Averaged observations and turnpike phenomenon for parameter-dependent systems - Hernández Salinas, Martin; Lazar, Martin; Zamorano, Sebastian (2024)
Our main contribution in this article is the achievement of the turnpike property in its integral and exponential forms for parameter-dependent systems with averaged observations in the cost functional. Namely, under suitable assumptions with respect to the matrices that defined the dynamics and the cost functional, we prove that the optimal control and state for the evolutionary problem converge in average to the optimal pair of an associated stationary problem. Moreover, we characterize the closeness between these two optimal solutions, proving that over a large time interval, they are exponentially close.<br
https://cris.fau.de/converis/publicweb/Publication/326485232
Keywords: Parameter dependent problem; Averaged observations; Turnpike property

AVERAGED TURNPIKE PROPERTY FOR DIFFERENTIAL EQUATIONS WITH RANDOM CONSTANT COEFFICIENTS - Hernández Salinas, Martin; Lecaros, Rodrigo; Zamorano, Sebastian (Mathematical Control and Related Fields, 2022)
This paper studies the integral turnpike and turnpike in average for a class of random ordinary differential equations. We prove that, under suitable assumptions on the matrices that define the system, the optimal solutions for an optimal distributed control tracking problem remain, in an averaged sense, sufficiently close to the associated random stationary optimal solution for the majority of the time horizon.
CRIS-Team WoS Importer:2022-04-29
Keywords: Turnpike property; exponential turnpike property; random ODE; long time behavior; model predictive control

Deep Neural Networks: Multi-Classification and Universal Approximation - Hernández Salinas, Martin; Zuazua Iriondo, Enrique (2025)
We demonstrate that a ReLU deep neural network with a width of 2 and a depth of 2N+4M−1 layers can achieve finite sample memorization for any dataset comprising N elements in Rd, where d≥1, and M classes, thereby ensuring accurate classification. By modeling the neural network as a time-discrete nonlinear dynamical system, we interpret the memorization property as a problem of simultaneous or ensemble controllability. This problem is addressed by constructing the network parameters inductively and explicitly, bypassing the need for training or solving any optimization problem. Additionally, we establish that such a network can achieve universal approximation in Lp(Ω;R+), where Ω is a bounded subset of Rd and p∈[1,∞), using a ReLU deep neural network with a width of d+1. We also provide depth estimates for approximating W1,p functions and width estimates for approximating Lp(Ω;Rm) for m≥1. Our proofs are constructive, offering explicit values for the biases and weights involved.<br
https://cris.fau.de/converis/publicweb/Publication/329303154
Keywords: Deep neuronal network; Finite sample memorization; Simultaneous controllability; Nonlinear discrete dynamics; Universal approximation theorem

Uniform Turnpike Property and Singular Limits - Hernández Salinas, Martin; Zuazua Iriondo, Enrique (Acta Applicandae Mathematicae, 2024)
Motivated by singular limits for long-time optimal control problems, we investigate a class of parameter-dependent parabolic equations. First, we prove a turnpike result, uniform with respect to the parameters within a suitable regularity class and under appropriate bounds. The main ingredient of our proof is the justification of the uniform exponential stabilization of the corresponding Riccati equations, which is derived from the uniform null control properties of the model. Then, we focus on a heat equation with rapidly oscillating coefficients. In the one-dimensional setting, we obtain a uniform turnpike property with respect to the highly oscillatory heterogeneous medium. Afterward, we establish the homogenization of the turnpike property. Finally, our results are validated by numerical experiment
Keywords: Turnpike property; Rapidly oscillating linear parabolic equation; Optimal control problems; Long time behavior; Uniform controllability; Singular limits problems.