Dynamics and control of discrete chaotic systems of fractional-order have received considerable attention over the last few years. So far, nonlinear control laws have been mainly used for stabilizing at zero the chaotic dynamics of fractional maps. This article provides a further contribution to such research field by presenting simple linear control laws for stabilizing three fractional chaotic maps in regard to their dynamics. Specifically, a one-dimensional linear control law and a scalar control law are proposed for stabilizing at the origin the chaotic dynamics of the Zeraoulia-Sprott rational map and the Ikeda map, respectively. Additionally, a two-dimensional linear control law is developed to stabilize the chaotic fractional flow map. All the results have been achieved by exploiting new theorems based on the Lyapunov method as well as on the properties of the Caputo
*h*-difference operator. The relevant simulation findings are implemented to confirm the validity of the established linear control scheme.

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[1] C. Goodrich and A.C. Peterson: Discrete Fractional Calculus. Springer: Berlin, Germany, 2015, ISBN 978-3-319-79809-7.

[2] P. Ostalczyk: Discrete Fractional Calculus: Applications in Control and Image Processing. World Scientific, 2016.

[3] K. Oprzedkiewicz and K. Dziedzic: Fractional discrete-continuous model of heat transfer process. Archives of Control Sciences, 31(2), (2021), 287– 306, DOI: 10.24425/acs.2021.137419.

[4] T. Kaczorek and A. Ruszewski: Global stability of discrete-time nonlinear systems with descriptor standard and fractional positive linear parts and scalar feedbacks. Archives of Control Sciences, 30(4), (2020), 667–681, DOI: 10.24425/acs.2020.135846.

[5] J.B. Diaz and T.J. Olser: Differences of fractional order. Mathematics of Computation, 28 (1974), 185–202, DOI: 10.1090/S0025-5718-1974-0346352-5.

[6] F.M. Atici and P.W. Eloe: A transform method in discrete fractional calculus. International Journal of Difference Equations, 2 (2007), 165–176.

[7] F.M. Atici and P.W. Eloe: Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory of Differential Equations, Spec. Ed. I, 3 , (2009), 1–12.

[8] G. Anastassiou: Principles of delta fractional calculus on time scales and inequalities. Mathematical and Computer Modelling, 52(3-4), (2010), 556– 566, DOI: 10.1016/j.mcm.2010.03.055.

[9] T. Abdeljawad: On Riemann and Caputo fractional differences. Computers and Mathematics with Applications, 62(3), (2011), 1602–1611, DOI: 10.1016/j.camwa.2011.03.036.

[10] M. Edelman, E.E.N. Macau, and M.A.F. Sanjun (Eds.): Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives. Springer International Publishing, 2018.

[11] G.C. Wu, D. Baleanu, and S.D. Zeng: Discrete chaos in fractional sine and standard maps. Physics Letters A, 378(5-6), (2014), 484–487, DOI: 10.1016/j.physleta.2013.12.010.

[12] G.C. Wu and D. Baleanu: Discrete fractional logistic map and its chaos. Nonlinear Dynamics, 75(1-2), (2014), 283–287, DOI: 10.1007/s11071-013-1065-7.

[13] T. Hu: Discrete chaos in fractional Henon map. Applied Mathematics, 5(15), (2014), 2243–2248, DOI: 10.4236/am.2014.515218.

[14] G.C. Wu and D. Baleanu: Discrete chaos in fractional delayed logistic maps. Nonlinear Dynamics, 80 (2015), 1697–1703, DOI: 10.1007/s11071-014-1250-3.

[15] M.K. Shukla and B.B. Sharma: Investigation of chaos in fractional order generalized hyperchaotic Henon map. International Journal of Electronics and Communications, 78 (2017), 265–273, DOI: 10.1016/j.aeue.2017.05.009.

[16] A. Ouannas, A.A. Khennaoui, S. Bendoukha, and G. Grassi: On the dynamics and control of a fractional form of the discrete double scroll. International Journal of Bifurcation and Chaos, 29(6), (2019), DOI: 10.1142/S0218127419500780.

[17] L. Jouini, A. Ouannas, A.A. Khennaoui, X. Wang, G. Grassi, and V.T. Pham: The fractional form of a new three-dimensional generalized Henon map. Advances in Difference Equations, 122 (2019), DOI: 10.1186/s13662-019-2064-x.

[18] F. Hadjabi, A. Ouannas,N. Shawagfeh, A.A. Khennaoui, and G. Grassi: On two-dimensional fractional chaotic maps with symmetries. Symmetry, 12(5), (2020), DOI: 10.3390/sym12050756.

[19] D. Baleanu, G.C. Wu, Y.R. Bai, and F.L. Chen: Stability analysis of Caputo-like discrete fractional systems. Communications in Nonlinear Science and Numerical Simulation, 48 (2017), 520–530, DOI: 10.1016/j.cnsns.2017.01.002.

[20] A-A. Khennaoui, A. Ouannas, S. Bendoukha, G. Grassi, X. Wang, V-T. Pham, and F.E. Alsaadi: Chaos, control, and synchronization in some fractional-order difference equations. Advances in Difference Equations, 412 (2019), DOI: 10.1186/s13662-019-2343-6.

[21] A. Ouannas, A.A. Khennaoui, G. Grassi, and S. Bendoukha: On chaos in the fractional-order Grassi-Miller map and its control. Journal of Computational and Applied Mathematics, 358(2019), 293–305, DOI: 10.1016/j.cam.2019.03.031.

[22] A. Ouannas, A.A. Khennaoui, S. Momani, G. Grassi and V.T. Pham: Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization. AIP Advances, 10 (2020), DOI: 10.1063/5.0004884.

[23] A. Ouannas, A-A. Khennaoui, S. Momani, G. Grassi, V-T. Pham, R. El- Khazali, and D. Vo Hoang: A quadratic fractional map without equilibria: Bifurcation, 0–1 test, complexity, entropy, and control. Electronics, 9 (2020), DOI: 10.3390/electronics9050748.

[24] A. Ouannas, A-A. Khennaoui, S. Bendoukha, Z.Wang, and V-T. Pham: The dynamics and control of the fractional forms of some rational chaotic maps. Journal of Systems Science and Complexity, 33 (2020), 584–603, DOI: 10.1007/s11424-020-8326-6.

[25] A-A. Khennaoui, A. Ouannas, S. Bendoukha, G. Grassi, R.P. Lozi, and V-T. Pham: On fractional-order discrete-time systems: Chaos, stabilization and synchronization. Chaos, Solitons and Fractals, 119(C), (2019), 150– 162, DOI: 10.1016/j.chaos.2018.12.019.

[26] A. Ouannas, A-A. Khennaoui, Z. Odibat, V-T. Pham, and G. Grassi: On the dynamics, control and synchronization of fractional-order Ikeda map. Chaos, Solitons and Fractals, 123(C), (2015), 108–115, DOI: 10.1016/j.chaos.2019.04.002.

[27] A. Ouannas, F. Mesdoui, S. Momani, I. Batiha, and G. Grassi: Synchronization of FitzHugh-Nagumo reaction-diffusion systems via onedimensional linear control law. Archives of Control Sciences, 31(2), 2021, 333–345, DOI: 10.24425/acs.2021.137421.

[28] Y. Li, C. Sun, H. Ling, A. Lu, and Y. Liu: Oligopolies price game in fractional order system. Chaos, Solitons and Fractals, 132(C), (2020), DOI: 10.1016/j.chaos.2019.109583.

[29] D. Mozyrska and E. Girejko: Overview of fractional h-difference operators. In Advances in harmonic analysis and operator theory. Birkhäuser, Basel, 2013, 253–268.

Go to article
[2] P. Ostalczyk: Discrete Fractional Calculus: Applications in Control and Image Processing. World Scientific, 2016.

[3] K. Oprzedkiewicz and K. Dziedzic: Fractional discrete-continuous model of heat transfer process. Archives of Control Sciences, 31(2), (2021), 287– 306, DOI: 10.24425/acs.2021.137419.

[4] T. Kaczorek and A. Ruszewski: Global stability of discrete-time nonlinear systems with descriptor standard and fractional positive linear parts and scalar feedbacks. Archives of Control Sciences, 30(4), (2020), 667–681, DOI: 10.24425/acs.2020.135846.

[5] J.B. Diaz and T.J. Olser: Differences of fractional order. Mathematics of Computation, 28 (1974), 185–202, DOI: 10.1090/S0025-5718-1974-0346352-5.

[6] F.M. Atici and P.W. Eloe: A transform method in discrete fractional calculus. International Journal of Difference Equations, 2 (2007), 165–176.

