Zoran Tomljanović


Associate Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
phone: +385-31-224-817
fax: +385-31-224-801
email:  ztomljan @ mathos.hr
office:  3 (first floor)


Research Interests

Numerical linear algebra
Numerical mathematics
Damping optimization in mechanical systems
Control Theory

Matrix Equations


  • PhD in Mathematics, Department of Mathematics, University of Zagreb, May 2011.

PhD Thesis: Optimal damping for vibrating systems using dimension reduction  (PDF)

  • 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. Truhar, Z. Tomljanović, M. Puvača, An Efficient Approximation For Optimal Damping In Mechanical Systems, International journal of numerical analysis and modeling 14/2 (2017), 201-217
    This paper is concerned with an efficient algorithm for damping optimization in mechanical systems with a prescribed structure. Our approach is based on the minimization of the total energy of the system which is equivalent to the minimization of the trace of the corresponding Lyapunov equation. Thus, the prescribed structure in our case means that a mechanical system is close to a modally damped system. Although our approach is very efficient (as expected) for mechanical systems close to modally damped system, our experiments show that for some cases when systems are not modally damped, the proposed approach provides efficient approximation of optimal damping.
  2. I. Kuzmanović, Z. Tomljanović, N. Truhar, Damping optimization over the arbitrary time of the excited mechanical system, Journal of Computational and Applied Mathematics, 304 (2016), 120-129
    In this paper we consider damping optimization in mechanical system excited by an external force. We use optimization criteria based on minimizing average energy amplitude and average displacement amplitude over the arbitrary time. As the main result we derive explicit formulas for objective functions. These formulas can be implemented efficiently and accelerate optimization process significantly, which is illustrated in a numerical example.
  3. P. Benner, P. Kurschner, Z. Tomljanović, N. Truhar, Semi-active damping optimization of vibrational systems using the parametric dominant pole algorithm, Journal of Applied Mathematics and Mechanics 96/5 (2016), 604-619
    We consider the problem of determining an optimal semi-active damping of vibrating systems. For this damping optimization we use a minimization criterion based on the impulse response energy of the system. The optimization approach yields a large number of Lyapunov equations which have to be solved. In this work, we propose an optimization approach that works with reduced systems which are generated using the parametric dominant pole algorithm. This optimization process is accelerated with a modal approach while the initial parameters for the parametric dominant pole algorithm are chosen in advance using residual bounds. Our approach calculates a satisfactory approximation of the impulse response energy while providing a significant acceleration of the optimization process. Numerical results illustrate the effectiveness of the proposed algorithm.
  4. N. Truhar, Z. Tomljanović, K. Veselić, Damping optimization in mechanical systems with external force, Applied mathematics and computation 250 (2015), 270-279
    We consider a mechanical system excited by external force. Model of such a system is described by the system of ordinary differential equations: $M ddot x(t) + D dot x(t) + K x(t) = {hat f}(t)$, where matrices $M, K$ (mass and stiffness) are positive definite and the vector ${hat f} $ corresponds to an external force. The damping matrix D is assumed to be positive semidefinite and has a small rank. We introduce two criteria that allow damping optimization of mechanical system excited by an external force. Since in general a damping optimization is a very demanding problem, we provide a new formulas which have been used for efficient damping optimization. The efficiency of new formulas is illustrated with a numerical experiment.
  5. M. Rukav, K. Stražanac, N. Šuvak, Z. Tomljanović, Markov decision processes in minimization of expected costs, Croatian Operational Research Review 5/2 (2014), 247-257
    Basics of Markov decision processes will be introduced in order to obtain the optimization goal function for minimizing the long-run expected cost. We focus on minimization of such cost of the farmer's policy consisting of different decisions in specic states regarding both milk quality and quantity (lactation states) produced by a dairy cow. The transition probability matrix of the Markov process, used here for modeling of transitions of a dairy cow from one state to another, will be estimated from the data simulated from the lactation model that is often used in practice. We want to choose optimal actions in the states of this Markov process regarding the farmer's costs. This problem can be solved by exhaustive enumeration of all possible cases in order to obtain the optimal policy. How- ever, this is feasible only for a small number of states. Generally, this problem can be approached in the linear programming setting which yields an efficient solution. In order to demonstrate and compare these two approaches, we present an example based on the simulated data regarding milk quality and quantity.


  • 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 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



Committee Memberships and organization





  • Mathematical Communications
  • Osječki matematički list


Workshop and Conference Talks


Seminar talks


  • Optimization of semi-active damping and viscous damping in excited mechanical systems, Matrix Computation Seminar,   10 November 2015, Departments of Mathematics at Virginia Tech, USA
  • Optimization of semi-active damping and external damping in mechanical systems with external force, Absolventen-Seminar  - Numerische Mathematik, 30 June 2015, Berlin, Germany
  • Damping optimization in mechanical systems with external force, CSC 

    Seminar at MPI Magdeburg, Germany, 5 May 2015

  • Optimizacija prigušenja u mehaničkim sustavima sa vanjskom silom, Department of Mathematics, University of Osijek, Optmization and applications seminar, 15 April 2015
  • Optimization of semi-active damping and viscous damping in mechanical systems with external force, October 21, 2014, University in Innsbruck, Department of Mathematics, Austria
  • Optimization of mechanical systems using dimension reduction, October 17, 2013, Forschungsseminar Scientific Computing, Technische Universitat Chemnitz, Germany
  • Optimalno prigušenje kod vibracijskih sistema koristeći redukciju dimenzije, March 17, 2011, Seminar za numeričku matematiku i računarstvo, PMF-Mathematical Departments, University of Zagreb, Croatia
  • Optimizacija prigušenja u vibracijskim sistemima pomoću redukcije dimenzije sistema, April 29, 2009,Optmization and applications seminar, University of Osijek, Department of Mathematics, Croatia
  • An efficient algorithm for solving and optimizing some types of Lyapunov, May 15, 2007, TU Chemnitz, Research Seminar Numerics, Germany


