TY - JOUR A1 - Bárcena-Petisco, J.A. A1 - Cavalcante, M. A1 - Coclite, G.M. A1 - de Nitti, N. A1 - Zuazua, E. T1 - Control of Hyperbolic and Parabolic Equations on Networks and Singular limits N2 - We study the controllability properties of the transport equation and of parabolic equations posed on a tree. Using a control localized on the exterior nodes, we prove that the hyperbolic and the parabolic systems are null-controllable. The hyperbolic proof relies on the method of characteristics, the parabolic one on duality arguments and Carleman inequalities. We also show that the parabolic system may not be controllable if we do not act on all exterior vertices because of symmetries. Moreover, we estimate the cost of the null-controllability of transport-diffusion equations with diffusivity ε > 0ε>0 and study its asymptotic behavior when ε → 0^+ε→0 + . We prove that the cost of the controllability decays for a time sufficiently large and explodes for short times. This is done by duality arguments allowing to reduce the problem to obtain observability estimates which depend on the viscosity parameter. These are derived by using Agmon and Carleman inequalities. Y1 - 2021 ER - TY - JOUR A1 - Joheac, J A1 - Trelat, E A1 - Zuazua, E T1 - Nonnegative control of finite-dimensional linear systems N2 - We consider the controllability problem for finite-dimensional linear autonomous control systems with nonnegative controls. Despite the Kalman condition, the unilateral nonnegativity control constraint may cause a positive minimal controllability time. When this happens, we prove that, if the matrix of the system has a real eigenvalue, then there is a minimal time control in the space of Radon measures, which consists of a finite sum of Dirac impulses. When all eigenvalues are real, this control is unique and the number of impulses is less than half the dimension of the space. We also focus on the control system corresponding to a finite-difference spatial discretization of the one-dimensional heat equation with Dirichlet boundary controls, and we provide numerical simulations. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1016/j.anihpc.2020.07.004 VL - 38 IS - 2 SP - 301 EP - 346 ER - TY - INPR A1 - Wang, G A1 - Zhang, Y A1 - Zuazua, E T1 - Flow decomposition for heat equations with memory N2 - 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. KW - Heat equations with memory KW - decomposition of the flow KW - hybrid parabolic-hyperbolic behavior Y1 - 2021 ER - TY - INPR A1 - Barcena-Petisco, J.A. A1 - Zuazua, E T1 - Averaged dynamics and control for heat equations with random diffusion N2 - This paper deals with the averaged dynamics for heat equations in the degenerate case where the diffusivity coefficient, assumed to be constant, is allowed to take the null value. First we prove that the averaged dynamics is analytic. This allows to show that, most often, the averaged dynamics enjoys the property of unique continuation and is approximately controllable. We then determine if the averaged dynamics is actually null controllable or not depending on how the density of averaging behaves when the diffusivity vanishes. In the critical density threshold the dynamics of the average is similar to the \frac{1}{2}-fractional Laplacian, which is wellknown to be critical in the context of the controllability of fractional diffusion processes. Null controllability then fails (resp. holds) when the density weights more (resp. less) in the null diffusivity regime than in this critical regime. Y1 - 2021 ER - TY - INPR A1 - Biccari, U A1 - Zuazua, E T1 - Multilevel control by duality N2 - We discuss the multilevel control problem for linear dynamical systems, consisting in designing a piece-wise constant control function taking values in a finite-dimensional set. In particular, we provide a complete characterization of multilevel controls through a duality approach, based on the minimization of a suitable cost functional. In this manner we build optimal multi-level controls and characterize the time needed for a given ensemble of levels to assure the controllability of the system. Moreover, this method leads to efficient numerical algorithms for computing multilevel controls. Y1 - 2021 ER - TY - JOUR A1 - Ko, D A1 - Zuazua, E T1 - Model predictive control with random batch methods for a guiding problem N2 - We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space. The numerical simulation of such models quickly becomes unfeasible for a large number of interacting agents. To reduce the computational cost, we use the Random Batch Method (RBM), which provides a computationally feasible approximation of the dynamics. At each time step, the RBM randomly divides the set of particles into small subsets (batches), considering only the interactions inside each batch. Due to the averaging effect, the RBM approximation converges to the exact dynamics as the time discretization gets finer. We propose an algorithm that leads to the optimal control of a fixed RBM approximated trajectory using a classical gradient descent. The resulting control is not optimal for the original complete system, but rather for the reduced RBM model. We then adopt a Model Predictive Control (MPC) strategy to handle the error in the dynamics. While the system evolves in time, the MPC strategy consists in periodically updating the state and computing the optimal control over a long-time horizon, which is implemented recursively in a shorter time-horizon. This leads to a semi-feedback control strategy. Through numerical experiments we show that the combination of RBM and MPC leads to a significant reduction of the computational cost, preserving the capacity of controlling the overall dynamics. