1,721,063 research outputs found
Homogeneous Tangent Vectors and High Order Necessary Conditions for Optimal Controls
No full textSummary:
The author introduces and analyzes "homogeneous" tangent vectors which provide high-order approximations to the attainable set for an affine control system of the form . Homogeneous tangent vectors are defined relative to one-parameter families of dilations on . Adjoint equations associated with the corresponding homogeneous variational equation are derived and used to transport homogeneous tangent vectors along the flow of a reference trajectory. These constructions are then used to derive a homogeneous high-order test for optimality of control problems in Mayer form without terminal constraints. Essentially, it is shown that if is a homogeneous tangent vector with respect to a dilation generated by a control variation, then it is a necessary condition for optimality that , where denotes the solution of the corresponding homogeneous adjoint equation
Decomposition of homogeneous vector fields of degree one and representation of the flow
Full text linkSummary:
The paper gives a decomposition theorem for the elements of the nonsemisimple Lie algebra of the vector fields on that are homogeneous of degree one with respect to a dilation Each is proved to be equal to with and linear semisimple. As a consequence, the author proves that "in absence of esonance" the vector field is equivalent to its linear part. Finally, the above results are applied to obtain a representation formula for the trajectories of a vector field and those of the affine control system with constant of minimum degree
Normal Forms for Vector Fields with respect to an Arbitrary Dilation
No full textSummary:
The paper is devoted to the study of real analytic vector fields, which can be expanded in terms of homogeneous fields of degree greater than or equal to one with respect to an arbitrary dilation. Necessary and sufficient conditions are given for the existence of a local coordinate change that transforms the field into the homogeneous field of degree one
On the convergence rate of the Glimm scheme for general nonlinear hyperbolic systems
No full text[RIASSUNTO (ABSTRACT)] L'autore era invited speaker del Workshop ed ha tenuto un seminario l'8 dicembre 2008. Un sunto della conferenza e' stato pubblicato nel volume n. 56 della serie "Oberwolfach Reports". Ulteriori in formazioni sul Workshop sono disponibili sul sito:
http://www.mfo.de/cgi-bin/tagung_espe?type=21&tnr=0850
Organizers:
Constantine Dafermos (Brown University), Dietmar Kröner (Freiburg University),
Randall J. LeVeque (University of Washington)
A wavefront tracking algorithm for N×N nongenuinely nonlinear conservation laws
No full textAbstractWe introduce a wavefront tracking algorithm for N×N hyperbolic systems of conservation lawsut+F(u)x=0, that admits characteristic fields that are neither genuinely nonlinear nor linearly degenerate in the sense of Lax. Instead we assume that, for any nongenuinely nonlinear ith characteristic family, the derivative of the ith eigenvalue λi(u) of DF(u) in the direction of the ith right eigenvector ri(u), vanishes on a single (N−1)-dimensional hypersurface in the u-space, transversal to the field ri(u). Systems that fulfill this type of assumptions are of particular interest in studying elastodynamic or rigid heat conductors at low temperature. The first proof of the existence of weak solutions for nongenuinely nonlinear systems was given by T. P. Liu (Mem. Amer. Math. Soc.30 (1981), no. 240), based on a Glimm scheme. Our construction here provides an alternative method for establishing the global existence of weak solutions for such systems. Moreover, relying on the stability analysis developed in Ancona and Marson, preprint S.I.S.S.A.-I.S.A.S. 27/99/11, 1999, and preprint, 2000, we show that these solutions are entropy admissible in the sense of Lax
Well-posedness for general 2 x 2 conservation laws
No full textWe consider the Cauchy problem for a strictly hyperbolic
system of conservation laws in one space dimension
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which is neither linearly degenerate nor genuinely non-linear.
We make the following assumption on the characteristic fields.
If denotes the -th right eigenvector of
and the corresponding eigenvalue, then
the set is
a smooth curve
in the -plane that is transversal to the vector field
Systems of conservation laws that fulfill such assumptions arise in studying
elastodynamics
or rigid heat conductors at low temperature.
For such systems we prove the existence
of a closed domain \ \Cal D
\subset L^1, \ containing all functions with sufficiently
small total variation, and of a uniformly Lipschitz continuous
semigroup S:{\Cal D} \times [0,+\infty)\rightarrow \Cal D
with the following properties.
Each trajectory \ \ of is a weak
solution of (1). Viceversa, if a piecewise Lipschitz,
entropic solution of (1) exists for
then it coincides with the trajectory of ,
i.e.
This result yields the uniqueness and continuous dependence of
weak, entropy-admissible solutions of the Cauchy problem (1)
with small initial data, for systems satysfying the above assumption
On the attainable set for scalar nonlinear conservation laws with boundary control
No full textSummary:
The paper treats the initial boundary value problem for a scalar conservation law with strictly convex flux function. The boundary data is a Lebesgue-measurable and bounded function regarded as a control and constrained to remain in a prescribed set of admissible controls. A time being fixed, the authors characterize the set consisting of the corresponding entropy solutions at the time . Under natural assumptions on , it is proven that is a compact subset of . Such a compactness property provides the key information in order to establish the existence of solutions for a class of optimisation problems. Finally the results are applied by the authors to an optimisation problem concerning a model of traffic flow on a highway
A locally quadratic Glimm functional and sharp convergence rate of the Glimm scheme for nonlinear hyperbolic systems
No full textConsider the Cauchy problem for a strictly hyperbolic,
quasilinear system in one space dimension
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u_t+A(u) u_x=0,\qquad u(0,x)=\bar u(x),
\eqno (1)
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where is a smooth matrix-valued map, and the initial
data is assumed to have small total variation. We
investigate the rate of convergence of approximate solutions of (1)
constructed by the Glimm scheme, under the assumption that, letting
, denote the -th eigenvalue and a
corresponding eigenvector of , respectively, for each -th
characteristic family the linearly degenerate manifold
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is either the whole space, or it is empty, or it consists of a finite
number of smooth, -dimensional, connected, manifolds that are
transversal to the characteristic vector field .
We introduce a Glimm type functional which is the sum of the cubic
interaction potential defined in \cite{sie}, and of a quadratic term
that takes into account interactions of waves of the same family with
strength smaller than some fixed threshold parameter. Relying on an
adapted wave tracing method, and on the decrease amount of such a
functional, we obtain the same type of error estimates valid for Glimm
approximate solutions of hyperbolic systems satisfying the classical
Lax assumptions of genuine nonlinea\-ri\-ty or linear degeneracy of
the characteristic families
On the Glimm functional for general hyperbolic systems
No full textIn this note we discuss the features of a quadratic interaction
potential for first order hyperbolic systems in one space dimension,
aiming to suggest the need of defining a non local functional depending
on the global wave structure of the solution in order to obtain
a quadratic Glimm type functional for general hyperbolic systems
Uniqueness and Stability of Solutions for Temple Class Systems with Boundary and Properties of the Attainable Sets
No full textSummary:
The authors consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws on the quarter t-x plane
where . For a class of initial and boundary data in with possibly unbounded variation, they construct a flow of solutions that depend continuously, in the distance, both on the initial data and on the boundary data. Moreover, we show that each trajectory of such flow provides the unique weak solution of the corresponding initial-boundary value problem that satisfies an entropy condition of Oleinik type.
Next, they study the initial-boundary value problem from the point of view of control theory, taking the initial data fixed and considering, in connection with a prescribed set of boundary data regarded as admissible controls, the set of attainable profiles at a fixed time and at a fixed point ,
establishing closure and compactness of such sets in the topolog..
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