arXiv subjects

Bogdan Gheorghe

Publications and source records attributed to Bogdan Gheorghe.

10 recordsLinked to original sources

arXiv2026

Data-Driven Tube-Based Zonotopic Predictive Control With Nonconvex Layered Terminal Sets

This paper presents a data-driven tube-based zonotopic predictive control (DTZPC) framework with nonconvex layered terminal sets. Existing DTZPC schemes with closed-loop guarantees typically rely on a single ellipsoidal terminal set, which can be conservative and thereby limit feasibility. We propose a layered terminal-set design that decouples stability certification, feasibility enlargement, and motion-region screening into three components with distinct roles. First, an offline-designed feedback gain together with a contractive constrained zonotope provides a terminal ingredient for stability certification, while avoiding probabilistic feedback synthesis in high-dimensional DTZPC. Second, we derive a data-driven characterization of the inverse admissible closed-loop model set, avoiding the conservatism of interval-matrix relaxation and inversion. Combined with exact set multiplication, this yields inner and outer approximations of the maximal robust positively invariant (MRPI) set under fixed closed-loop dynamics. The inner approximation serves as a nonconvex terminal set to enlarge feasibility, whereas the outer approximation provides certified motion-region descriptions for fast screening and monitoring. Numerical examples demonstrate tighter inverse-set enclosures and improved feasibility over existing convex-terminal DTZPC schemes.

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arXiv2026

Inclusion conditions for the Constrained Polynomial Zonotopic case

Set operations are well understood for convex sets but become considerably more challenging in the non-convex case due to the loss of structural properties in their representation. Constrained polynomial zonotopes (CPZs) offer an effective compromise, as they can capture complex, typically non-convex geometries while maintaining an algebraic structure suitable for further manipulation. Building on this, we propose novel nonlinear encodings that provide sufficient conditions for testing inclusion between two CPZs and adapt them for seamless integration within optimization frameworks.

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arXiv2026

On maximal positive invariant set computation for rank-deficient linear systems

The maximal positively invariant (MPI) set is obtained through a backward reachability procedure involving the iterative computation and intersection of predecessor sets under state and input constraints. However, standard static feedback synthesis may place some of the closed-loop eigenvalues at zero, leading to rank-deficient dynamics. This affects the MPI computation by inducing projections onto lower-dimensional subspaces during intermediate steps. By exploiting the Schur decomposition, we explicitly address this singular case and propose a robust algorithm that computes the MPI set in both polyhedral and constrained-zonotope representations.

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arXiv2025

Tight displacement-based formation control under bounded disturbances. A set-theoretic perspective

This paper investigates the synthesis of controllers for displacement-based formation control in the presence of bounded disturbances, specifically focusing on uncertainties originating from measurement noise. While the literature frequently addresses such problems using stochastic frameworks, this work proposes a deterministic methodology grounded in set-theoretic concepts. By leveraging the principles of set invariance, we adapt the theory of ultimate boundedness to the specific dynamics of displacement-based formations. This approach provides a rigorous method for analyzing the system's behavior under persistent disturbances. Furthermore, this set-theoretic framework allows for the optimized selection of the proposed control law parameters to guarantee pre-specified performance bounds. The efficacy of the synthesized controller is demonstrated in the challenging application of maintaining tight formations in a multi-obstacles environment.

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arXiv2020

The special fiber of the motivic deformation of the stable homotopy category is algebraic

For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/τ\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/τ$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $BP_*BP$-comodules that are concentrated in even degrees. We prove that $\widehat{S^{0,0}}/τ\text{-}\mathbf{Mod}_{harm}^b$ is equivalent to $\mathcal{D}^b({{BP}_*{BP}\text{-}\mathbf{Comod}}^{ev})$ as stable $\infty$-categories equipped with $t$-structures. As an application, for each prime $p$, we prove that the motivic Adams spectral sequence for $\widehat{S^{0,0}}/τ$, which converges to the motivic homotopy groups of $\widehat{S^{0,0}}/τ$, is isomorphic to the algebraic Novikov spectral sequence, which converges to the classical Adams-Novikov $E_2$-page for the sphere spectrum $\widehat{S^0}$. This isomorphism of spectral sequences allows Isaksen and the second and third authors to compute the stable homotopy groups of spheres at least to the 90-stem, with ongoing computations into even higher dimensions.