[7] F.M. Atici and P.W. Eloe: Discrete fractional calculus with the nabla operator. Electronic Journal of Qualitative Theory of Differential Equations, Spec. Ed. I, 3 , (2009), 1–12.

[8] G. Anastassiou: Principles of delta fractional calculus on time scales and inequalities. Mathematical and Computer Modelling, 52(3-4), (2010), 556– 566, DOI: 10.1016/j.mcm.2010.03.055.

[9] T. Abdeljawad: On Riemann and Caputo fractional differences. Computers and Mathematics with Applications, 62(3), (2011), 1602–1611, DOI: 10.1016/j.camwa.2011.03.036.

[10] M. Edelman, E.E.N. Macau, and M.A.F. Sanjun (Eds.): Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives. Springer International Publishing, 2018.

[11] G.C. Wu, D. Baleanu, and S.D. Zeng: Discrete chaos in fractional sine and standard maps. Physics Letters A, 378(5-6), (2014), 484–487, DOI: 10.1016/j.physleta.2013.12.010.

[12] G.C. Wu and D. Baleanu: Discrete fractional logistic map and its chaos. Nonlinear Dynamics, 75(1-2), (2014), 283–287, DOI: 10.1007/s11071-013-1065-7.

[13] T. Hu: Discrete chaos in fractional Henon map. Applied Mathematics, 5(15), (2014), 2243–2248, DOI: 10.4236/am.2014.515218.

[14] G.C. Wu and D. Baleanu: Discrete chaos in fractional delayed logistic maps. Nonlinear Dynamics, 80 (2015), 1697–1703, DOI: 10.1007/s11071-014-1250-3.

[15] M.K. Shukla and B.B. Sharma: Investigation of chaos in fractional order generalized hyperchaotic Henon map. International Journal of Electronics and Communications, 78 (2017), 265–273, DOI: 10.1016/j.aeue.2017.05.009.

[16] A. Ouannas, A.A. Khennaoui, S. Bendoukha, and G. Grassi: On the dynamics and control of a fractional form of the discrete double scroll. International Journal of Bifurcation and Chaos, 29(6), (2019), DOI: 10.1142/S0218127419500780.

[17] L. Jouini, A. Ouannas, A.A. Khennaoui, X. Wang, G. Grassi, and V.T. Pham: The fractional form of a new three-dimensional generalized Henon map. Advances in Difference Equations, 122 (2019), DOI: 10.1186/s13662-019-2064-x.

[18] F. Hadjabi, A. Ouannas,N. Shawagfeh, A.A. Khennaoui, and G. Grassi: On two-dimensional fractional chaotic maps with symmetries. Symmetry, 12(5), (2020), DOI: 10.3390/sym12050756.

[19] D. Baleanu, G.C. Wu, Y.R. Bai, and F.L. Chen: Stability analysis of Caputo-like discrete fractional systems. Communications in Nonlinear Science and Numerical Simulation, 48 (2017), 520–530, DOI: 10.1016/j.cnsns.2017.01.002.

[20] A-A. Khennaoui, A. Ouannas, S. Bendoukha, G. Grassi, X. Wang, V-T. Pham, and F.E. Alsaadi: Chaos, control, and synchronization in some fractional-order difference equations. Advances in Difference Equations, 412 (2019), DOI: 10.1186/s13662-019-2343-6.

[21] A. Ouannas, A.A. Khennaoui, G. Grassi, and S. Bendoukha: On chaos in the fractional-order Grassi-Miller map and its control. Journal of Computational and Applied Mathematics, 358(2019), 293–305, DOI: 10.1016/j.cam.2019.03.031.

[22] A. Ouannas, A.A. Khennaoui, S. Momani, G. Grassi and V.T. Pham: Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization. AIP Advances, 10 (2020), DOI: 10.1063/5.0004884.

[23] A. Ouannas, A-A. Khennaoui, S. Momani, G. Grassi, V-T. Pham, R. El- Khazali, and D. Vo Hoang: A quadratic fractional map without equilibria: Bifurcation, 0–1 test, complexity, entropy, and control. Electronics, 9 (2020), DOI: 10.3390/electronics9050748.

[24] A. Ouannas, A-A. Khennaoui, S. Bendoukha, Z.Wang, and V-T. Pham: The dynamics and control of the fractional forms of some rational chaotic maps. Journal of Systems Science and Complexity, 33 (2020), 584–603, DOI: 10.1007/s11424-020-8326-6.

[25] A-A. Khennaoui, A. Ouannas, S. Bendoukha, G. Grassi, R.P. Lozi, and V-T. Pham: On fractional-order discrete-time systems: Chaos, stabilization and synchronization. Chaos, Solitons and Fractals, 119(C), (2019), 150– 162, DOI: 10.1016/j.chaos.2018.12.019.

[26] A. Ouannas, A-A. Khennaoui, Z. Odibat, V-T. Pham, and G. Grassi: On the dynamics, control and synchronization of fractional-order Ikeda map. Chaos, Solitons and Fractals, 123(C), (2015), 108–115, DOI: 10.1016/j.chaos.2019.04.002.

[27] A. Ouannas, F. Mesdoui, S. Momani, I. Batiha, and G. Grassi: Synchronization of FitzHugh-Nagumo reaction-diffusion systems via onedimensional linear control law. Archives of Control Sciences, 31(2), 2021, 333–345, DOI: 10.24425/acs.2021.137421.

[28] Y. Li, C. Sun, H. Ling, A. Lu, and Y. Liu: Oligopolies price game in fractional order system. Chaos, Solitons and Fractals, 132(C), (2020), DOI: 10.1016/j.chaos.2019.109583.

[29] D. Mozyrska and E. Girejko: Overview of fractional h-difference operators. In Advances in harmonic analysis and operator theory. Birkhäuser, Basel, 2013, 253–268.

Keywords:
bounded hybrid system
safety property
delta-reachability
multidirectional efficiency analysis

Hybrid systems (HS) are roughly described as a set of discrete state transitions and continuous dynamics modeled by differential equations. Parametric HS may be constructed by having parameters on the differential equations, initial conditions, jump conditions, or a combination of the previous ones. In real applications, the best solution is obtained by a set of metrics functional over the set of solutions generated from a finite set of parameters. This paper examines the choice of parameters on delta-reachability bounded hybrid systems.We present an efficient model based on the tool pHL-MT to benchmark the HS solutions (based on dReach), and a non-parametric frontier analysis approach, relying on multidirectional efficiency analysis (MEA). Three numerical examples of epidemic models with variable growth infectivity are presented, namely: when the variable of infected individuals oscillates around some endemic (non-autonomous) equilibrium; when there is an asymptotically stable non-trivial attractor; and in the presence of bump functions.

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[1] M. Althoff and J.M. Dolan: Online verification of automated road vehicles using reachability analysis. IEEE Trans. on Robotics, 30(4), (2014), 903– 918, DOI:
10.1109/TRO.2014.2312453.

[2] P. Bogetoft and J.L. Hougaard: Efficiency evaluations based on potential (non-proportional) improvements. Journal of Productivity Analysis, 12(3), (1999), 233–247, DOI: 10.1023/A:1007848222681.

[3] X. Chen, E. Abraham and S. Sankaranarayanan: Flow*: An analyzer for non-linear hybrid systems. In: Proc. of CAV’13. LNCS, 8044, 258–263, Springer, 2013.

[4] E.M. Clarke and S. Gao: Model checking hybrid systems. In: Margaria T., Steffen B. (eds): Leveraging Applications of Formal Methods, Verification and Validation. Specialized Techniques and Applications. ISoLA 2014. Lecture Notes in Computer Science, 8803, 385–386, Springer, Berlin, Heidelberg, 2014.

[5] M. Franzle, C. Herde, S. Ratschan, T. Schubert and T. Teige: Efficient solving of large non-linear arithmetic constraint systems with complex Boolean structure. Journal on Satisfiability, Boolean Modeling and Computation, 1 (2007), 209–236, DOI: 10.3233/SAT190012.

[6] G. Frehse, C.L. Guernic, A. Donze, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang and O. Maler: SpaceEx: Scalable verification of hybrid systems. In: Proc. of CAV’11. LNCS, 6806, 379–395, Springer, 2011.

[7] S. Gao: Computable analysis, decision procedures, and hybrid automata: A new framework for the formal verification of cyber-physical systems. Ph.D. thesis, Carnegie Mellon University, 2012.

[8] S. Gao, S. Kong and E.M. Clarke: dReal: An SMT solver for nonlinear theories over the reals. In: M.P. Bonacina (ed.) CADE 2013. LNCS (LNAI), 7898, 208–214, Springer, Heidelberg (2013). DOI: 10.1007/978-3-642-38574-2.

