Odjel za matematiku

Zoran Tomljanović


Associate Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia

Google Scholar Profile

phone: +385-31-224-827
fax: +385-31-224-801
email:  ztomljan @ mathos.hr
office:  18 (ground floor)


Research Interests

Numerical linear algebra
Damping optimization in mechanical systems
Control Theory

Matrix Equations


  • PhD in Mathematics, Department of Mathematics, University of Zagreb, May 2011.
  • BSc in Mathematics, Department of Mathematics, University of Zagreb, Croatia, December 2005.
  • 1997-2001 Mathematical Gymnasium at high school in Našice


Journal Publications

  1. N. Jakovčević Stor, I. Slapničar, Z. Tomljanović, Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems, Mathematics 10/5 (2022), 1-17
    We propose a fast algorithm for computing optimal viscosities of dampers of a linear vibrational system. We are using a standard approach where the vibrational system is first modeled using the second-order structure. This structure yields a quadratic eigenvalue problem which is then linearized. Optimal viscosities are those for which the trace of the solution of the Lyapunov equation with the linearized matrix is minimal. Here, the free term of the Lyapunov equation is a low-rank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires O(n3) floating-point operations. In our approach, we transform the linearized matrix into an eigenvalue problem of a diagonal-plus-low-rank matrix whose eigenvectors have a Cauchy-like structure. Our algorithm is based on a new fast eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast multiplication of linked Cauchy-like matrices, yielding computation of optimal viscosities for each choice of external dampers in O(kn2) operations, k being the number of dampers. The accuracy of our algorithm is compatible with the accuracy of the standard approach.
  2. I. Nakić, D. Tolić, Z. Tomljanović, I. Palunko, Numerically Efficient H∞ Analysis of Cooperative Multi-Agent Systems, Journal of The Franklin Institute (2022), prihvaćen za objavljivanje
    This article proposes a numerically efficient approach for computing the maximal (or minimal) impact one agent has on the cooperative system it belongs to. For example, if one is able to disturb/bolster merely one agent in order to maximally disturb/bolster the entire team, which agent to choose? We quantify the agent-to-system impact in terms of $H_{infty}$ norm whereas output synchronization is taken as the underlying cooperative control scheme. The agent dynamics are homogeneous, second order and linear whilst communication graphs are weighted and undirected. We devise simple sufficient conditions on agent dynamics, topology and output synchronization parameters rendering all agent-to-system $H_{infty}$ norms to attain their maxima in the origin (that is, when constant disturbances are applied). Essentially, we quickly identify bottlenecks and weak/strong spots in multi-agent systems without resorting to intense computations, which becomes even more important as the number of agents grows. Our analyses also provide directions towards improving communication graph design and tuning/selecting cooperative control mechanisms. Lastly, numerical examples with a large number of agents and experimental verification employing off-the-shelf nano quadrotors are provided.
  3. N. Truhar, Z. Tomljanović, R. Li, Perturbation Theory for Hermitian Quadratic Eigenvalue Problem -- Damped and Simultaneously Diagonalizable Systems, Applied mathematics and computation 371 (2020)
    The main contribution of this paper is a novel approach to the perturbation theory of a structured Hermitian quadratic eigenvalue problems $(lambda^2 M + lambda D + K) x=0$. We propose a new concept without linearization, considering two structures: general quadratic eigenvalue problems (QEP) and simultaneously diagonalizable quadratic eigenvalue problems (SDQEP). Our first two results are upper bounds for the difference $left| | X_2^* M widetilde{;X};_1 |_F^2 - | X_2^* M {;X};_1 |_F^2 right|$, and for $| X_2^* M widetilde X_1 - X_2^* M X_1|_F$, where the columns of $X_1=[x_1, ldots, x_k]$ and $X_2=[x_{;k+1};, ldots, x_n]$ are linearly