Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.4205

Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations

Giordano Pola·Paulo Tabuada
View original source ↗Read source document ↗Export metadata ↓

Abstract

Symbolic models are abstract descriptions of continuous systems in which symbols represent aggregates of continuous states. In the last few years there has been a growing interest in the use of symbolic models as a tool for mitigating complexity in control design. In fact, symbolic models enable the use of well known algorithms in the context of supervisory control and algorithmic game theory, for controller synthesis. Since the 1990's many researchers faced the problem of identifying classes of dynamical and control systems that admit symbolic models. In this paper we make a further progress along this research line by focusing on control systems affected by disturbances. Our main contribution is to show that incrementally globally asymptotically stable nonlinear control systems with disturbances admit symbolic models. When specializing these results to linear systems, we show that these symbolic models can be easily constructed.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Giordano Pola, Paulo Tabuada. 2007-07-28. Symbolic Models for Nonlinear Control Systems: Alternating Approximate Bisimulations. https://arxiv.org/abs/0707.4205

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

ItsDEAL: Inexact two-level smoothing descent algorithms for weakly convex optimization

This paper addresses nonconvex optimization problems via a two-level smoothing framework, where the high-order Moreau envelope (HOME) provides a smooth approximation of weakly convex cost functions. We study the differentiability and weak smoothness of HOME. Building on the inexact two-level smoothing optimization framework (ItsOPT), we propose an inexact two-level smoothing descent algorithm (ItsDEAL) by: (i) approximately solving the proximal subproblem to obtain an inexact first-order oracle for HOME at the lower level; and (ii) applying an inexact first-order method at the upper level. We derive computable sufficient conditions for lower-level inexactness, enabling approximate high-order proximal-points to be certified without access to the exact proximal-point or HOME value. In particular, we study a parameter-free Hölderian inexact gradient method and an inexact descent method with Armijo line search, both exploiting the weak smoothness of HOME. We establish subsequential convergence under mild inexactness assumptions, and global and linear convergence under the Kurdyka--Łojasiewicz (KL) property. Preliminary experiments on ridge-regularized hinge-loss classification, a high-dimensional MCP-regularized classification model, and robust matrix completion illustrate the numerical behavior of the proposed methods and the role of the HOME order.

math.OC↗
arXiv2026

OFF Newton-type Methods for Unconstrained Optimization with Inexact Derivatives

We study objective-function-free (OFF) Newton-type methods for smooth unconstrained optimization with inexact gradients and Hessians. We propose two methods for first- and second-order stationarity, coupling explicit step-size rules with derivative-accuracy and linear-solver residual conditions to guarantee sufficient decrease without evaluating the objective function. The OFF regularized Newton method finds an $\varepsilon$-first-order stationary point in $\mathcal{O}(\varepsilon^{-2})$ iterations. Under strong convexity, a local Newton variant converges with \(Q\)-order at least $1+θ$, where \(θ\in (0, 1]\). For second-order stationarity, we combine regularized Newton steps with approximate eigenvectors associated with the minimum eigenvalue. With high probability, the resulting method finds an $(\varepsilon, \sqrt{L_h\varepsilon})$-second-order stationary point in $\mathcal{O}(\varepsilon^{-3/2})$ iterations, using a total of $\widetilde{\mathcal{O}}(\varepsilon^{-7/4})$ approximate gradient evaluations and Hessian-vector products. For finite-sum problems, we develop subsampling schemes that yield best-iterate convergence in expectation in the nonconvex first-order setting and local \(R\)-linear convergence with high probability in the strongly convex setting, followed by \(Q\)-order at least \(1+θ\) once full sampling is reached. The subsampled second-order method achieves a high-probability $\widetilde{\mathcal{O}}(\varepsilon^{-7/2})$ total sample-operation bound, counting component gradient evaluations and component Hessian-vector products.

math.OC↗
arXiv2026

An efficient mixed-integer linear programming formulation for solving influence diagrams

Influence diagrams represent decision-making problems with interdependencies between random events, decisions, and consequences. Traditionally, they have been solved using tailored algorithms that determine the expected utility-maximizing decision strategy. Towards this end, most recent state-of-the-art solution approaches convert them into mixed-integer linear programming (MILP) models that can be solved using off-the-shelf MILP solvers. Computationally, these approaches are most efficient for influence diagrams that represent periodic (or sequential) decision processes, which can be cast as partially observable Markov Decision Processes. However, they are inefficient for problems that lack this periodic structure or whose influence diagrams have large state spaces. The efficient MILP formulation presented in this paper is specifically designed for influence diagrams that are challenging for earlier MILP formulations. Additionally, this formulation can be adapted to optimize conditional value-at-risk and to incorporate chance and logical constraints, thereby retaining the modeling flexibility of MILP-based methods. Finally, computational experiments on a range of problems from the literature are reported to show that the proposed formulation is much more efficient than earlier MILP formulations, particularly in problems which cannot be modeled as partially observable Markov decision processes and which lead to influence diagrams whose decision or chance nodes may have large state spaces.

math.OC↗