# Martin LazarUniversity of Dubrovnik · Department of Electrical Engineering and Computing

Martin Lazar

Professor

## About

38

Publications

4,636

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307

Citations

Citations since 2017

Introduction

Additional affiliations

June 2017 - July 2017

October 2016 - September 2020

March 2013 - September 2013

## Publications

Publications (38)

The paper provides strong convergence of solutions to a sequence of linear-quadratic (LQ) optimization problems defined in an abstract functional framework. Each problem is accompanied by the constraint of reaching a given target within a prescribed precision. We show that the problems are well posed and characterise their solutions. The main resul...

The paper investigates switches of circulation orientation in inland basins, either at the surface or near the bottom. The study is based on an analytical 2D model used to simulate thermohaline circulation in lakes and inland seas. The model allows different density profiles varying in both horizontal and vertical directions. By assuming some simpl...

We analyse the turnpike properties for a general, infinite dimensional, linear-quadratic (LQ) optimal control problem, both in the deterministic and in the stochastic case. The novelty of the paper is twofold. Firstly, it obtains positive turnpike results for systems that are (partially) uncontrollable. Secondly, it provides turnpike results for av...

In this paper we develop a procedure to deal with a family of parameter-dependent ill-posed problems, for which the exact solution in general does not exist. The original problems are relaxed by considering corresponding approximate ones, whose optimal solutions are well defined, where the optimality is determined by the minimal norm requirement. T...

In this chapter we provide an overview of recent progress on the problem of controllability of parameter dependent systems. We explore different control notions successfully developed through the last decade. The aim of the control function is to steer the system to a state satisfying some properties prescribed either at some time instant T>0 or du...

The paper investigates switches of circulation orientation in inland basins, either at the surface or near the bottom. The study is based on an analytical 2D model used to simulate thermohaline circulation in lakes and inland seas. The model allows different density profiles varying in both horizontal and vertical directions. By assuming some simpl...

In this paper we consider a constrained parabolic optimal control problem. The cost functional is quadratic and it combines the distance of the trajectory of the system from the desired evolution profile together with the cost of a control. The constraint is given by a term measuring the distance between the final state and the desired state toward...

The aim of this paper is to provide an efficient method for solving a
family of parameter dependent, algebraic Lyapunov equations in an infinite dimensional
setting. Our analysis is based on previous work on reduced modeling
and (weak) greedy algorithms for parameter dependent PDEs and abstract equations
in Banach spaces. The major contribution is...

We construct an algorithm for solving a constrained optimal control problem for a first-order evolutionary system governed by a positive self-adjoint operator. The problem consists in identifying distributed control that minimizes a given cost functional, which comprises a cost of the control and a trajectory regulation term, while steering the fin...

In this chapter we provide an overview of recent progress on the problem of controllability of parameter dependent systems. We explore a different control notions successfully developed through the last decade. The aim of the control function is to steer the system to a state satisfying some properties prescribed either at some time instant T > 0 o...

In this article we study a linear control system with unknown parameter. However, we assume that the possible realisations of this parameter are finite and each realisation can appear with a known probability. We aim to design a control independent of the parameter such that the expectation of the system's realisation reach a given target at a fina...

We extend results obtained by Francfort (An introduction to H-measures and their applications. Variational problems in materials science. Birkhäuser, Basel, pp 85–110, 2006) to parabolic H-measures developed by Antonić and Lazar (J Funct Anal 265:1190–1239, 2013). The well known theory of pseudodifferential operators is extended to parabolic classe...

In this article we consider a linear finite dimensional system. Our aim is to design a control such that the output of the system reach a given target at a final time T>0. This notion is known as output controllability.
We extend this notion to the one of long-time output controllability. More precisely, we consider the question: is it possible to...

We extend results obtained by Francfort (2006) to parabolic H-measures developed by Antoni´cAntoni´c and Lazar (2013). The well known theory of pseudodifferential operators is extended to parabolic classes of symbols and operators and used to obtain results applicable to a wide class of partial differential equations. The main result is the propaga...

In this paper, we deal with the approximation of optimal controls for parameter-dependent elliptic and parabolic equations. We adapt well-known results on greedy algorithms to approximate in an efficient way the optimal controls for parameterized elliptic control problems. Our results yield an optimal approximation procedure that, in particular, pe...

Practical applications of semiclassical measures are tightly connected with a so called oscillatory property, prevailing leakage of information related to high frequencies. In this paper we propose a complementary, concentratory property which prevents loss of information related to low frequencies. We demonstrate that semiclassical measures attain...

Microlocal defect functionals (H-measures, H-distributions,
semiclassical measures etc.) are objects which determine, in some
sense, the lack of strong compactness for weakly convergent ${\rm L}^p$
sequences. Recently, Luc Tartar introduced one-scale H-measures, a
generalisation of H-measures with a characteristic length, which also
comprehend t...

We prove existence of solutions to Cauchy problem for scalar conservation
laws with non-degenerate discontinuous flux
$$
\partial_t u+ \hbox{div}f(t,\mathbf{x},u)=s(t,\mathbf{x},u), \quad
t\geq 0, \mathbf{x}\in \mathbb{R}^d,
$$
where for every $(t,\mathbf{x})\in \mathbb{R}^+\times \mathbb{R}$,
the flux $f(t,\mathbf{x},\cdot) \in \hbox{Lip}(\mathbb{...

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 depending PDEs or, more gene...

A method is developed for analysing asymptotic behaviour of
terms involving an arbitrary integer order powers of L p 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 hig...