independent right eigenvectors and $M$ is positive definite Hermitian matrix. As an application of these results we present an eigenvalue perturbation bound for SDQEP. The third result is a lower and an upper bound for $|sin{;Theta(mathcal{;X};_1, widetilde{;mathcal{;X};};_1)}; |_F$, where $Theta$ is a matrix of canonical angles between the eigensubspaces $mathcal{;X};_1 $ and $widetilde{;mathcal{;X};};_1$, $mathcal{;X};_1 $ is spanned by the set of linearly independent right eigenvectors of SDQEP and $widetilde{;mathcal{;X};};_1$ is spanned by the corresponding perturbed eigenvectors. The quality of the mentioned results have been illustrated by numerical examples.
  4. C. Beattie, S. Gugercin, Z. Tomljanović, Sampling-free model reduction of systems with low-rank parameterization, Advances in Computational Mathematics 46/6 (2020), 1-34
    We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that allow for low-rank variation in the state matrix. Usual approaches for parametric model reduction typically involve exploring the parameter space to identify representative parameter values and the associated models become the principal focus of model reduction methodology. These models are then combined in various ways in order to interpolate the response. The initial exploration of the parameter space can be a forbiddingly expensive task. A different approach is proposed here that requires neither parameter sampling nor parameter space exploration. Instead, we represent the system response function as a composition of four subsystem response functions that are nonparametric with a purely parameter-dependent function. One may apply any one of a number of standard (non-parametric) model reduction strategies to reduce the subsystems independently, and then conjoin these reduced models with the underlying parameterization to obtain the overall parameterized response. Our approach has elements in common with the parameter mapping approach of Baur et al. (PAMM 14(1), 19–22 2014) but offers greater flexibility and potentially greater control over accuracy. In particular, a data-driven variation of our approach is described that exercises this flexibility through the use of limited frequency-sampling of the underlying non-parametric models. The parametric structure of our system representation allows for a priori guarantees of system stability in the resulting reduced models across the full range of parameter values. Incorporation of system theoretic error bounds allows us to determine appropriate approximation orders for the non-parametric systems sufficient to yield uniformly high accuracy across the parameter range. We illustrate our approach on a class of structural damping optimization problems and on a benchmark model of thermal conduction in a semiconductor chip. The parametric structure of our reduced system representation lends itself very well to the development of optimization strategies making use of efficient cost function surrogates. We discuss this in some detail for damping parameter and location optimization for vibrating structures.
  5. Z. Tomljanović, M. Voigt, Semi-active H∞-norm damping optimization by adaptive interpolation, Numerical Linear Algebra with Applications 27/4 (2020), 1-17
    In this work we consider the problem of semi-active damping optimization of mechanical systems with fixed damper positions. Our goal is to compute a damping that is locally optimal with respect to the H∞-norm of the transfer function from the exogenous inputs to the performance outputs. We make use of a new greedy method for computing the H∞-norm of a transfer function based on rational interpolation. In this paper, this approach is adapted to parameter-dependent transfer functions. The interpolation leads to parametric reduced-order models that can be optimized more efficiently. At the optimizers we then take new interpolation points to refine the reduced-order model and to obtain updated optimizers. In our numerical examples we show that this approach normally converges fast and thus can highly accelerate the optimization procedure. Another contribution of this work are heuristics for choosing initial interpolation points.



