arXiv subjects

Arthur Castello B. de Oliveira

Publications and source records attributed to Arthur Castello B. de Oliveira.

6 recordsLinked to original sources

arXiv2026

Safe-by-Design: Approximate Nonlinear Model Predictive Control with Real Time Feasibility

This paper proposes a computationally lightweight, continuous-time receding-horizon nonlinear model predictive control (MPC) approach with infinitesimal prediction horizon. Unlike discrete-time one-step MPC schemes, this methodology permits the formulation of small-sized convex quadratic pro- grams (QPs) for feedback yet recovers the theoretical guarantees of quasi-infinite horizon MPC. The convex QP comes with a small computational footprint, which is advantageous for real- time application where runtime guarantees must be given. We demonstrate the effectiveness of the proposed approach when compared to other constrained control techniques through numerical experiments for nonlinear constrained spacecraft control.

math.OC↗

arXiv2025

On the (almost) Global Exponential Convergence of the Overparameterized Policy Optimization for the LQR Problem

In this work we study the convergence of gradient methods for nonconvex optimization problems -- specifically the effect of the problem formulation to the convergence behavior of the solution of a gradient flow. We show through a simple example that, surprisingly, the gradient flow solution can be exponentially or asymptotically convergent, depending on how the problem is formulated. We then deepen the analysis and show that a policy optimization strategy for the continuous-time linear quadratic regulator (LQR) (which is known to present only asymptotic convergence globally) presents almost global exponential convergence if the problem is overparameterized through a linear feed-forward neural network (LFFNN). We prove this qualitative improvement always happens for a simplified version of the LQR problem and derive explicit convergence rates for the gradient flow. Finally, we show that both the qualitative improvement and the quantitative rate gains persist in the general LQR through numerical simulations.

math.OC↗

arXiv2025

Convergence Analysis of Gradient Flow for Overparameterized LQR Formulations

Motivated by the growing use of artificial intelligence (AI) tools in control design, this paper analyses the intersection between results from gradient methods for the model-free linear quadratic regulator (LQR), and linear feedforward neural networks (LFFNNs), More specifically, it looks into the case where one wants to find a LFFNN feedback that minimizes a LQR cost. This paper starts by analyzing the structure of the gradient expression for the parameters of each layer, which implies a key conservation law of the system. This conservation law is then leveraged to generalize existing results on boundedness and global convergence of solutions to critical points, and invariance of the set of stabilizing networks under the training dynamics. This is followed by an analysis of the case where the LFFNN has a single hidden layer, for which the paper proves that the training converges not only to the critical points, but to the optimal feedback control law for all but a set of Lebesgue measure zero of the initializations. These theoretical results are followed by an extensive analysis of a simple version of the problem -- the ``vector case'' -- proving the theoretical properties of accelerated convergence and small-input input-to-state stability (ISS) for this simpler example. Finally, the paper presents numerical evidence of faster convergence of the training of general LFFNNs when compared to non-overparameterized formulations, showing that the acceleration of the solution is observable even when the gradient is not explicitly computed, but estimated from evaluations of the cost function.

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arXiv2025

Remarks on the Polyak-Lojasiewicz inequality and the convergence of gradient systems

This work explores generalizations of the Polyak-Lojasiewicz inequality (PLI) and their implications for the convergence behavior of gradient flows in optimization problems. Motivated by the continuous-time linear quadratic regulator (CT-LQR) policy optimization problem -- where only a weaker version of the PLI is characterized in the literature -- this work shows that while weaker conditions are sufficient for global convergence to, and optimality of the set of critical points of the cost function, the "profile" of the gradient flow solution can change significantly depending on which "flavor" of inequality the cost satisfies. After a general theoretical analysis, we focus on fitting the CT-LQR policy optimization problem to the proposed framework, showing that, in fact, it can never satisfy a PLI in its strongest form. We follow up our analysis with a brief discussion on the difference between continuous- and discrete-time LQR policy optimization, and end the paper with some intuition on the extension of this framework to optimization problems with L1 regularization and solved through proximal gradient flows.

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arXiv2023

On the ISS Property of the Gradient Flow for Single Hidden-Layer Neural Networks with Linear Activations

Recent research in neural networks and machine learning suggests that using many more parameters than strictly required by the initial complexity of a regression problem can result in more accurate or faster-converging models -- contrary to classical statistical belief. This phenomenon, sometimes known as ``benign overfitting'', raises questions regarding in what other ways might overparameterization affect the properties of a learning problem. In this work, we investigate the effects of overfitting on the robustness of gradient-descent training when subject to uncertainty on the gradient estimation. This uncertainty arises naturally if the gradient is estimated from noisy data or directly measured. Our object of study is a linear neural network with a single, arbitrarily wide, hidden layer and an arbitrary number of inputs and outputs. In this paper we solve the problem for the case where the input and output of our neural-network are one-dimensional, deriving sufficient conditions for robustness of our system based on necessary and sufficient conditions for convergence in the undisturbed case. We then show that the general overparametrized formulation introduces a set of spurious equilibria which lay outside the set where the loss function is minimized, and discuss directions of future work that might extend our current results for more general formulations.

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arXiv2023

Edge Selection in Bilinear Dynamical Networks

We develop some basic principles for the design and robustness analysis of a continuous-time bilinear dynamical network, where an attacker can manipulate the strength of the interconnections/edges between some of the agents/nodes. We formulate the edge protection optimization problem of picking a limited number of attack-free edges and minimizing the impact of the attack over the bilinear dynamical network.In particular, the $\mathcal{H}_2$-norm of bilinear systems is known to capture robustness and performance properties analogous to its linear counterpart and provides valuable insights for identifying which edges are most sensitive to attacks. The exact optimization problem is combinatorial in the number of edges, and brute-force approaches show poor scalability. However, we show that the $\mathcal{H}_2$-norm as a cost function is supermodular and, therefore, allows for efficient greedy approximations of the optimal solution. We illustrate and compare the effectiveness of our theoretical findings via numerical simulations.

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