrained optimal control theory; they have been employed to ensure the validity of the Maximum Principle in normal form, to establish regularity properties of the value function, to justify interpreting the value function as a unique solution of the Hamilton-Jacobi equation, and for other purposes. A range of estimates are required, which differ according the metrics used to measure the `distance' and the modulus $\theta(h)$ of state constraint violation $h$ in terms of which the estimates are expressed. Recent research has shown that simple linear estimates are valid when the state constraint set $A$ has smooth boundary, but do not generalize to a setting in which the boundary of $A$ has corners. Indeed, for a velocity set $F$ which does not depend on $(t,x)$ and for state constraints taking the form of the intersection of two closed spaces (the simplest case of a boundary with corners), the best distance estimates we can hope for, involving the $W^{1,1,}$ metric on state trajectories, is a super-linear estimate expressed in terms of the $h|\log(h)|$ modulus. But, distance estimates involving the $h|\log (h)|$ modulus are not in general valid when the velocity set $F(.,x)$ is required merely to be continuous, while not even distance estimates involving the weaker, Hölder modulus $h^{\alpha}$ (with $\alpha$ arbitrarily small) are in general valid, when $F(.,x)$ is allowed to be discontinuous. This paper concerns the validity of distance estimates when the velocity set $F(t,x)$ is $(t,x)$-dependent and satisfy standard hypotheses on the velocity set (linear growth, Lipschitz $x$-dependence and an inward pointing condition). Hypotheses are identified for the validity of distance estimates, involving both the $h|\log(h)|$ and linear moduli, within the framework of control systems described by a controlled differential equation and state constraint sets having a functional inequality representation.
On the motion planning of the ball with a trailer
Nicolas Boizot and Jean-Paul Gauthier
2013, 3(3): 269-286 doi: 10.3934/mcrf.2013.3.269 +[Abstract](276) +[PDF](2330.4KB)
Abstract:
This paper is about motion planing for kinematic systems, and more particularly $\epsilon$-approximations of non-admissible trajectories by admissible ones. This is done in a certain optimal sense.
    The resolution of this motion planing problem is showcased through the thorough treatment of the ball with a trailer kinematic system, which is a non-holonomic system with flag of type $(2,3,5,6)$.
The Serret-Andoyer Riemannian metric and Euler-Poinsot rigid body motion
Bernard Bonnard, Olivier Cots and Nataliya Shcherbakova
2013, 3(3): 287-302 doi: 10.3934/mcrf.2013.3.287 +[Abstract](363) +[PDF](389.2KB)
Abstract:
The Euler-Poinsot rigid body motion is a standard mechanical system and is the model for left-invariant Riemannian metrics on $SO(3)$. In this article, using the Serret-Andoyer variables we parameterize the solutions and compute the Jacobi fields in relation with the conjugate locus evaluation. Moreover the metric can be restricted to a 2D surface and the conjugate points of this metric are evaluated using recent work [4] on surfaces of revolution.
Phantom tracking method, homogeneity and rapid stabilization
Jean-Michel Coron
2013, 3(3): 303-322 doi: 10.3934/mcrf.2013.3.303 +[Abstract](283) +[PDF](441.2KB)
Abstract:
In this paper we explain on various examples the ``phantom tracking'' method, a method which can be used to stabilize nonlinear control systems modeled by ordinary differential equations or partial differential equations. We show how it can handle global controllability, homogeneity issues or fast stabilization.
Asymptotic stability of uniformly bounded nonlinear switched systems
Philippe Jouan and Said Naciri
2013, 3(3): 323-345 doi: 10.3934/mcrf.2013.3.323 +[Abstract](338) +[PDF](408.4KB)
Abstract:
We study the asymptotic stability properties of nonlinear switched systems under the assumption of the existence of a common weak Lyapunov function.
    We consider the class of nonchaotic inputs, which generalize the different notions of inputs with dwell-time, and the class of general ones. For each of them we provide some sufficient conditions for asymptotic stability in terms of the geometry of certain sets.
Affine-quadratic problems on Lie groups
Velimir Jurdjevic
2013, 3(3): 347-374 doi: 10.3934/mcrf.2013.3.347 +[Abstract](328) +[PDF](526.7KB)
Abstract:
This paper focuses on a class of left invariant variational problems on a Lie group$\ G\ $whose Lie algebra $\mathfrak{g}$ admits Cartan decomposition $ \mathfrak{g}=\mathfrak{p}+\mathfrak{k}$ with the usual Lie algebraic conditions \begin{equation*} \lbrack \mathfrak{p},\mathfrak{p]\subseteq k\ },\ \mathfrak{[p},\mathfrak{ k]\subseteq p},\mathfrak{\ [k},\mathfrak{k]\subseteq k.} \end{equation*}         The Maximum Principle of optimal control leads to the Hamiltonians $H$ on $ \mathfrak{g\ }$that admit spectral parameter representations with important contributions to the theory of integrable Hamiltonian systems. Particular cases will be singled out that provides natural explanations for the classical results of Fock and Moser linking Kepler's problem to the geodesics on spaces of constant curvature, C.L. Jacobi's geodesic problem on an ellipsoid and J.Moser's work on integrability based on isospectral methods. The paper also shows the relevance of this class of Hamiltonians to the elastic curves on spaces of constant curvature.

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