  • Accelerated solution of optimal damping problems, -- scientific project; supported by the DAAD for period 2021--2022 (principal investigator together with Jens Saak); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany 

  • Vibration Reduction in Mechanical Systems -- scientific project (IP-2019-04-6774, VIMS). This project has been fully supported by Croatian Science Foundation for the period 01.01.2020.--31.12.2023. (principal investigator) 

  • Control of Dynamical Systems -- scientific project (IP-2016-06-2468, ConDyS). This project has been fully supported by Croatian Science Foundation for the period 01.03.2017.--28.02.2021. (investigator)

  • Robustness optimization of damped mechanical systems, -- scientific project; supported by the DAAD for period 2017--2018 (principal investigator together with Matthias Voigt); cooperation with TU Berlin, Germany

  • Optimization of parameter dependent mechanical systems  -- scientific project (IP-2014-09-9540; OptPDMechSys). This project has been fully supported by Croatian Science Foundation for the period 01.07.2015.--30.06.2019. (investigator)

  • Damping optimization in mechanical systems excited with external force -- scientific project; supported by the J. J. Strossmayer University of Osijek for period 2015 (principal investigator)

  • Mixed Integer Nonlinear Programming (MINLP) for damper optimization -- scientific project; supported by the DAAD for period 2015--2016 (investigator); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg 

  • European Model Reduction Network (EU-MORNET). Funded by: COST (European Cooperation in Science and Technology) (investigator).

  • Optimization of semi-active damping in vibrational systems -- scientific project; supported by the J. J. Strossmayer University of Osijek for period 2014 (principal investigator); cooperation with Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

  • Optimal Damping of Vibrating Systems -- scientific project; supported by the DAAD for period 2013--2014 (investigator) 

  • Passive control of mechanical models -- scientific project No.235-2352818-1042 of the Croatian Ministry of Science, Education and Sports for period 2007.-- (investigator) 

  • Optimization algorithms for determination of optimal damping in mechanical systems -- scientific project; supported by the Croatian Science Foundation for period 2008--2009 (principal investigator)  

Professional Activities

Professional Societiey Membership
  • International Association of Applied Mathematics and Mechanics, GAMM
  • GAMM Activity Group Applied and Numerical Linear Algebra, GAMM ANLA
  • Croatian Mathematical Society, HMD
  • Croatian Operational Research Society, CRORS
  • Society for Industrial and Applied Mathematics, SIAM



Committee Memberships and organization


  • Co-organizer of the 3rd Workshop on Optimal Control of Dynamical Systems and applications, 28-31 March 2022 at Department of Mathematics, J. J. Strossmayer University of Osijek: webpage

  • Co-organizer of the Workshop on Optimal Control of Dynamical Systems and applications, 5-6 November 2020 at Department of Mathematics, J. J. Strossmayer University of Osijek: webpage  

  • Co-organizer of the Tenth Conference on Applied Mathematics and Scientific Computing 14-18 September 2020, Brijuni, Croatia. In 2020 we have a special section on optimal control of dynamical systems and applications, coorganized with the Department of Mathematics, University of Osijek., webpage 

  • Co-organizer of International Workshop on Optimal Control of Dynamical Systems and applications, 20-22 June 2018 at Department of Mathematics, J. J. Strossmayer University of Osijek,  webpage  

  • Co-organizer of Workshop on Model Reduction Methods and Optimization, 20-21 September 2016, in Opatija, Croatia, webpage 

  • Co-organizer of The third International School on Model Reduction for Dynamical Control Systems, 5 - 10 October 2015, in Dubrovnik, Croatia, webpage

  • Co-organizer of the DAAD International School on Linear Optimal Control of Dynamic Systems, 23 - 28 September 2013, Osijek, webpage

  • Co-organizer of the Summer School on Numerical Linear Algebra for Dynamical and High-Dimensional Problems, October 10-15, 2011, Trogir, Croatia, webpage




  • Mathematical Communications
  • Osječki matematički list


Workshop and Conference Talks


Seminar talks



Study Visits Abroad and Professional Improvement:

  • visiting researcher at Department of Mathematics, Carlos III University of Madrid, Spain 06/09/2016 - 15/09/2016
  • visiting researcher at Departments of Mathematics at Virginia Tech, USA,  21/04/2019-27/04/2019, 02/10/2017-14/10/2017, 4/11/2015-20/11/2015
  • visiting researcher at TU Berlin, Germany 11/11/2018-15/11/2018, 16/07/2018-22/07/2018, 07/01/2018-14/01/2018,  21/06/2017 - 29/6/2017, 30/03/2017 - 07/04/2017, 23/02/2016 - 25/02/2016, 29/6/2015 - 9/7/2015
  • visiting researcher at University in Innsbruck, Department of Mathematics 19/10/2014-22/10/2014
  • visiting researcher at Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Germany 12/7/2018-19/7/2017, 22/11 /16 - 26/11/16, 29/06/16 - 09/07/16, 15/02/16 - 23/02/16, 04/27/15 - 05/06/15,02/04/14 - 08/4/14,   5/2/2013 - 20/2/2013, 3/7/2013 - 30/7/2013, 26/8/2012 -29/9/2012, 29/8/2010 - 19/9/2010,
  • visiting researcher at TU Chemnitz, Germany 13/5/2007 - 14/6/2007, 10/4/2008 - 8/5/2008, 10/5/2009 - 10/7/2009, 24/1/2010 - 24/2/2010, 9/10/2013 - 20/10/2013