Study Visits Abroad and Professional Improvement:

  • 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 4/11/2015-20/11/2015
  • visiting researcher at TU Berlin, Germany 21/06/2016 - 29/6/2016 ,30/03/2016 - 07/04/2016, 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 22/11 /16 - 26/11/16, 29/06/16 - 09/07/16, 15/02/16 - 23/02/16, 29/8/2010 - 19/9/2010, 26/8/2012 -29/9/2012, 5/2/2013 - 20/2/2013, 3/7/2013 - 30/7/2013, 02/04/14 - 04/14, 04/27/15 - 05/06/15
  • 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
  •  Chairman of the Mathematical Colloquium in Osijek, since 2017 -
  •  Erasmus+ coordinator on Department of Mathematics, since 2013 -


Konzultacije (Office Hours):

Termini sljedećih konzultacija:

  • utorak 19.9.2017. u 8:30 sati
  • utorak 26.9.2017. u 8:30 sati
  • zbog službenog putovanja slijedeće konzultacije će biti u petak 20.10. 2017. u 13 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
  • Svojstveni problem i metoda potencija
    - definirati svojstveni problem
    - obraditi metodu potencija, inverznu metodu potencija i metodu potencija s šiftom
    - implementirati i ilustrirati na primjeru
  • SVD dekompozicija i primjene
    -obraditi pojam singularne dekompozicije matrice
    -obraditi glavne teorijske rezultate s naglaskom na primjenu
    -na primjerima ilustrirati neke od primjena poput: kompresije slike, redukcije modela i sl.
  • 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
  • Hijerarhijsko odlučivanje (zauzeto)
    - izgraditi model za rješavanje višekriterijskih problema odlučivanja;
    - primjeniti neku od metoda za hijerarhijsko odlučivanje na primjerima
  • Rezononcija
    - obraditi pojam rezonancije kod mehaničkih sustava
    - na primjerima sa dvije ili tri mase ilustrirati rezonanciju
  • 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
    - definirati pojam NP-potpnog problema
    - obraditi pojam redukcije i detaljno ilustrirati redukciju na primjeru


Nastavne aktivnosti u zimskom semestru Akademske 2016/2017

Matematička teorija računarstva, predavanja i vježbe

ponedjeljkom od 10 do 14 sati

Građevinski fakultet Osijek

        Matematika na prvoj godini Arhitekture i urbanizma

Matematika (na razlikovnoj godini)

 PhD study, Parameter Dependent Nonlinear Eigenvalue Problems (with Prof. N. Truhar)

Nastavne aktivnosti u ljetnom semestru Akademske 2016/2017

Metode optimizacije, predavanja i vježbe

srijedom 8:00 - 10:00

Računarski praktikum, predavanja i vježbe

petkom 8:00 - 11:00

Teorija odlučivanja, predavanja i vježbe

utorkom  11:30 - 13:00

Nastavne aktivnosti u prošlosti (Past Courses)

Matematička teorija računarstva (Mathematical theory of computation(exercises and lectures, winter semester  13/14, 14/15, 15/16, 16/17)

Teorija odlučivanja (Decision theory) (exercises and lectures, winter semester  13/14, 14/15, 15/16, 16/17)

Metode optimizacije (Optimization methods), (vježbe, winter semestar 08/09, 09/10, 10/11, 11/12, 12/13, 13/14, 16/17)

Numerička linearna algebra (Numerical linear algebra), (exercises, winter semester 06/07, 08/09, 09/10, 10/11, 11/12, 12/13, summer semester 07/08)

Računarski praktikum, (exercises and lectures, ljetni semestar 08/09, 09/10, 11/12, 12/13, 13/14, 14/15,15/16, 16/17)

Metodika nastave informatike, (exercises and lectures, winter semester 11/12, 12/13, 13/14, 14/15, 15/16)

Elementarna geometrija (Elementary geometry), (exercises, ljetni semestar 05/06, 06/07)

Obične diferencijalne jednadžbe (Ordinary differential equations), (exercises, zimski semestar 06/07, 07/08)

Numerička matematika (Numerical mathematics), (exercises, zimski semestar 07/08, ljetni semestar 08/09)

Integralni račun (Integral calculus), (vježbe, ljetni semestar 09/10, 10/11)

Odjel za fiziku (Department of Physics)

Diferencijalne jednadžbe (Differential equations), vježbe, ljetni semestar 06/07, 10/11, 11/12

Geometrija ravnine i prostora - uvod u algebru (Geometry of plain and space - introduction to algebra), (vježbe, zimski semestar 06/07, 07/08, 08/09, 09/10, 10/11)

Ekonomski fakultet (Faculty of Economycs)

Matematika (Mathematics), (vježbe, zimski semestar 07/09, 08/09, 09/10, 10/11, 11/12, predavanja, zimski semestar 12/13)

Prehrambeno tehnološki fakultet

Inženjerska matematika , (vježbe 08/09, 10/11, 11/12, 12/13) Primijenjena matematika (vježbe 08/09, 09/10,11/12, 12/13, 13/14)

Građevinski fakultet Osijek

Matematika (na razlikovnoj godini), exercises and lectures 12/13, 13/14

Matematika (na Arhitekturi i urbanizmu) (winter semester 16/17)


Poslovna matematika, predavanja i vježbe 12/13, 13/14