[9] HyCreate: A tool for overapproximating reachability of hybrid automata, http://stanleybak.com/projects/hycreate/hycreate.html.

[10] S. Kong, S. Gao, W. Chen and E. Clarke: dReach: δ-reachability analysis for hybrid systems. In Proc. International Conference on Tools and Algorithms for the Construction and Analysis of Systems, 2015. Available: http://link.springer.com/10.1007/978-3-662-46681-0.

[11] J. Lygeros, C. Tomlin and S. Sastry: Hybrid systems: modeling analysis and control. Electronic Research Laboratory, University of California, Berkeley, CA, Tech. Rep. UCB/ERL M, 2008.

[12] A. Platzer and J. Quesel: Keymaera: A hybrid theorem prover for hybrid systems (system description). In: Proc. of IJCAR’08.LNCS, 5195, 171–178, Springer, 2008.

[13] S. Ratschan and Z. She: Safety verification of hybrid systems by constraint propagation based abstraction refinement. In: Proc. of HSCC’05. LNCS, 3414, 573–589, Springer, 2005.

[14] E.M. Rocha: Oscillatory behaviour on a non-autonomous hybrid SIRmodel. In: M. Chaves and M. Martins (eds.), Molecular Logic and Computational Synthetic Biology. MLCSB 2018. Lecture Notes in Computer Science, 11415, Springer, Cham, 2019.

[15] L. Zhang, Z. She, S. Ratschan, H. Hermanns and E. Hahn: Safety verification for probabilistic hybrid systems. In: Proc. International Conference on Computer Aided Verification, (2010), 196–211.

Go to article
[2] P. Bogetoft and J.L. Hougaard: Efficiency evaluations based on potential (non-proportional) improvements. Journal of Productivity Analysis, 12(3), (1999), 233–247, DOI: 10.1023/A:1007848222681.

[3] X. Chen, E. Abraham and S. Sankaranarayanan: Flow*: An analyzer for non-linear hybrid systems. In: Proc. of CAV’13. LNCS, 8044, 258–263, Springer, 2013.

[4] E.M. Clarke and S. Gao: Model checking hybrid systems. In: Margaria T., Steffen B. (eds): Leveraging Applications of Formal Methods, Verification and Validation. Specialized Techniques and Applications. ISoLA 2014. Lecture Notes in Computer Science, 8803, 385–386, Springer, Berlin, Heidelberg, 2014.

[5] M. Franzle, C. Herde, S. Ratschan, T. Schubert and T. Teige: Efficient solving of large non-linear arithmetic constraint systems with complex Boolean structure. Journal on Satisfiability, Boolean Modeling and Computation, 1 (2007), 209–236, DOI: 10.3233/SAT190012.

[6] G. Frehse, C.L. Guernic, A. Donze, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang and O. Maler: SpaceEx: Scalable verification of hybrid systems. In: Proc. of CAV’11. LNCS, 6806, 379–395, Springer, 2011.

[7] S. Gao: Computable analysis, decision procedures, and hybrid automata: A new framework for the formal verification of cyber-physical systems. Ph.D. thesis, Carnegie Mellon University, 2012.

[8] S. Gao, S. Kong and E.M. Clarke: dReal: An SMT solver for nonlinear theories over the reals. In: M.P. Bonacina (ed.) CADE 2013. LNCS (LNAI), 7898, 208–214, Springer, Heidelberg (2013). DOI: 10.1007/978-3-642-38574-2.

[9] HyCreate: A tool for overapproximating reachability of hybrid automata, http://stanleybak.com/projects/hycreate/hycreate.html.

[10] S. Kong, S. Gao, W. Chen and E. Clarke: dReach: δ-reachability analysis for hybrid systems. In Proc. International Conference on Tools and Algorithms for the Construction and Analysis of Systems, 2015. Available: http://link.springer.com/10.1007/978-3-662-46681-0.

[11] J. Lygeros, C. Tomlin and S. Sastry: Hybrid systems: modeling analysis and control. Electronic Research Laboratory, University of California, Berkeley, CA, Tech. Rep. UCB/ERL M, 2008.

[12] A. Platzer and J. Quesel: Keymaera: A hybrid theorem prover for hybrid systems (system description). In: Proc. of IJCAR’08.LNCS, 5195, 171–178, Springer, 2008.

[13] S. Ratschan and Z. She: Safety verification of hybrid systems by constraint propagation based abstraction refinement. In: Proc. of HSCC’05. LNCS, 3414, 573–589, Springer, 2005.

[14] E.M. Rocha: Oscillatory behaviour on a non-autonomous hybrid SIRmodel. In: M. Chaves and M. Martins (eds.), Molecular Logic and Computational Synthetic Biology. MLCSB 2018. Lecture Notes in Computer Science, 11415, Springer, Cham, 2019.

[15] L. Zhang, Z. She, S. Ratschan, H. Hermanns and E. Hahn: Safety verification for probabilistic hybrid systems. In: Proc. International Conference on Computer Aided Verification, (2010), 196–211.

The paper is concerned with the presentation and analysis of the Dynamic Matrix Control (DMC) model predictive control algorithm with the representation of the process input trajectories by parametrised sums of Laguerre functions. First the formulation of the DMCL (DMC with Laguerre functions) algorithm is presented. The algorithm differs from the standard DMC one in the formulation of the decision variables of the optimization problem – coefficients of approximations by the Laguerre functions instead of control input values are these variables. Then the DMCL algorithm is applied to two multivariable benchmark problems to investigate properties of the algorithm and to provide a concise comparison with the standard DMC one. The problems with difficult dynamics are selected, which usually leads to longer prediction and control horizons. Benefits from using Laguerre functions were shown, especially evident for smaller sampling intervals.

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[1] T.L. Blevins, G.K. McMillan, W.K. Wojsznis, and M.W. Brown: Advanced Control Unleashed. The ISA Society, Research Triangle Park, NC, 2003.

[2] T.L. Blevins,W.K.Wojsznis and M.Nixon: Advanced ControlFoundation. The ISA Society, Research Triangle Park, NC, 2013.

[3] E.F. Camacho and C. Bordons: Model Predictive Control. Springer Verlag, London, 1999.

[4] M. Ławrynczuk: Computationally Efficient Model Predictive Control Algorithms: A Neural Network Approach, Studies in Systems, Decision and Control. Vol. 3. Springer Verlag, Heidelberg, 2014.

[5] M. Ławrynczuk: Nonlinear model predictive control for processes with complex dynamics: parametrisation approach using Laguerre functions. International Journal of Applied Mathematics and Computer Science, 30(1), (2020), 35–46, DOI: 10.34768/amcs-2020-0003.

[6] J.M. Maciejowski: Predictive Control. Prentice Hall, Harlow, England, 2002.

[7] R. Nebeluk and P. Marusak: Efficient MPC algorithms with variable trajectories of parameters weighting predicted control errors. Archives of Control Sciences, 30(2), (2020), 325–363, DOI: 10.24425/acs.2020.133502.

[8] S.J. Qin and T.A.Badgwell:Asurvey of industrial model predictive control technology. Control Engineering Practice, 11(7), (2003), 733–764, DOI: 10.1016/S0967-0661(02)00186-7.

[9] J. B. Rawlings and D. Q. Mayne: Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, 2009.

[10] J.A. Rossiter: Model-Based Predictive Control. CRC Press, Boca Raton – London – New York – Washington, D.C., 2003.

[11] P. Tatjewski: Advanced Control of Industrial Processes. Springer Verlag, London, 2007.

[12] P. Tatjewski: Advanced control and on-line process optimization in multilayer structures. Annual Reviews in Control, 32(1), (2008), 71–85, DOI: 10.1016/j.arcontrol.2008.03.003.

[13] P. Tatjewski: Disturbance modeling and state estimation for offset-free predictive control with state-spaced process models. International Journal of Applied Mathematics and Computer Science, 24(2), (2014), 313–323, DOI: 10.2478/amcs-2014-0023.

[14] P. Tatjewski: Offset-free nonlinear Model Predictive Control with statespace process models. Archives of Control Sciences, 27(4), (2017), 595–615, DOI: 10.1515/acsc-2017-0035.

[15] P. Tatjewski: DMC algorithm with Laguerre functions. In Advanced, Contemporary Control, Proceedings of the 20th Polish Control Conference, pages 1006–1017, Łódz, Poland, (2020).

[16] G. Valencia-Palomo and J.A. Rossiter: Using Laguerre functions to improve efficiency of multi-parametric predictive control. In Proceedings of the 2010 American Control Conference, Baltimore, (2010).