This paper deals with an optimal control problem of Bolza type for a class of parabolic equations.
It consists in �nding the initial datum that minimises a cost functional, which comprises
an energy term on the control and a regulation term given by a distributed cost on the state,
and such that the �nal state lies within a prescribed distance to a...

We introduce a new family of functional spaces which incorporate Bochner spaces L p (R m ; E), with E being an appropriate Banach space, and to which we extend the H-distributions. We use the developed theory to prove a general version of the velocity averaging lemma in a heterogeneous L p , p ≤ 2 setting.

We analyse stability of observability estimates for solutions to wave and
Scr\" odinger equations subjected to additive perturbations. The paper
generalises the recent averaged observability/control result by allowing for
systems consisting of operators of different types. The method also applies to
the simultaneous observability problem by which o...

We analyze the problem of averaged observability and control of wave equations.
This topic is motivated by the control of parameter-dependent systems of wave equations. We look for controls ensuring the controllability of the averages of the states with respect to the parameter. This turns out to be equivalent to the problem of averaged observati...

Assume that $(u_n)$ is a sequence of solutions to equations with fractional
derivatives, weakly converging to zero in ${\rm L}^p(R^{d+m})$, with $p\leq 2$.
We prove that the sequence of averaged quantities $(\int \rho(y) u_n(x,y)
dy)$ is strongly precompact in $L^1_{loc}(R^d)$ for any $\rho\in C_c{\R^m}$,
provided that restrictive non-degeneracy co...

Classical H-measures introduced by L. Tartar [Proc. R. Soc. Edinb., Sect. A, Math. 115, No. 3–4, 193–230 (1990; Zbl 0774.35008)] and independently by P. Gérard [Commun. Partial Differ. Equations 16, No. 11, 1761–1794 (1991; Zbl 0770.35001)] are not well suited for the study of parabolic equations. Recently, several parabolic variants have been prop...

We examine necessary and sufficient conditions under which a continuous bilinear functional B on Lp(Rd)⊗ELp(Rd)⊗E, p>1p>1, E being a separable Banach space, can be continuously extended to a linear functional on Lp(Rd;E)Lp(Rd;E). The extension enables a generalisation of the H-distribution concept, allowing us to obtain a (heterogeneous) velocity a...

We prove that the sequence of averaged quantities
$\int_{\R^m}u_n(\mx,\msnop)$ $\rho(\msnop)d\msnop$, is strongly precompact in
$\Ldl\Rd$, where $\rho\in \Ldc{\R^m}$, and $u_n\in
\Ld{\R^m; \pL s\Rd}$, $s\geq 2$, are weak solutions to differential operator
equations with variable coefficients. In particular, this includes differential
operators of h...

We prove that the sequence of averaged quantities ∫Rm un(t,x,y)v(y)dy is strongly precompact in L2loc(R1+d), where u ε L2c (Rm), and un ε L2(R1+d x Rm) are solutions to strictly parabolic transport equations with flux explicitly depending on space and time. In order to obtain the result, we use a recently introduced parabolic variant of H-measures.

H-measures, as originally introduced by Luc Tartar and (independently) Patrick Gérard are well suited for hyperbolic problems. For parabolic problems, some variants should be considered, which would be better adapted to parabolic problems.Recently, we introduced a few parabolic scalings and corresponding variant H-measures, including the existence...

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 approxima...

H-measures, as originally introduced by Luc Tartar and Patrick Gérard, are suited to hyperbolic problems. However, they turned
out not to be well adjusted to the study of parabolic equations. A variant of H-measures is proposed, which is much better
adapted to such kind of problems. We present the new parabolic scaling and the main ingredients for...

A new, parabolic variant of H-measures, as well as its application to the advection-diffusion equation is presented. By the means of localisation property, the strong compactness for gradients of solutions is obtained in the case of variable coefficients. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Since their introduction H-measures have been mostly used in problems related to propagation effects for hyperbolic equations and systems. In this study we give an attempt to apply the H-measure theory to other types of equations. Through a number of examples we present how do the differences between parabolic and hyperbolic equations reflect in th...

Two diagnostic models, reproducing circulation generated in a marginal sea by variable density, have been developed. The models’ domain is a 2D transverse section for which analytical solutions have been obtained. They describe the winter situation in the northern Adriatic, with a strong vertical mixing present and the density maximum dominating th...

Starting from the method for computing microlocal energy density, which was developed independently by Francfort and Murat, and Gérard for the linear wave equation, we compute that very density for the hyperbolic system
We express the energy limit for the sequence of initial problems in terms of the energy of initial conditions. The basic tool we u...

In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limit. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) effects.
A number of results in that direction has been obtained for linear problems.
Tartar...

## Projects

Projects (2)

The project focus on the highly relevant control topics attracting significant attention from theoretical, numerical and applicational points of view. The scientific activities of the project are divided into 6 working packages:
WP1. Control of parameter dependent problems
WP2. Control of vibrational systems
WP3. Adaptive Dual Control under realistic communications
WP4. The cost of controlling hyperbolic partial differential equations in a multiscale setting
WP5. Optimal design and small amplitude homogenisation
WP6. Modeling, system analysis and control of autonomous systems
Within these working packages we aim to cover a range of important issues in modern control, including the following topics: the extension of the greedy control results in several directions, extension of the results for the homogeneous vibrational systems to the systems with external input, the interplay between adaptive dual control and degraded information, estimates for the cost of null-controllability for hyperbolic PDEs on multiscale domains, optimal design result for two phase composites by means of small amplitude homogenisation techniques up to the third (and higher) order expansion, and application of novel control concepts on models representing real-life systems as well as controllability studies which will be useful in the design process of this systems.
http://condys.unidu.hr/