Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics

Producción CientíficaWe study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carathéodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carathéodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide results of continuous dependence of the solutions as well as of continuity of the skew-product semiflows generated by Carathéodory delay differential equations when the considered phase space is a Sobolev space.MINECO/FEDER MTM2015-66330-PH2020-MSCA-ITN-2014 643073 CRITICS

On a two-point boundary value problem for second order singular equations

summary:The problem on the existence of a positive in the interval $\mathopen ]a,b\mathclose [$ solution of the boundary value problem $$u^{\prime \prime }=f(t,u)+g(t,u)u^{\prime };\quad u(a+)=0, \quad u(b-)=0$$ is considered, where the functions $f$ and $g\:\mathopen ]a,b\mathclose [\times \mathopen ]0,+\infty \mathclose [ \rightarrow \mathbb R$ satisfy the local Carathéodory conditions. The possibility for the functions $f$ and $g$ to have singularities in the first argument (for $t=a$ and $t=b$) and in the phase variable (for $u=0$) is not excluded. Sufficient and, in some cases, necessary and sufficient conditions for the solvability of that problem are established

The Carath\'eodory-Fej\'er Interpolation Problems and the von-Neumann Inequality

The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. 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 $\mathbb D^n.$ In the special case of $n=2$ (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 $n=2,$ 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 $H^2(\mathbb T^2)$ then becomes apparent. Many of our results remain valid for any $n\in \mathbb N,$ however, the computations are somewhat cumbersome for $n>2$ and are omitted. The inequality $\lim_{n\to \infty}C_2(n)\leq 2 K^\mathbb C_G$, where $K_G^\mathbb C$ is the complex Grothendieck constant and $$C_2(n)=\sup\big\{\|p(\boldsymbol T)\|:\|p\|_{\mathbb D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1 \big\}$$ is due to Varopoulos. Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples $\boldsymbol T:=(T_1,\ldots,T_n)$ of contractions. We show that $$\lim_{n\to \infty}C_2(n)\leq \frac{3\sqrt{3}}{4} K^\mathbb C_G$$ obtaining a slight improvement in the inequality of Varopoulos. We show that the normed linear space $\ell^1(n),$ $n>1,$ has no isometric embedding into $k\times k$ complex matrices for any $k\in \mathbb N$ 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

Limits of conformal images and conformal images of limits for planar random curves

Consider a chordal random curve model on a planar graph, in the scaling limit when a fine-mesh graph approximates a simply-connected planar domain. The well-known precompactness conditions of Kemppainen and Smirnov show that certain "crossing estimates" guarantee the subsequential weak convergence of the random curves in the topology of unparametrized curves, as well as in a topology inherited from curves on the unit disc via conformal maps. We complement this result by proving that proceeding to weak limit commutes with changing topology, i.e., limits of conformal images are conformal images of limits, without imposing any boundary regularity assumptions on the domains where the random curves lie. Treating such rough boundaries becomes necessary, e.g., in convergence proofs to multiple SLEs. The result in this generality has not been explicated before and is not trivial, which we demonstrate by giving warning examples and deducing strong consequences.Comment: 34 pages, 11 figures. v2: minor improvement

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