[17] B. Wahlberg: System identification using the Laguerre models. IEEE Transactions on Automatic Control, 36(5), (1991), 551–562, DOI: 10.1109/9.76361.

[18] L. Wang: Discrete model predictive controller design using Laguerre functions. Journal of Process Control, 14(2), (2004), 131–142, DOI: 10.1016/S0959-1524(03)00028-3.

[19] L. Wang: Model Predictive Control System Design and Implementation using MATLAB. Springer Verlag, London, 2009.

[20] R. Wood and M. Berry: Terminal composition control of a binary distillation column. Chemical Engineering Science, 28(9), (1973), 1707–1717, DOI: 10.1016/0009-2509(73)80025-9.

Go to article
[2] T.L. Blevins,W.K.Wojsznis and M.Nixon: Advanced ControlFoundation. The ISA Society, Research Triangle Park, NC, 2013.

[3] E.F. Camacho and C. Bordons: Model Predictive Control. Springer Verlag, London, 1999.

[4] M. Ławrynczuk: Computationally Efficient Model Predictive Control Algorithms: A Neural Network Approach, Studies in Systems, Decision and Control. Vol. 3. Springer Verlag, Heidelberg, 2014.

[5] M. Ławrynczuk: Nonlinear model predictive control for processes with complex dynamics: parametrisation approach using Laguerre functions. International Journal of Applied Mathematics and Computer Science, 30(1), (2020), 35–46, DOI: 10.34768/amcs-2020-0003.

[6] J.M. Maciejowski: Predictive Control. Prentice Hall, Harlow, England, 2002.

[7] R. Nebeluk and P. Marusak: Efficient MPC algorithms with variable trajectories of parameters weighting predicted control errors. Archives of Control Sciences, 30(2), (2020), 325–363, DOI: 10.24425/acs.2020.133502.

[8] S.J. Qin and T.A.Badgwell:Asurvey of industrial model predictive control technology. Control Engineering Practice, 11(7), (2003), 733–764, DOI: 10.1016/S0967-0661(02)00186-7.

[9] J. B. Rawlings and D. Q. Mayne: Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, 2009.

[10] J.A. Rossiter: Model-Based Predictive Control. CRC Press, Boca Raton – London – New York – Washington, D.C., 2003.

[11] P. Tatjewski: Advanced Control of Industrial Processes. Springer Verlag, London, 2007.

[12] P. Tatjewski: Advanced control and on-line process optimization in multilayer structures. Annual Reviews in Control, 32(1), (2008), 71–85, DOI: 10.1016/j.arcontrol.2008.03.003.

[13] P. Tatjewski: Disturbance modeling and state estimation for offset-free predictive control with state-spaced process models. International Journal of Applied Mathematics and Computer Science, 24(2), (2014), 313–323, DOI: 10.2478/amcs-2014-0023.

[14] P. Tatjewski: Offset-free nonlinear Model Predictive Control with statespace process models. Archives of Control Sciences, 27(4), (2017), 595–615, DOI: 10.1515/acsc-2017-0035.

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Keywords:
Duhamel’s integral
Duhamel’s principle
Duhamel’s formula
Laplace transformation
semigroup of operators
Leibniz integral rule
Volterra integral equation
Caputo fractional derivative

The paper is a newapproach to the Duhamel integral. It contains an overviewof formulas and applications of Duhamel’s integral as well as a number of new results on the Duhamel integral and principle. Basic definitions are recalled and formulas for Duhamel’s integral are derived via Laplace transformation and Leibniz integral rule. Applications of Duhamel’s integral for solving certain types of differential and integral equations are presented. Moreover, an interpretation of Duhamel’s formula in the theory of operator semigroups is given. Some applications of Duhamel’s formula in control systems analysis are discussed. The work is also devoted to the usage of Duhamel’s integral for differential equations with fractional order derivative.

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[16] M.I. Gomoyunov: Optimal control problems with a fixed terminal time in linear fractional-order systems. Archives of Control Sciences, 30(4), (2020), 721–744, DOI: 10.24425/acs.2020.135849.

[17] R. Gorenflo and F. Mainardi: Fractional calculus: Integral and differential equations of fractional order. In: A. Carpinteri and F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997.

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[20] V.A. Ilin, V.A. Sadovniczij and B.H. Sendov: Mathematical Analysis, Continuation Course. Moscow University Press, Moscow, 1987, (in Russian).

[21] T. Kaczorek: Realization problem for fractional continuous-time systems. Archives of Control Sciences, 18(1), (2008), 43–58.

[22] T. Kaczorek: A new method for computation of positive realizations of fractional linear continuous-time systems. Archives of Control Sciences, 28(4), (2018), 511–525, DOI: 10.24425/acs.2018.125481.

[23] K.S. Kazakov: Dynamic response of a single degree of freedom (SDOF) system in some special load cases, based on the Duhamel integral. In: Proceedings of the EngOpt 2008: International Conference on Engineering Optimization, Rio de Janeiro, Brazil, 2008.

[24] N.S. Khabeev: Duhamel integral form for the interface heat flux between bubble and liquid. International Journal of Heat andMass Transfer, 50(25), (2007), 5340–5343, DOI: 10.1016/j.ijheatmasstransfer.2007.06.012.

[25] J. Klamka and B. Sikora:Newcontrollability criteria for fractional systems with varying delays. Theory and applications of non-integer order systems in Lecture Notes in Electrical Engineering, 407, (2017), 333–344.

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[30] M.A. Lavrentev and B.V. Shabat: Methods of the Theory of Functions of a Complex Variable. Nauka, Moscow, 1973, (in Russian).

[31] Sz. Lubecki: Duhamel’s theorem for time-dependent thermal boundary conditions. In: R.B Hetnarski (Ed.), Encyclopedia of Thermal Stresses, Springer Netherlands, New Delhi, 2014.

[32] E. Łobos and B. Sikora: Advanced Calculus – Selected Topics. Silesian University of Technology Press, Gliwice, 2009.

[33] Z. Łuszczki: Application of general Bernstein polynomials to proving some theorem on partial derivatives. Prace Matematyczne, 2(2), (1958), 355–360, (in Polish).

[34] J. Mikusinski: On continuous partial derivatives of function of several variables. Prace Matematyczne, 7(1), (1962), 55–58, (in Polish).

[35] S. Mischler: An introduction to evolution PDEs, Chapter 3 – Evolution Equation and semigroups, Dauphine Université Paris, published online: www.ceremade.dauphine.fr/~mischler/Enseignements/M2evol1516/chap3.pdf.

[36] N.Nguyen Du Vi, D. Dinh Thanh and T. Vu Kim:Weighted norminequalities for derivatives and their applications. Kodai Mathematical Journal, 36(2), (2013), 228–245.

[37] J. Osiowski: An Outline of the Operator Calculus.Wydawnictwo Naukowo- Techniczne, Warsaw, 1981, (in Polish).

[38] G. Petiau: Theory of Bessel’s Functions. Centre National de la Recherche Scientifique, Paris, 1955, (in French).

[39] K. Rogowski: Reachability of standard and fractional continuous-time systems with constant inputs. Archives of Control Sciences, 26(2), (2016), 147–159, DOI: 10.1515/acsc-2016-0008.

[40] F.A. Shelkovnikov and K.G. Takaishvili: Collection of Problems on Operational Calculus. Higher School Press, Moscow, 1976, (in Russian).

[41] B. Sikora: Controllability of time-delay fractional systems with and without constraints. IET Control Theory & Applications, 10(3), (2019), 320–327.

[42] B. Sikora: Controllability criteria for time-delay fractional systems with retarded state. International Journal of Applied Mathematics and Computer Science, 26(6), (2019), 521–531.

[43] B. Sikora and J. Klamka: Constrained controllability of fractional linear systems with delays in control. Systems & Control Letters, 106, (2017), 9–15.

[44] B. Sikora and J. Klamka: Constrained controllability of fractional linear systems with delays in control. Kybernetika, 53(2), (2017), 370–381.

[45] B. Sikora: On application of Rothe’s fixed point theorem to study the controllability of fractional semilinear systems with delays. Kybernetika, 55(4), (2019), 675–698.

[46] B. Sikora and N. Matlok: On controllability of fractional positive continuous-time linear systems with delay. Archives of Control Sciences, 31(1), (2021), 29–51, DOI: 10.24425/acs.2021.136879.

[47] J. Tokarzewski: Zeros and the output-zeroing problem in linear fractionalorder systems. Archives of Control Sciences, 18(4), (2008), 437–451.

[48] S. Umarov: On fractional Duhamel’s principle and its applications. Journal of Differential Equations, 252(10), (2012), 5217–5234.

