Friedrich-Alexander-Universität Erlangen-Nürnberg
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In this article, we explore the effects of memory terms in continuous-layer Deep Residual Networks by studying Neural ODEs (NODEs). We investigate two types of models. On one side, we consider the case of Residual Neural Networks with dependence on multiple layers, more precisely Momentum ResNets. On the other side, we analyse a Neural ODE with auxiliary states playing the role of memory states. We examine the interpolation and universal approximation properties for both architectures through a simultaneous control perspective. We also prove the ability of the second model to represent sophisticated maps, such as parametrizations of time-dependent functions. Numerical simulations complement our study.
We analyze the sidewise controllability for the variable coefficients one-dimensional wave equation. The control is acting on one extreme of the string with the aim that the solution tracks a given path at the otherfree end. This sidewise control problem is also often referred to as nodal profile or tracking control. First, the problem is reformulated as a dual observability property for the corresponding adjoint system. Using sidewiseenergy propagation arguments the sidewise observability is shown to hold, ina sufficiently large time, in the class of BV-coefficients. We also present a number of open problems and perspectives for further research.
Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval.
We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systems.
We build up a decomposition for the flow generated by the heat equation with a real analytic memory kernel. It
consists of three components: The first one is of parabolic nature; the second one gathers the hyperbolic component
of the dynamics, with null velocity of propagation; the last one exhibits a finite smoothing effect. This decomposition reveals the hybrid parabolic-hyperbolic nature of the flow and clearly illustrates the significant impact of the memory term on the parabolic behavior of the system in the absence of memory terms.
This paper is devoted to analysing the explicit slow decay rate and turnpike in the infinite-horizon linear quadratic optimal control problems for hyperbolic systems. Assume that some weak observability or controllability are satisfied, by which, the lower and upper bounds of the corresponding algebraic Riccati operator are estimated, respectively. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed.
We then apply these results to the LQ optimal control problems constraint to networks of onedimensional wave equations and also some multi-dimensional ones with local controls which lack of GCC (Geometric Control Condition).