673 research outputs found
Characterization of well-posedness of piecewise linear systems
One of the basic issues in the study of hybrid systems is the well-posedness (existence and uniqueness of solutions) problem of discontinuous dynamical systems. The paper addresses this problem for a class of piecewise-linear discontinuous systems under the definition of solutions of Caratheodory. The concepts of jump solutions or of sliding modes are not considered here. In this sense, the problem to be discussed is one of the most basic problems in the study of well-posedness for discontinuous dynamical systems. First, we derive necessary and sufficient conditions for bimodal systems to be well-posed, in terms of an analysis based on lexicographic inequalities and the smooth continuation property of solutions. Next, its extensions to the multimodal case are discussed. As an application to switching control, in the case that two state feedback gains are switched according to a criterion depending on the state, we give a characterization of all admissible state feedback gains for which the closed loop system remains well-pose
Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals
Among the fastest methods for solving stiff PDE are exponential integrators, which require the evaluation of , where is a negative definite matrix and is the exponential function or one of the related `` functions'' such as . Building on previous work by Trefethen and Gutknecht, Gonchar and Rakhmanov, and Lu, we propose two methods for the fast evaluation of that are especially useful when shifted systems can be solved efficiently, e.g. by a sparse direct solver. The first method method is based on best rational approximations to on the negative real axis computed via the Carathéodory-Fejér procedure, and we conjecture that the accuracy scales as , where is the number of complex matrix solves. In particular, three matrix solves suffice to evaluate to approximately six digits of accuracy. The second method is an application of the trapezoid rule on a Talbot-type contour
Fine properties of functions with bounded variation in Carnot-Carath\'eodory spaces
We study properties of functions with bounded variation in
Carnot-Ca\-ra\-th\'eo\-do\-ry spaces. We prove their almost everywhere
approximate differentiability and we examine their approximate discontinuity
set and the decomposition of their distributional derivatives. Under an
additional assumption on the space, called property , we show that
almost all approximate discontinuities are of jump type and we study a
representation formula for the jump part of the derivative
The Carath\'eodory-Fej\'er Interpolation Problems and the von-Neumann Inequality
The validity of the von-Neumann inequality for commuting - tuples of
matrices remains open for . We give a partial answer to
this question, which is used to obtain a necessary condition for the
Carath\'{e}odory-Fej\'{e}r interpolation problem on the polydisc
In the special case of (which follows from Ando's theorem as well), this
necessary condition is made explicit. An alternative approach to the
Carath\'{e}odory-Fej\'{e}r interpolation problem, in the special case of
adapting a theorem of Kor\'{a}nyi and Puk\'{a}nzsky is given. As a consequence,
a class of polynomials are isolated for which a complete solution to the
Carath\'{e}odory-Fej\'{e}r interpolation problem is easily obtained. A natural
generalization of the Hankel operators on the Hardy space of
then becomes apparent. Many of our results remain valid for any however, the computations are somewhat cumbersome for and are
omitted. The inequality , where
is the complex Grothendieck constant and
is due to Varopoulos. Here the
supremum is taken over all complex polynomials in variables of degree
at most and commuting - tuples of
contractions. We show that obtaining a slight improvement in the inequality of Varopoulos.
We show that the normed linear space has no isometric
embedding into complex matrices for any and
discuss several infinite dimensional operator space structures on it.Comment: This is my thesis submitted to Indian Institute of Science, Bangalore
on 20th July, 201
An Approach to Studying Quasiconformal Mappings on Generalized Grushin Planes
We demonstrate that the complex plane and a class of generalized Grushin
planes , where is a function satisfying specific requirements, are
quasisymmetrically equivalent. Then using conjugation we are able to develop an
analytic definition of quasisymmetry for homeomorphisms on spaces. In the
last section we show our analytic definition of quasisymmetry is consistent
with earlier notions of conformal mappings on the Grushin plane. This leads to
several characterizations of conformal mappings on the generalized Grushin
planes
Quasilinear problems involving a perturbation with quadratic growth in the gradient and a noncoercive zeroth order term
In this paper we consider the problem u in H^1_0 (Omega), - div (A(x) Du) =
H(x, u, Du) + f(x) + a_0 (x) u in D'(Omega), where Omega is an open bounded set
of R^N, N \geq 3, A(x) is a coercive matrix with coefficients in
L^\infty(Omega), H(x, s, xi) is a Carath\'eodory function which satisfies for
some gamma > 0 -c_0 A(x) xi xi \leq H(x, s, xi) sign (s) \leq gamma A(x) xi xi
a.e. x in Omega, forall s in R, forall xi in R^N, f belongs to L^{N/2} (Omega),
and a_0 \geq 0 to L^q (Omega ), q > N/2. For f and a_0 sufficiently small, we
prove the existence of at least one solution u of this problem which is
moreover such that e^{delta_0 |u|} - 1 belongs to H^1_0 (Omega) for some
delta_0 \geq gamma, and which satisfies an a priori estimate.Comment: 37 pages, 2 figure
Maximizing the probability of attaining a target prior to extinction
We present a dynamic programming-based solution to the problem of maximizing
the probability of attaining a target set before hitting a cemetery set for a
discrete-time Markov control process. Under mild hypotheses we establish that
there exists a deterministic stationary policy that achieves the maximum value
of this probability. We demonstrate how the maximization of this probability
can be computed through the maximization of an expected total reward until the
first hitting time to either the target or the cemetery set. Martingale
characterizations of thrifty, equalizing, and optimal policies in the context
of our problem are also established.Comment: 22 pages, 1 figure. Revise
Invariant subspaces for operators whose spectra are Carathéodory regions
AbstractIn this paper it is shown that if an operator T satisfies ‖p(T)‖⩽‖p‖σ(T) for every polynomial p and the polynomially convex hull of σ(T) is a Carathéodory region whose accessible boundary points lie in rectifiable Jordan arcs on its boundary, then T has a nontrivial invariant subspace. As a corollary, it is also shown that if T is a hyponormal operator and the outer boundary of σ(T) has at most finitely many prime ends corresponding to singular points on ∂D and has a tangent at almost every point on each Jordan arc, then T has a nontrivial invariant subspace
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