[49] S. Umarov and E. Saydamatov: A fractional analog of the Duhamel principle. Fractional Calculus & Applied Analysis, 9(1), (2006), 57–70.

[50] S. Umarov and E. Saydamatov: A generalization of Duhamel’s principle for differential equations of fractional order. Doklady Mathematics, 75(1), (2007), 94–96.

[51] C. Villani: Birth of a Theorem, Grasset & Fasquelle, 2012, (in French).

Go to article
[2] R. Almeida, R. Kamocki, A.B. Malinowska and T. Odzijewicz: On the existence of optimal consensus control for the fractional Cucker– Smale model. Archives of Control Sciences, 30(4), (2020), 625–651, DOI: 10.24425/acs.2020.135844.

[3] J.J. Benedetto and W. Czaja: Integration and Modern Analysis. Brikhäuser, Boston, 2009.

[4] N. Bourbaki: Fonctions d’une Variable Réele. Hermann, Paris, 1976.

[5] M. Buslowicz: Robust stability of a class of uncertain fractional order linear systems with pure delay. Archives of Control Sciences, 25(2), (2015), 177–187.

[6] A.I. Daniliu: Impulse Series Method in the Dynamic Analysis of Structures. In: Proceedings of the Eleventh World Conference on Earthquake Engineering, paper no 577, (1996).

[7] S. Das: Functional Fractional Calculus. Second edition, Springer, Berlin, 2011.

[8] L. Debnath and D. Bhatta: Integral Transforms and their Applications. Third edition, CRC Press – Taylor & Francis Group, Boca Raton, 2015.

[9] V.A. Ditkin and A.P. Prudnikov: Integral Transforms and Operational Calculus. Pergamon Press, Oxford, 1965.

[10] G. Doetsch: Handbuch der Laplace-Transformation. Band I, Brikhäuser, Basel, 1950.

[11] D.T. Duc and N.D.V. Nhan: Norm inequalities for new convolutions and their applications. Applicable Analysis and Discrete Mathematics, 9(1), (2015), 168–179, DOI: 10.2298/AADM150109001D.

[12] K.J. Engel and R. Nagel: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York, 2000.

[13] K.J. Engel and R. Nagel: A Short Course on Operator Semigroups. Springer, New York, 2006.

[14] T.M. Flett: Differential analysis. Cambridge University Press, Cambridge, 2008.

[15] J.F. Gomez-Aguilar, H. Yepez-Martinez, C. Calderon-Ramon, I. Cruz- Orduna, R.F. Escobar-Jimenez and V.H. Olivares-Peregrino: Modeling of mass-spring-damper system by fractional derivatives with and without a singular kernel. Entropy, 17(9), (2015), 5340–5343, DOI: 10.3390/e17096289.

[16] M.I. Gomoyunov: Optimal control problems with a fixed terminal time in linear fractional-order systems. Archives of Control Sciences, 30(4), (2020), 721–744, DOI: 10.24425/acs.2020.135849.

[17] R. Gorenflo and F. Mainardi: Fractional calculus: Integral and differential equations of fractional order. In: A. Carpinteri and F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien, 1997.

[18] R. Grzymkowski and R. Witula: Complex functions and Laplace transform in examples and problems. Pracownia Komputerowa Jacka Skalmierskiego, Gliwice, 2010, (in Polish).

[19] V.A. Ilin, V.A. Sadovniczij and B.H. Sendov: Mathematical Analysis, Initial Course. Moscow University Press, Moscow, 1985, (in Russian).

[20] V.A. Ilin, V.A. Sadovniczij and B.H. Sendov: Mathematical Analysis, Continuation Course. Moscow University Press, Moscow, 1987, (in Russian).

[21] T. Kaczorek: Realization problem for fractional continuous-time systems. Archives of Control Sciences, 18(1), (2008), 43–58.

[22] T. Kaczorek: A new method for computation of positive realizations of fractional linear continuous-time systems. Archives of Control Sciences, 28(4), (2018), 511–525, DOI: 10.24425/acs.2018.125481.

[23] K.S. Kazakov: Dynamic response of a single degree of freedom (SDOF) system in some special load cases, based on the Duhamel integral. In: Proceedings of the EngOpt 2008: International Conference on Engineering Optimization, Rio de Janeiro, Brazil, 2008.

[24] N.S. Khabeev: Duhamel integral form for the interface heat flux between bubble and liquid. International Journal of Heat andMass Transfer, 50(25), (2007), 5340–5343, DOI: 10.1016/j.ijheatmasstransfer.2007.06.012.

[25] J. Klamka and B. Sikora:Newcontrollability criteria for fractional systems with varying delays. Theory and applications of non-integer order systems in Lecture Notes in Electrical Engineering, 407, (2017), 333–344.

[26] M. Klimek: On Solutions of Linear Fractional Differential Equations of a Variational Type. Wydawnictwo Politechniki Częstochowskiej, Częstochowa, 2009.

[27] M.I. Kontorowicz: The Operator Calculus and Processes in Electrical Systems. Wydawnictwo Naukowo-Techniczne, Warsaw, 1968, (in Polish).

[28] M.I. Krasnov , A.I. Kiselev and G.I. Makarenko: Functions of a Complex Variable, Operational Calculus, and Stability Theory, Problems and Exercises. Mir Publishers, Moscow, 1984.

[29] L.D. Kudryavtsev: Course of Mathematical Analysis. Volume 2, Higher School Press, Moscow, 1988, (in Russian).

[30] M.A. Lavrentev and B.V. Shabat: Methods of the Theory of Functions of a Complex Variable. Nauka, Moscow, 1973, (in Russian).

[31] Sz. Lubecki: Duhamel’s theorem for time-dependent thermal boundary conditions. In: R.B Hetnarski (Ed.), Encyclopedia of Thermal Stresses, Springer Netherlands, New Delhi, 2014.

[32] E. Łobos and B. Sikora: Advanced Calculus – Selected Topics. Silesian University of Technology Press, Gliwice, 2009.

[33] Z. Łuszczki: Application of general Bernstein polynomials to proving some theorem on partial derivatives. Prace Matematyczne, 2(2), (1958), 355–360, (in Polish).

[34] J. Mikusinski: On continuous partial derivatives of function of several variables. Prace Matematyczne, 7(1), (1962), 55–58, (in Polish).

[35] S. Mischler: An introduction to evolution PDEs, Chapter 3 – Evolution Equation and semigroups, Dauphine Université Paris, published online: www.ceremade.dauphine.fr/~mischler/Enseignements/M2evol1516/chap3.pdf.

[36] N.Nguyen Du Vi, D. Dinh Thanh and T. Vu Kim:Weighted norminequalities for derivatives and their applications. Kodai Mathematical Journal, 36(2), (2013), 228–245.

[37] J. Osiowski: An Outline of the Operator Calculus.Wydawnictwo Naukowo- Techniczne, Warsaw, 1981, (in Polish).

[38] G. Petiau: Theory of Bessel’s Functions. Centre National de la Recherche Scientifique, Paris, 1955, (in French).

[39] K. Rogowski: Reachability of standard and fractional continuous-time systems with constant inputs. Archives of Control Sciences, 26(2), (2016), 147–159, DOI: 10.1515/acsc-2016-0008.

[40] F.A. Shelkovnikov and K.G. Takaishvili: Collection of Problems on Operational Calculus. Higher School Press, Moscow, 1976, (in Russian).

[41] B. Sikora: Controllability of time-delay fractional systems with and without constraints. IET Control Theory & Applications, 10(3), (2019), 320–327.

[42] B. Sikora: Controllability criteria for time-delay fractional systems with retarded state. International Journal of Applied Mathematics and Computer Science, 26(6), (2019), 521–531.

[43] B. Sikora and J. Klamka: Constrained controllability of fractional linear systems with delays in control. Systems & Control Letters, 106, (2017), 9–15.

[44] B. Sikora and J. Klamka: Constrained controllability of fractional linear systems with delays in control. Kybernetika, 53(2), (2017), 370–381.

[45] B. Sikora: On application of Rothe’s fixed point theorem to study the controllability of fractional semilinear systems with delays. Kybernetika, 55(4), (2019), 675–698.

[46] B. Sikora and N. Matlok: On controllability of fractional positive continuous-time linear systems with delay. Archives of Control Sciences, 31(1), (2021), 29–51, DOI: 10.24425/acs.2021.136879.

[47] J. Tokarzewski: Zeros and the output-zeroing problem in linear fractionalorder systems. Archives of Control Sciences, 18(4), (2008), 437–451.

[48] S. Umarov: On fractional Duhamel’s principle and its applications. Journal of Differential Equations, 252(10), (2012), 5217–5234.

