Optimal location of controllers for the one-dimensional wave equation

In this paper, we consider the homogeneous one-dimensional wave equation defined on (0,π). For every subset ωâŠ[0,π] of positive measure, every T≥2π, and all initial data, there exists a unique control of minimal norm in L2(0,T;L2(ω)) steering the system exactly to zero. In this article we consider two optimal design problems. Let L∈(0,1). The first problem is to determine the optimal shape and position of ω in order to minimize the norm of the control for given initial data, over all possible measurable subsets ω of [0,π] of Lebesgue measure Lπ. The second problem is to minimize the norm of the control operator, over all such subsets. Considering a relaxed version of these optimal design problems, we show and characterize the emergence of different phenomena for the first problem depending on the choice of the initial data: existence of optimal sets having a finite or an infinite number of connected components, or nonexistence of an optimal set (relaxation phenomenon). The second problem does not admit any optimal solution except for L=1/2. Moreover, we provide an interpretation of these problems in terms of a classical optimal control problem for an infinite number of controlled ordinary differential equations. This new interpretation permits in turn to study modal approximations of the two problems and leads to new numerical algorithms. Their efficiency will be exhibited by several experiments and simulations

Optimal shape and location of sensors or actuators in PDE models

We investigate the problem of optimizing the shape and location of sensors and actuators for evolution systems driven by distributed parameter systems or partial differential equations (PDE). We consider wave, Schrödinger and heat equations on an arbitrary domain Ω, in any space dimension, and with suitable boundary conditions (if there is a boundary) which can be of Dirichlet, Neumann, mixed or Robin type. This kind of problem is frequently encountered in applications where one aims, for instance, at maximizing the quality of reconstruction of the solution, using only a partial observation. From the mathematical point of view, using probabilistic considerations we model this problem as that of maximizing the so-called randomized observability constant, over all possible subdomains of Ω having a prescribed measure. The spectral analysis of this problem reveals intimate connections with the theory of quantum chaos. More precisely, we provide a solution to this problem when the domain Ω satisfies suitable quantum ergodicity assumptions

Complexity and regularity of maximal energy domains for the wave equation with fixed initial data

We consider the homogeneous wave equation on a bounded open connected subset Ω of IRn. Some initial data being specified, we consider the problem of determining a measurable subset ω of Ω maximizing the L2-norm of the restriction of the corresponding solution to ω over a time interval [0, T], over all possible subsets of Ω having a certain prescribed measure. We prove that this problem always has at least one solution and that, if the initial data satisfy some analyticity assumptions, then the optimal set is unique and moreover has a finite number of connected components. In contrast, we construct smooth but not analytic initial conditions for which the optimal set is of Cantor type and in particular has an infinite number of connected components

On the stabilization problem for nonholonomic distributions

Abstract. Let M be a smooth connected complete manifold of dimension n, and be a smooth nonholonomic distribution of rank m ≤ n on M. We prove that if there exists a smooth Riemannian metric on for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of on M. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the corresponding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and we establish fine properties of optimal trajectories

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya

Preliminaries Consider the local SR-geometry (U, D, g), where U is a neighborhood of 0 ∈ R 3 , D is a Martinet-type distribution, which can be taken in the normal form D = Ker ω, ω = dz − y 2 2 dx, and g is a C ω metric on D, which can be written (see Expanding F 1 and F 2 in Taylor series according to the previous weights and identifying at the order p two elements whose Taylor series are the same at the order p, we obtain the following normal forms of order −1 and 0: • Normal form of order −1: (flat case); • Normal form of order 0: 2 dx 2 + (1 + βx + γy) 2 dy 2 , α, β, γ ∈ R. 1.1. Geodesics equations. The energy minimization problem equivalent to the SR-problem is the following optimal control problem: from Pontryagin's maximum principle [9], minimizing solutions are solutions of the following equations: where H ν is the pseudo-Hamiltonian where ν is a constant normalized to 0 or 1/2. A solution of the previous equations is called an extremal; when ν = 1/2 (resp. ν = 0), the solutions are called normal (resp. abnormal), and their projections onto the state space are called the geodesics. They can be easily computed

Greedy optimal control for elliptic problems and its application to turnpike problems

This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-018-1005-zWe adapt and apply greedy methods to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, performs better than simply sampling the parameter-space to compute controls for each parameter value. The same method can be adapted for parabolic control problems, but this leads to greedy selections of the realizations of the parameters that depend on the initial datum under consideration. The turnpike property (which ensures that parabolic optimal control problems behave nearly in a static manner when the control horizon is long enough) allows using the elliptic greedy choice of the parameters in the parabolic setting too. We present various numerical experiments and an extensive discussion of the efficiency of our methodology for parabolic control and indicate a number of open problems arising when analyzing the convergence of the proposed algorithmsThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 694126-DyCon). Part of this research was done while the second author visited DeustoTech and Univesity of Deusto with the support of the DyCon project. The second author was also partially supported by Croatian Science Foundation under ConDyS Project, IP-2016-06-2468. The work of the third author was partially supported by the Grants MTM2014-52347, MTM2017-92996 of MINECO (Spain) and ICON of the French AN