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arXiv2018

C-motivic modular forms

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and we construct a "motivic modular forms" spectrum over C.

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arXiv2017

Exotic Motivic Periodicities

One can attempt to study motivic homotopy groups by mimicking the classical (non-motivic) chromatic approach. There are however major differences, which makes the motivic story more complicated and still not well understood. For example, classically the $p$-local sphere spectrum $S^0_{(p)} $ admits an essentially unique non-nilpotent self-map, which is not the case motivically, since Morel showed that the first Hopf map $η\colon S^{1,1} \to S^{0,0}$ is non-nilpotent. In the same way that the non-nilpotent self-map $2 = v_0 \in π_{\ast,\ast}(S^{0,0})$ starts the usual chromatic story of $v_n$-periodicity, there is a similar theory starting with the non-nilpotent element $η\in π_{\ast,\ast}(S^{0,0})$, which Andrews-Miller denoted by $η= w_0$. In this paper we investigate the beginning of the motivic story of $w_n$-periodicity when the base scheme is $\mathbf{Spec} \! \ \mathbb{C}$. In particular, we construct motivic fields $K(w_n)$ designed to detect such $w_n$-periodic phenomena, in the same way that $K(n)$ detects $v_n$-periodic phenomena. In the hope of detecting motivic nilpotence, we also construct a more global motivic spectrum $wBP$ with homotopy groups $π_{\ast,\ast}(wBP) \cong \mathbb{F}_2[w_0, w_1, \ldots]$.

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arXiv2017

The Motivic Cofiber of $τ$

Consider the Tate twist $τ\in H^{0,1}(S^{0,0})$ in the mod 2 cohomology of the motivic sphere. After 2-completion, the motivic Adams spectral sequence realizes this element as a map $τ\colon S^{0,-1} \to S^{0,0}$, with cofiber $Cτ$. We show that this motivic 2-cell complex can be endowed with a unique $E_{\infty}$ ring structure. Moreover, this promotes the known isomorphism $π_{\ast,\ast} Cτ\cong \mathrm{Ext}^{\ast,\ast}_{BP_{\ast}BP}(BP_{\ast},BP_{\ast})$ to an isomorphism of rings which also preserves higher products. We then consider the closed symmetric monoidal category $({ }_{Cτ}\textbf{Mod}, - \wedge_{Cτ} -)$ which lives in the kernel of Betti realization. Given a motivic spectrum $X$, the $Cτ$-induced spectrum $X \wedge Cτ$ is usually better behaved and easier to understand than $X$ itself. We specifically illustrate this concept in the examples of the mod 2 Eilenberg-Maclane spectrum $H\mathbb{F}_2$, the mod 2 Moore spectrum $S^{0,0}/2$ and the connective hermitian $K$-theory spectrum $kq$.

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arXiv2016

The Picard group of motivic A(1)

We show that the Picard group $Pic(A(1))$ of the stable category of modules over $\mathbb{C}$-motivic $A(1)$ is isomorphic to $\mathbb{Z}^4$. By comparison, the Picard group of classical $A(1)$ is $\mathbb{Z}^2 \oplus \mathbb{Z}/2$. One extra copy of $\mathbb{Z}$ arises from the motivic bigrading. The joker is a well-known exotic element of order $2$ in the Picard group of classical $A(1)$. The $\mathbb{C}$-motivic joker has infinite order.

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arXiv2015

The Structure of Motivic Homotopy Groups

We study the stable motivic homotopy groups $π_{s,w}$ of the 2-completion of the motivic sphere spectrum over $\mathbb{C}$. When arranged in the $(s,w)$-plane, these groups break into four different regions: a vanishing region, an $η$-local region that is entirely known, a $τ$-local region that is identical to classical stable homotopy groups, and a region that is not well-understood.

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