[49] S. Umarov and E. Saydamatov: A fractional analog of the Duhamel principle. Fractional Calculus & Applied Analysis, 9(1), (2006), 57–70.

[50] S. Umarov and E. Saydamatov: A generalization of Duhamel’s principle for differential equations of fractional order. Doklady Mathematics, 75(1), (2007), 94–96.

[51] C. Villani: Birth of a Theorem, Grasset & Fasquelle, 2012, (in French).

Keywords:
fractional optimal control problem
discrete time singular system
fractional derivative
Hamiltonian technique

In this work, we present optimal control formulation and numerical algorithm for fractional order discrete time singular system (DTSS) for fixed terminal state and fixed terminal time endpoint condition. The performance index (PI) is in quadratic form, and the system dynamics is in the sense of Riemann-Liouville fractional derivative (RLFD). A coordinate transformation is used to convert the fractional-order DTSS into its equivalent non-singular form, and then the optimal control problem (OCP) is formulated. The Hamiltonian technique is used to derive the necessary conditions. A solution algorithm is presented for solving the OCP. To validate the formulation and the solution algorithm, an example for fixed terminal state and fixed terminal time case is presented.

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[8] R.K. Biswas and S. Sen: Fractional optimal control problems with specified final time. Journal of Computational and Nonlinear Dynamics, 6(021009), (2010), DOI: 10.1115/1.4002508.

[9] R.K. Biswas and S. Sen: Free final time fractional optimal control problems. Journal of the Franklin Institute, 351(2), (2014), 941–951, DOI: 10.1016/j.jfranklin.2013.09.024.

[10] R.K. Biswas and S. Sen: Numerical method for solving fractional optimal control problems. In: Proceedings of the ASME IDETC/CIE Conference, (2010), 1205–120, DOI: 10.1115/DETC2009-87008.

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[12] Y. Ding, Z. Wang, and H. Ye: Optimal control of a fractional-order HIVimmune system with memory. IEEE Transactions on Control Systems Technology, 20(3), (2012), 763–769, DOI: 10.1109/TCST.2011.2153203.

[13] T. Chiranjeevi and R.K. Biswas: Closed-form solution of optimal control problem of a fractional order system. Journal of King Saud University – Science, 31(4), (2019), 1042–1047, DOI: 10.1016/j.jksus.2019.02.010.

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[16] S. Yousefi, A. Lotfi, and M. Dehghan: The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems. Journal of Vibration and Control, 17(13), (2011), 2059–2065, DOI: 10.1177/1077546311399950.

[17] M. Gomoyunov: Optimal control problems with a fixed terminal time in linear fractional-order systems. Archives of Control Sciences, 30(2), (2019), 295–324, DOI: 10.24425/acs.2020.135849.

[18] T. Chiranjeevi and R.K. Biswas: Discrete-time fractional optimal control. Mathematics, 5(2), (2017), DOI: 10.3390/math5020025.

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[20] T. Chiranjeevi, R.K. Biswas, and N.R. Babu: Effect of initialization on optimal control problem of fractional order discrete-time system. Journal of Interdisciplinary Mathematics, 23(1), (2020), 293–302, DOI: 10.1080/09720502.2020.1721924.

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[25] R.K. Biswas and S. Sen: Fractional optimal control problems: a pseudostate- space approach. Journal of Vibration and Control, 17(7), (2011), 1034–1041, DOI: 10.1177/1077546310373618.

[26] R.K. Biswas and S. Sen: Fractional optimal control within Caputo’s derivative. In: Proceedings of the ASME IDETC/CIE Conference, (2012), 353– 360, DOI: 10.1115/DETC2011-48045.

[27] T. Chiranjeevi, R.K. Biswas, and C. Sethi: Optimal control of fractional order singular system. The International Journal of Electrical Engineering & Education, p. 0020720919833031, (2019), DOI: 10.1177/0020720919833031.

[28] T. Chiranjeevi and R.K. Biswas: Numerical approach to the fractional optimal control problem of continuous-time singular system. In: Advances in Electrical Control and Signal Systems, Singapore, (2020), 239–248, DOI: 10.1007/978-981-15-5262-5_16.

[29] T. Chiranjeevi and R.K. Biswas: Linear quadratic optimal control problem of fractional order continuous-time singular system. Procedia Computer Science, 171 (2020), 1261–1268, DOI: 10.1016/j.procs.2020.04.134.

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A problem of optimization for production and storge costs is studied. The problem consists in manufacture of
*n *types of products, with some given restrictions, so that the total production and storage costs are minimal. The mathematical model is built using the framework of driftless control affine systems. Controllability is studied using Lie geometric methods and the optimal solution is obtained with Pontryagin Maximum Principle. It is proved that the economical system is not controllable, in the sense that we can only produce a certain quantity of products. Finally, some numerical examples are given with graphical representation.

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[1] A. Agrachev and Y.L. Sachkov: Control theory from the geometric viewpoint. Encyclopedia of Mathematical Sciences, Control Theory and Optimization, 87, Springer, 2004.

[2] K.J. Arrow: Applications of control theory of economic growth. Mathematics of Decision Sciences, 2, AMS, 1968.

[3] S. Axsater: Control theory concepts in production and inventory control. International Journal of Systems Science, 16(2), (1985), 161–169, DOI: 10.1080/00207728508926662.

[4] R. Bellmann: Adaptive control processes: a guided tour. Princeton Univ. Press: Princeton, 1972.

[5] S. Benjaafar, J.P. Gayon, and S. Tepe: Optimal control of a productioninventory system with customer impatience. Operations Research Letters, 38(4), (2010), 267–272, DOI: 10.1016/j.orl.2010.03.008.

[6] R. Brocket: Lie algebras and Lie groups in control theory. In: Mayne D.Q., Brockett R.W. (eds) Geometric Methods in System Theory. NATO Advanced Study Institutes Series (Series C – Mathematical and Physical Sciences), vol. 3. Springer, Dordrecht, 1973, 43–82, DOI: 10.1007/978-94-010-2675-8_2.

[7] M. Caputo: Foundations of Dynamic Economic Analysis: Optimal Control Theory and Applications. Cambridge Univ. Press, 2005, DOI: 10.1017/CBO9780511806827.

[8] M. Danahe, A. Chelbi, and N. Rezg: Optimal production plan for a multiproducts manufacturing system with production rate dependent failure rate. International Journal of Production Research, 50(13), (2012), 3517–3528, DOI: 10.1080/00207543.2012.671585.

[9] G. Feichtinger and R. Hartl: Optimal pricing and production in an inventory model. European Journal of Operational Research, 19 (1985), 45–56, DOI: 10.1016/0377-2217(85)90307-8.

[10] C. Gaimon: Simultaneous and dynamic price, production, inventory and capacity decisions. European Journal of Operational Research, 35 (1988), 426–441.

[11] J.P. Gayon, S. Vercraene, and S.D. Flapper: Optimal control of a production-inventory system with product returns and two disposal options. European Journal of Operational Research, 262(2), (2017), 499–508, DOI: 10.1016/j.ejor.2017.03.018.

[12] C. Hermosilla, R. Vinter, and H. Zidani: Hamilton–Jacobi–Bellman equations for optimal control processes with convex state constraints. Systems & Control Letters, 109 (2017), 30–36, DOI: 10.1016/j.sysconle.2017.09.004.

[13] V. Jurdjevic: Geometric Control Theory. Cambridge Studies in Advanced Mathematics, 52, Cambridge Univ. Press, 1997, DOI: 10.1017/CBO9780511530036.

[14] M.I. Kamien and N.L. Schwartz: Dynamic optimization: The Calculus of Variations and Optimal Control in Economics and Management, 31 Elsevier, 1991.

[15] K. Kogan and E. Khmelnitsky: An optimal control model for continuous time production and setup scheduling. International Journal of Production Research, 34(3), (1996), 715–725.

[16] Y. Qiu, J. Qiao, and P. Pardalos: Optimal production, replenishment, delivery, routing and inventory management policies for products with perishable inventory. Omega-International Journal of Management Science, 82 (2019), 193–204, DOI: 10.1016/j.omega.2018.01.006.

[17] S.M. LaValle: Planning Algorithms. Cambridge University Press, 2006.

[18] M. Li and Z. Wang: An integrated replenishment and production control policy under inventory inaccuracy and time-delay. Computers&Operations Research, 88 (2017), 137–149, DOI: 10.1016/j.cor.2017.06.014.

[19] B. Li and A. Arreola-Risa: Optimizing a production-inventory system under a cost target. Computers&Operations Research, 123 (2020), 105015, DOI: 10.1016/j.cor.2020.105015.

