Reachability problems for PAMs

Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other systems. The reachability problem for one-dimentional PAM is still open even if we define it with only two intervals. As the main contribution of this paper we introduce new techniques for solving reachability problems based on p-adic norms and weights as well as showing decidability for two classes of maps. Then we show the connections between topological properties for PAM's orbits, reachability problems and representation of numbers in a rational base system. Finally we show a particular instance where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional part in that sequence.Comment: 16 page

Optimal and H∞H_\infty Control of Stochastic Reaction Networks

Stochastic reaction networks is a powerful class of models for the representation a wide variety of population models including biochemistry. The control of such networks has been recently considered due to their important implications for the control of biological systems. Their optimal control, however, has been relatively few studied until now. The continuous-time finite-horizon optimal control problem is formulated first and explicitly solved in the case of unimolecular reaction networks. The problems of the optimal sampled-data control, the continuous $H_\infty$ control, and the sampled-data $H_\infty$ control of such networks are addressed next. The results in the unimolecular case take the form of nonstandard Riccati differential equations or differential Lyapunov equations coupled with difference Riccati equations, which can all be solved numerically by backward-in-time integration.Comment: 39 page

A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces

The present paper presents a counterexample to the sequentially weak density of smooth maps between two manifolds $M$ and $N$ in the Sobolev space $W^{1, p} (M, N)$, in the case $p$ is an integer. It has been shown that, if $p<\dim M$ is not an integer and the $[p]$-th homotopy group $\pi_{[p]}(N)$ of $N$ is not trivial, $[p]$ denoting the largest integer less then $p$, then smooth maps are not sequentially weakly dense in $W^{1, p} (M, N)$ for the strong convergence. On the other, in the case $p< \dim M$ is an integer, examples have been provided where smooth maps are actually sequentially weakly dense in $W^{1, p} (M, N)$ with $\pi_{p}(N)\not = 0$. This is the case for instance for $M= \mathbb B^m$, the standard ball in $\mathbb R^m$, and $N=\mathbb S^p$ the standard sphere of dimension $p$, for which $\pi_{p}(N) =\mathbb Z$. The main result of this paper shows however that such a property does not holds for arbitrary manifolds $N$ and integers $p$.Our counterexample deals with the case $p=3$, $\dim M\geq 4$ and $N=\mathbb S^2$, for which the homotopy group $\pi_3(\mathbb S^2)=\mathbb Z$ is related to the Hopf fibration.Comment: 68 page

Virtual Control Contraction Metrics: Convex Nonlinear Feedback Design via Behavioral Embedding

This paper proposes a novel approach to nonlinear state-feedback control design that has three main advantages: (i) it ensures exponential stability and $\mathcal{L}_2$-gain performance with respect to a user-defined set of reference trajectories, and (ii) it provides constructive conditions based on convex optimization and a path-integral-based control realization, and (iii) it is less restrictive than previous similar approaches. In the proposed approach, first a virtual representation of the nonlinear dynamics is constructed for which a behavioral (parameter-varying) embedding is generated. Then, by introducing a virtual control contraction metric, a convex control synthesis formulation is derived. Finally, a control realization with a virtual reference generator is computed, which is guaranteed to achieve exponential stability and $\mathcal{L}_2$-gain performance for all trajectories of the targeted reference behavior. Connections with the linear-parameter-varying (LPV) theory are also explored showing that the proposed methodology is a generalization of LPV state-feedback control in two aspects. First, it is a unified generalization of the two distinct categories of LPV control approaches: global and local methods. Second, it provides rigorous stability and performance guarantees when applied to the true nonlinear system, while such properties are not guaranteed for tracking control using LPV approaches