Fakultet primijenjene matematike i informatike

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,
  • MSc in Mathematics, Department of Mathematics, University of Zagreb, Croatia, December 2005,
  • 1997-2001 Mathematical Gymnasium at high school in Našice


Journal Publications

  1. Z. Tomljanović, Damping optimization of the excited mechanical system using dimension reduction, Mathematics and Computers in Simulation 207 (2023), 24-40
    We consider a mechanical system excited by a periodic external force. The main problem is to determine the best damping matrix to be able to minimize the system average displacement amplitude. Damping optimization usually includes optimization of damping positions and corresponding damping viscosities. Since the objective function is non-convex, a standard optimization approach requires a large number of objective function evaluations. We first propose a dimension reduction approach that calculates approximation of the average displacement amplitude and additionally we efficiently use a low rank update structure that appears in the external damping matrix. Moreover, an error bound which allows determination of appropriate approximation orders is derived and incorporated within the optimization method. We also present a theoretical error bound that allows determination of effective damping positions. The methodology proposed here provides a significant acceleration of the optimization process. The gain in efficiency is illustrated in numerical experiments.
  2. N. Jakovčević Stor, T. Mitchell, Z. Tomljanović, M. Ugrica, Fast optimization of viscosities for frequency-weighted damping of second-order systems, Journal of Applied Mathematics and Mechanics 103/5 (2023), 1-21
    We consider frequency-weighted damping optimization for vibrating systems described by a second-order differential equation. The goal is to determine viscosity values such that eigenvalues are kept away from certain undesirable areas on the imaginary axis. To this end, we present two complementary techniques. First, we propose new frameworks using nonsmooth constrained optimization problems, whose solutions both damp undesirable frequency bands and maintain the stability of the system. These frameworks also allow us to weight which frequency bands are the most important to damp. Second, we also propose a fast new eigensolver for the structured quadratic eigenvalue problems that appear in such vibrating systems. In order to be efficient, our new eigensolver exploits special properties of diagonal-plus-rank-one complex symmetric matrices, which we leverage by showing how each quadratic eigenvalue problem can be transformed into a short sequence of such linear eigenvalue problems. The result is an eigensolver that is substantially faster than standard techniques. By combining this new solver with our new optimization frameworks, we obtain our overall algorithm for fast computation of optimal viscosities. The efficiency and performance of our new approach are verified and illustrated on several numerical examples.
  3. 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.
  4. I. Nakić, D. Tolić, Z. Tomljanović, I. Palunko, Numerically Efficient H∞ Analysis of Cooperative Multi-Agent Systems, Journal of The Franklin Institute 359/16 (2022), 9110-9128
    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.
  5. 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.



  • 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


  • UPCOMING: Co-organizer of the the Winter School on Model Reduction for Optimization and Control that will be held on 19 - 23 February 2024 in Dubrovnik, Croatia: webpage

  • 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 10th 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




  • LAA
  • APOM
  • Mathematical Communications
  • Osječki matematički list


Workshop and Conference Talks


Seminar talks



Study Visits Abroad and Professional Improvement:

  • visiting researcher at Charles University, Faculty of Mathematics and Physics, Prague 20/2/2022-27/2/2022
  • 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,  6/11/2022-12/11/2022, 10/9/2021-18/9/2021, 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, starting from 2010 almost every year there was a visit between a week up to one month.
  • 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 Research at J. J. Strossmayer University of Osijek, Department of Mathematics, since October 2021 -
  • Deputy Head of Department for Education and Students at J. J. Strossmayer University of Osijek, Department of Mathematics, from October 2017 - October 2021
  • Moderator of Optimization and application seminar, since October 2017 -
  • Chairman of the Mathematical Colloquium in Osijek, since January 2017 -
  • Erasmus+ coordinator in the Department of Mathematics, since 2013 - 2017


Konzultacije (Office Hours):

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

  • srijedom  u 10:00 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
  • 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
  • Iterativne metode za rješavanje linearnih sustava
    - napraviti osnovni pregled iterativnih metode za sustave 
    - imlementirati neku od metoda te napraviti ilustraraciju na numeričkim primjerima 


Nastavne aktivnosti u zimskom semestru Akademske 2022./2023.


Linearna algebra I, predavanja

srijedom od 8-10, 

Redukcija modela i aproksimacijski pristupi, predavanja

utorkom od 10-12, 


Nastavne aktivnosti u ljetnom semestru Akademske 2022./2023.


Osnove teorije upravljanja s primjenama, predavanja


Teorijske osnove računalne znanosti, predavanja i vježbe