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1142/S0218202521500329 VL - 31 IS - 8 SP - 1569 EP - 1592 ER - TY - INPR A1 - Heiland, J A1 - Zuazua, E T1 - Classical system theory revisited for Turnpike in standard state space systems and impulse controllable descriptor systems N2 - The concept of turnpike connects the solution of long but finite time horizon optimal control problems with steady state optimal controls. A key ingredient of the analysis of turnpike phenomena is the linear quadratic regulator problem and the convergence of the solution of the associated differential Riccati equation as the terminal time approaches infinity. This convergence has been investigated in linear systems theory in the 1980s. We extend classical system theoretic results for the investigation of turnpike properties of standard state space systems and descriptor systems. We present conditions for turnpike phenomena in the non detectable case and for impulse controllable descriptor systems. For the latter, in line with the theory for standard linear systems,we establish existence and convergence of solutions to a generalized differential Riccati equation. KW - Riccati equations KW - descriptor systems KW - linear systems KW - long time behavior KW - optimal control Y1 - 2021 ER - TY - INPR A1 - Zhong-Jie, H A1 - Zuazua, E T1 - Slow decay and Turnpike for Infinite-horizon Hyperbolic LQ problems N2 - 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). Y1 - 2021 ER - TY - INPR A1 - Geshkovski, B A1 - Zuazua, E T1 - Optimal actuator design via Brunovsky’s normal form N2 - In this paper, by using the Brunovsky normal form, we provide a reformulation of the problem consisting in finding the actuator design which minimizes the controllability cost for finite-dimensional linear systems with scalar controls. Such systems may be seen as spatially discretized linear partial differential equations with lumped controls. The change of coordinates induced by Brunovsky’s normal form allows us to remove the restriction of having to work with diagonalizable system dynamics, and does not entail a randomization procedure as done in past literature on diffusion equations or waves. Instead, the optimization problem reduces to a minimization of the norm of the inverse of a change of basis matrix, and allows for an easy deduction of existence of solutions, and for a clearer picture of some of the problem’s intrinsic symmetries. Numerical experiments help to visualize these artifacts, indicate further open problems, and also show a possible obstruction of using gradient-based algorithms – this is alleviated by using an evolutionary algorithm. Y1 - 2021 ER - TY - INPR A1 - Biccari, U A1 - Warma, M A1 - Zuazua, E T1 - Control and Numerical approximation of Fractional Diffusion Equations N2 - The aim of this work is to give a broad panorama of the control properties of fractional diffusive models from a numerical analysis and simulation perspective. We do this by surveying several research results we obtained in the last years, focusing in particular on the numerical computation of controls, though not forgetting to recall other relevant contributions which can be currently found in the literature of this prolific field. Our reference model will be a non-local diffusive dynamics driven by the fractional Laplacian on a bounded domain ΩΩΩ. The starting point of our analysis will be a Finite Element approximation for the associated elliptic model in one and two space-dimensions, for which we also present error estimates and convergence rates in the L2L^2L2 and energy norm. Secondly, we will address two specific control scenarios: firstly, we consider the standard interior control problem, in which the control is acting from a small subset ω⊂Ωω ⊂ Ωω⊂Ω. Secondly, we move our attention to the exterior control problem, in which the control region O⊂ΩcO ⊂ Ω cO⊂Ωc is located outside ΩΩΩ. This exterior control notion extends boundary control to the fractional framework, in which the non-local nature of the models does not allow for controls supported on ∂Ω∂Ω∂Ω. We will conclude by discussing the interesting problem of simultaneous control, in which we consider families of parameter-dependent fractional heat equations and we aim at designing a unique control function capable of steering all the different realizations of the model to the same target configuration. In this framework, we will see how the employment of stochastic optimization techniques may help in alleviating the computational burden for the approximation of simultaneous controls. Our discussion is complemented by several open problems related with fractional models which are currently unsolved and may be of interest for future investigation. Y1 - 2021 ER -