Martin Lazar

Greedy controllability of finite dimensional linear systems

We analyse the problem of controllability for parameter dependent linear finite-dimensional systems. The goal is to identify the most distinguished realisations of those parameters so to better describe or approximate the whole range of controls. We adapt recent results on greedy and weak greedy algorithms for parameter dependent PDEs or, more generally, abstract equations in Banach spaces. Our results lead to optimal approximation procedures that, in particular, perform better than simply sampling the parameter-space to compute the controls for each of the parameter values. We apply these results for the approximate control of finite-difference approximations of the heat and the wave equation. The numerical experiments confirm the efficiency of the methods and show that the number of weak-greedy samplings that are required is particularly low when dealing with heat-like equations, because of the intrinsic dissipativity that the model introduces for high frequencies.

Microlocal Energy Density for Hyperbolic Systems

Starting from the method for computing microlocal energy density, which was developed independently by Francfort and Murat, and Gerard for the linear wave equation, we compute that very density for the hyperbolic system

Cyclonic versus Anticyclonic Circulation in Lakes and Inland Seas

A simple diagnostic model, reproducing circulation in lakes and marginal seas in which low-density waters are found close to the coasts while high-density waters dominate the offshore areas, is developed. An explicit solution is obtained for the central transverse section of an elongated basin, assuming that the Boussinesq and hydrostatic approximations are valid and that the alongshore variability vanishes. The model reveals cyclonic circulation that may either extend throughout the vertical (type C) or may top anticyclonic circulation developed in the bottom layer (type C/A). With the amplitude of the imposed density anomaly being fixed, the flow type is controlled by the frictional processes and by the basin dimensions. In a typical basin, type C/A flow is supported by weak bottom and vertical friction and by moderate lateral friction, unlike type C flow, which is supported by moderate bottom and vertical friction and by weak lateral friction. Strong frictional influence, especially in the basin interior, suppresses the flow everywhere. The flow is also suppressed in a basin that is narrow O(1 km) and shallow O(10 m), even without the frictional control being too strong. A basin that is narrow and deep favors type C/A flow, whereas a basin that is wide and shallow tends to support type C flow. The theoretical findings are related to observations, particularly those originating from the Adriatic Sea where type C flow prevails but may occasionally be replaced by type C/A flow, as well as to previous modeling results.

Exploring limit behaviour of non-quadratic terms via H-measures. Application to small amplitude homogenisation

A method is developed for analysing asymptotic behaviour of terms involving an arbitrary integer order powers of functions by means of H-measures. It is applied to the small amplitude homogenisation problem for a stationary diffusion equation, in which coefficients are assumed to be analytic perturbations of a constant, enabling formulæ for higher order correction terms in a general, non-periodic setting. Explicit expressions in terms of Fourier coefficients are obtained under periodicity assumption. The method allows of its generalisation and application to the corresponding non-stationary equation, as well as to some other small amplitude homogenisation problems.