[20] M. Ortega and L. Lin: Control theory applications to the productioninventory problem: a review. International Journal of Production Research, 42(11), (2004), 2303–2322, DOI: 10.1080/00207540410001666260.

[21] V. Pando and J. Sicilia: A new approach to maximize the profit/cost ratio in a stock-dependent demand inventory model. Computers & Operations Research, 120 (2020), 104940, DOI: 10.1016/j.cor.2020.104940.

[22] L. Popescu: Applications of driftles control affine sytems to a problem of inventory and production. Studies in Informatics and Control, 28(1), (2019), 25–34, DOI: 10.24846/v28i1y201903.

[23] L. Popescu: Applications of optimal control to production planning. Information Technology and Control, 49(1), (2020), 89–99, DOI: 10.5755/j01.itc.49.1.23891.

[24] L. Popescu, D. Militaru, and O. Mituca: Optimal control applications in the study of production management. International Journal of Computers, Communications & Control, 15(2), (2020), 3859, DOI: 10.15837/ijccc.2020.2.3859.

[25] A. Seierstad and K. Sydsater: Optimal Control Theory with Economic Applications. North-Holland, Amsterdam, NL, 1987.

[26] S.P. Sethi: Applications of the Maximum Principle to Production and Inventory Problems. Proceedings Third International Symposium on Inventories, Budapest, Aug. 27-31, (1984), 753–756.

[27] S.P. Sethi and G.L.Thompson: Optimal Control Theory: Applications to Management Science and Economics. Springer, New York, 2000.

[28] J.D. Schwartz and D.E. Rivera: A process control approach to tactical inventory management in production-inventory systems. International Journal of Production Economics, 125(1), (2010), 111–124, DOI: 10.1016/j.ijpe.2010.01.011.

[29] D.R. Towill, G.N. Evans, and P. Cheema: Analysis and design of an adaptive minimum reasonable inventory control system. Production Planning & Control, 8(6), (1997), 545–557, DOI: 10.1080/095372897234885.

[30] T.A. Weber, Optimal control theory with applications in economics. MIT Press, 2011.

Go to article
[2] K.J. Arrow: Applications of control theory of economic growth. Mathematics of Decision Sciences, 2, AMS, 1968.

[3] S. Axsater: Control theory concepts in production and inventory control. International Journal of Systems Science, 16(2), (1985), 161–169, DOI: 10.1080/00207728508926662.

[4] R. Bellmann: Adaptive control processes: a guided tour. Princeton Univ. Press: Princeton, 1972.

[5] S. Benjaafar, J.P. Gayon, and S. Tepe: Optimal control of a productioninventory system with customer impatience. Operations Research Letters, 38(4), (2010), 267–272, DOI: 10.1016/j.orl.2010.03.008.

[6] R. Brocket: Lie algebras and Lie groups in control theory. In: Mayne D.Q., Brockett R.W. (eds) Geometric Methods in System Theory. NATO Advanced Study Institutes Series (Series C – Mathematical and Physical Sciences), vol. 3. Springer, Dordrecht, 1973, 43–82, DOI: 10.1007/978-94-010-2675-8_2.

[7] M. Caputo: Foundations of Dynamic Economic Analysis: Optimal Control Theory and Applications. Cambridge Univ. Press, 2005, DOI: 10.1017/CBO9780511806827.

[8] M. Danahe, A. Chelbi, and N. Rezg: Optimal production plan for a multiproducts manufacturing system with production rate dependent failure rate. International Journal of Production Research, 50(13), (2012), 3517–3528, DOI: 10.1080/00207543.2012.671585.

[9] G. Feichtinger and R. Hartl: Optimal pricing and production in an inventory model. European Journal of Operational Research, 19 (1985), 45–56, DOI: 10.1016/0377-2217(85)90307-8.

[10] C. Gaimon: Simultaneous and dynamic price, production, inventory and capacity decisions. European Journal of Operational Research, 35 (1988), 426–441.

[11] J.P. Gayon, S. Vercraene, and S.D. Flapper: Optimal control of a production-inventory system with product returns and two disposal options. European Journal of Operational Research, 262(2), (2017), 499–508, DOI: 10.1016/j.ejor.2017.03.018.

[12] C. Hermosilla, R. Vinter, and H. Zidani: Hamilton–Jacobi–Bellman equations for optimal control processes with convex state constraints. Systems & Control Letters, 109 (2017), 30–36, DOI: 10.1016/j.sysconle.2017.09.004.

[13] V. Jurdjevic: Geometric Control Theory. Cambridge Studies in Advanced Mathematics, 52, Cambridge Univ. Press, 1997, DOI: 10.1017/CBO9780511530036.

[14] M.I. Kamien and N.L. Schwartz: Dynamic optimization: The Calculus of Variations and Optimal Control in Economics and Management, 31 Elsevier, 1991.

[15] K. Kogan and E. Khmelnitsky: An optimal control model for continuous time production and setup scheduling. International Journal of Production Research, 34(3), (1996), 715–725.

[16] Y. Qiu, J. Qiao, and P. Pardalos: Optimal production, replenishment, delivery, routing and inventory management policies for products with perishable inventory. Omega-International Journal of Management Science, 82 (2019), 193–204, DOI: 10.1016/j.omega.2018.01.006.

[17] S.M. LaValle: Planning Algorithms. Cambridge University Press, 2006.

[18] M. Li and Z. Wang: An integrated replenishment and production control policy under inventory inaccuracy and time-delay. Computers&Operations Research, 88 (2017), 137–149, DOI: 10.1016/j.cor.2017.06.014.

[19] B. Li and A. Arreola-Risa: Optimizing a production-inventory system under a cost target. Computers&Operations Research, 123 (2020), 105015, DOI: 10.1016/j.cor.2020.105015.

[20] M. Ortega and L. Lin: Control theory applications to the productioninventory problem: a review. International Journal of Production Research, 42(11), (2004), 2303–2322, DOI: 10.1080/00207540410001666260.

[21] V. Pando and J. Sicilia: A new approach to maximize the profit/cost ratio in a stock-dependent demand inventory model. Computers & Operations Research, 120 (2020), 104940, DOI: 10.1016/j.cor.2020.104940.

[22] L. Popescu: Applications of driftles control affine sytems to a problem of inventory and production. Studies in Informatics and Control, 28(1), (2019), 25–34, DOI: 10.24846/v28i1y201903.

[23] L. Popescu: Applications of optimal control to production planning. Information Technology and Control, 49(1), (2020), 89–99, DOI: 10.5755/j01.itc.49.1.23891.

[24] L. Popescu, D. Militaru, and O. Mituca: Optimal control applications in the study of production management. International Journal of Computers, Communications & Control, 15(2), (2020), 3859, DOI: 10.15837/ijccc.2020.2.3859.

[25] A. Seierstad and K. Sydsater: Optimal Control Theory with Economic Applications. North-Holland, Amsterdam, NL, 1987.

[26] S.P. Sethi: Applications of the Maximum Principle to Production and Inventory Problems. Proceedings Third International Symposium on Inventories, Budapest, Aug. 27-31, (1984), 753–756.

[27] S.P. Sethi and G.L.Thompson: Optimal Control Theory: Applications to Management Science and Economics. Springer, New York, 2000.

[28] J.D. Schwartz and D.E. Rivera: A process control approach to tactical inventory management in production-inventory systems. International Journal of Production Economics, 125(1), (2010), 111–124, DOI: 10.1016/j.ijpe.2010.01.011.

[29] D.R. Towill, G.N. Evans, and P. Cheema: Analysis and design of an adaptive minimum reasonable inventory control system. Production Planning & Control, 8(6), (1997), 545–557, DOI: 10.1080/095372897234885.

[30] T.A. Weber, Optimal control theory with applications in economics. MIT Press, 2011.

Small bucket models with many short fictitious micro-periods ensure high-quality schedules in multi-level systems, i.e., with multiple stages or dependent demand. In such models, setup times longer than a single period are, however, more likely. This paper presents new mixedinteger programming models for the proportional lot-sizing and scheduling problem (PLSP) with setup operations overlapping multiple periods with variable capacity.

A new model is proposed that explicitly determines periods overlapped by each setup operation and the time spent on setup execution during each period. The model assumes that most periods have the same length; however, a few of them are shorter, and the time interval determined by two consecutive shorter periods is always longer than a single setup operation. The computational experiments showthat the newmodel requires a significantly smaller computation effort than known models.

Go to article
A new model is proposed that explicitly determines periods overlapped by each setup operation and the time spent on setup execution during each period. The model assumes that most periods have the same length; however, a few of them are shorter, and the time interval determined by two consecutive shorter periods is always longer than a single setup operation. The computational experiments showthat the newmodel requires a significantly smaller computation effort than known models.