Service Activities
  • Deputy Head of Department for Education and Students, since October 2017 -
  • Moderator of Optimization and application seminar, since October 2017 -
  • Chairman of the Mathematical Colloquium in Osijek, since January 2017 -
  • Erasmus+ coordinator on Department of Mathematics, since 2013 - 2017


Konzultacije (Office Hours):

Termini sljedećih konzultacija (ured 18 u prizemlju Odjela za matematiku):.

  • utorak 13.9.2022. u 9 sati.
  • utorak 20.9.2022. u 9 sati.
  • utorak 27.9.2022. u 9 sati.

Teme diplomskih i završnih radova:

U nastavku se nalaze nazivi tema i kratki opis, a više informacija studenti mogu dobiti na konzultacijama. Mole se zaniteresirani studenti da se jave ukoliko su zainteresirani za neku od tema.

  • Numeričko rješavanje običnih diferencijalnih jednadžbi
    - obraditi osnovne metode: Eulerova i osnovne Runge Kutta metode
    - implementirati ih u Matlabu ili C-u i ilustrirati efikasnost na primjerima
  • QR dekompozicija s pivotiranjem 
    - obraditi QR dekompoziciju i QR dekompoziciju s pivotiranjem
    - implementirati ju u Matlabu
    - na primjerima pokazati osnovne primjene npr. na određivanje ranga matrice
  • Schurova dekompozicija i primjene
    - definirati definirati Schurovu dekompoziciju 
    - obraditi osnovna svojstva i primjene
    - implementirati i ilustrirati na primjeru
  • Udaljenost do neupravljivog sustava
    -obraditi pojam upravljivosti i važnost udaljenosti do neupravljivog sustava
    -obraditi glavne teorijske rezultate s naglaskom na primjenu u robotici i mehaničkim sustavima
    -implementirati metodu za  udaljenost do neupravljivog sustava te ju na primjerima ilustrirati
  • Metoda Gaussovih eliminacija s potpunim pivotiranjem
    - obraditi metodu Gaussovih eliminacija s potpunim pivotiranjem
    - u Matlabu napraviti ilustraciju metode kroz vizualizaciju koraka
    - napraviti vizualizaciju rjesenja dvije jednadzbe s dvije nepoznanice
  • AHP metoda za odlučivanje
    - izgraditi model za rješavanje višekriterijskih problema odlučivanja;
    - primjeniti AHP metodu za hijerarhijsko odlučivanje na primjerima
  • Iterativne metode za rješavanje linearnih sustava
    - napraviti osnovni pregled iterativnih metode za sustave 
    - imlementirati neku od metoda te napraviti ilustraraciju na numeričkim primjerima 
  • NP-potpuni problem i redukcija problema (zauzeto)
    - definirati pojam NP-potpnog problema
    - obraditi pojam redukcije i detaljno ilustrirati redukciju na primjeru


Nastavne aktivnosti u zimskom semestru Akademske 2021./2022.


Lijearna algebra I, predavanja

srijedom od 10-12, 


Nastavne aktivnosti u ljetnom semestru Akademske 2021./2022.


Redukcija modela i aproksimacijski pristupi, predavanja

Osnove teorije upravljanja s primjenama, predavanja


Teorijske osnove računalne znanosti, predavanja