[1] I. Barany, T.J. van Roy and L.A. Wolsey: Uncapacitated lot-sizing: The convex hull of solutions. Mathematical Programming Studies, 22 (1984), 32–43, DOI:
10.1007/BFb0121006.

[2] G. Belvaux and L.A. Wolsey: Modelling practical lot-sizing problems as mixed-integer programs. Management Science, 47(7), (2001), 993–1007, DOI: 10.1287/mnsc.47.7.993.9800.

[3] J.D. Blocher, S. Chand and K. Sengupta: The changeover scheduling problem with time and cost considerations: Analytical results and a forward algorithm. Operations Research, 47(7), (1999), 559-569, DOI: 10.1287/opre.47.4.559.

[4] W. Bozejko, M. Uchronski and M. Wodecki: Multi-machine scheduling problem with setup times. Archives of Control Sciences, 22(4), (2012), 441– 449, DOI: 10.2478/v10170-011-0034-y.

[5] W. Bozejko, A. Gnatowski, R. Idzikowski and M. Wodecki: Cyclic flow shop scheduling problem with two-machine cells. Archives of Control Sciences, 27(2), (2017), 151–167, DOI: 10.1515/acsc-2017-0009.

[6] D. Cattrysse, M. Salomon, R. Kuik and L. vanWassenhove: A dual ascent and column generation heuristic for the discrete lotsizing and scheduling problem with setup times. Management Science, 39(4), (1993), 477–486, DOI: 10.1287/mnsc.39.4.477.

[7] K. Copil, M. Worbelauer, H. Meyr and H. Tempelmeier: Simultaneous lotsizing and scheduling problems: a classification and review of models. OR Spectrum, 39(1), (2017), 1–64, DOI: 10.1007/s00291-015-0429-4.

[8] A. Drexl and K. Haase: Proportional lotsizing and scheduling. International Journal of Production Economics, 40(1), (1995), 73–87, DOI: 10.1016/0925-5273(95)00040-U.

[9] B. Fleischmann: The discrete lot-sizing and scheduling problem. European Journal of Operational Research, 44(3), (1990), 337-348, DOI: 10.1016/0377-2217(90)90245-7.

[10] K. Haase: Lotsizing and scheduling for production planning. Number 408 in Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, 1994.

[11] W. Kaczmarczyk: Inventory cost settings in small bucket lot-sizing and scheduling models. In Total Logistic Management Conference, Zakopane, Poland, November 25-28 2009.

[12] W. Kaczmarczyk: Modelling multi-period set-up times in the proportional lot-sizing problem. Decision Making in Manufacturing and Services, 3(1-2), (2009), 15–35, DOI: 10.7494/dmms.2009.3.2.15.

[13] W. Kaczmarczyk: Proportional lot-sizing and scheduling problem with identical parallel machines. International Journal of Production Research, 49(9), (2011), 2605–2623, DOI: 10.1080/00207543.2010.532929.

[14] W. Kaczmarczyk: Valid inequalities for proportional lot-sizing and scheduling problem with fictitious microperiods. International Journal of Production Economics, 219(1), (2020), 236–247, DOI: 10.1016/j.ijpe.2019.06.005.

[15] W.Kaczmarczyk: Explicit modelling of multi-period setup times in proportional lot-sizing problem with constant capacity. (2021), Preprint available at Research Square, DOI: 10.21203/rs.3.rs-1086310/v1.

[16] U.S. Karmarkar and L. Schrage: The deterministic dynamic product cycling problem. Operations Research, 33(2), (1985), 326–345, DOI: 10.1287/opre.33.2.326.

Go to article
[2] G. Belvaux and L.A. Wolsey: Modelling practical lot-sizing problems as mixed-integer programs. Management Science, 47(7), (2001), 993–1007, DOI: 10.1287/mnsc.47.7.993.9800.

[3] J.D. Blocher, S. Chand and K. Sengupta: The changeover scheduling problem with time and cost considerations: Analytical results and a forward algorithm. Operations Research, 47(7), (1999), 559-569, DOI: 10.1287/opre.47.4.559.

[4] W. Bozejko, M. Uchronski and M. Wodecki: Multi-machine scheduling problem with setup times. Archives of Control Sciences, 22(4), (2012), 441– 449, DOI: 10.2478/v10170-011-0034-y.

[5] W. Bozejko, A. Gnatowski, R. Idzikowski and M. Wodecki: Cyclic flow shop scheduling problem with two-machine cells. Archives of Control Sciences, 27(2), (2017), 151–167, DOI: 10.1515/acsc-2017-0009.

[6] D. Cattrysse, M. Salomon, R. Kuik and L. vanWassenhove: A dual ascent and column generation heuristic for the discrete lotsizing and scheduling problem with setup times. Management Science, 39(4), (1993), 477–486, DOI: 10.1287/mnsc.39.4.477.

[7] K. Copil, M. Worbelauer, H. Meyr and H. Tempelmeier: Simultaneous lotsizing and scheduling problems: a classification and review of models. OR Spectrum, 39(1), (2017), 1–64, DOI: 10.1007/s00291-015-0429-4.

[8] A. Drexl and K. Haase: Proportional lotsizing and scheduling. International Journal of Production Economics, 40(1), (1995), 73–87, DOI: 10.1016/0925-5273(95)00040-U.

[9] B. Fleischmann: The discrete lot-sizing and scheduling problem. European Journal of Operational Research, 44(3), (1990), 337-348, DOI: 10.1016/0377-2217(90)90245-7.

[10] K. Haase: Lotsizing and scheduling for production planning. Number 408 in Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, Berlin, 1994.

[11] W. Kaczmarczyk: Inventory cost settings in small bucket lot-sizing and scheduling models. In Total Logistic Management Conference, Zakopane, Poland, November 25-28 2009.

[12] W. Kaczmarczyk: Modelling multi-period set-up times in the proportional lot-sizing problem. Decision Making in Manufacturing and Services, 3(1-2), (2009), 15–35, DOI: 10.7494/dmms.2009.3.2.15.

[13] W. Kaczmarczyk: Proportional lot-sizing and scheduling problem with identical parallel machines. International Journal of Production Research, 49(9), (2011), 2605–2623, DOI: 10.1080/00207543.2010.532929.

[14] W. Kaczmarczyk: Valid inequalities for proportional lot-sizing and scheduling problem with fictitious microperiods. International Journal of Production Economics, 219(1), (2020), 236–247, DOI: 10.1016/j.ijpe.2019.06.005.

[15] W.Kaczmarczyk: Explicit modelling of multi-period setup times in proportional lot-sizing problem with constant capacity. (2021), Preprint available at Research Square, DOI: 10.21203/rs.3.rs-1086310/v1.

[16] U.S. Karmarkar and L. Schrage: The deterministic dynamic product cycling problem. Operations Research, 33(2), (1985), 326–345, DOI: 10.1287/opre.33.2.326.

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**Computer file** of the manuscript may be sent by e-mail to the address of Assistant Editor or acs@polsl.pl. Preferred text processors is TeX or LaTeX, however Word and other processors are also acceptable. In case of difficulties in processing the text, the author may be asked to supply the ASCII export of the original file.

**Manuscript**s sent via ordinary post should be typewritten double-spaced on one side of a standard size (A4) paper. Left side margin should be approximately 3cm (1.2'') wide. Each page should contain approximately 30 lines of 60 characters each. The manuscript including figures and tables together with their captions should be submitted. A separate signed letter giving the Author's preferred address for correspondence and return of proofs should be enclosed. Manuscript is the basis for editorial work.

**First page** should include the title of the paper, first name(s) and surname(s) of the Author(s), and a short summary (abstract), not longer than 20 lines.

**Keywords** of max. 5 - 7 items should be included in manuscript.

**Numeration**. All chapters, including the introduction, should be numbered in arabic numerals. Equations, tables and figures as well as theorems, corollaries, examples etc., should be numbered consecutively throughout the paper in arabic numerals, except in appendices. Appendices should be numbered with capital letters, and numeration should be closed within individual appendices.

**If the manuscript is not prepared with TeX**, mathematical expressions should be carefully written so as not to arouse confusion. Care should be taken that subscripts and superscripts are easily readable.

**Tables and figures** should be placed as desired by the Author within the text or on separate sheets with their suggested location indicated by the number of table or figure in the text. Figures, graphs and pictures (referred to as Fig. in the manuscript) should be numbered at the beginning of their caption following the figure. All figures should be prepared as PostScript EPS files or LaTeX picture files; in special cases, bitmaps of figure are also acceptable. The numbers and titles of tables should be placed above the main body of each table.

**References** should be listed alphabetically at the end of the manuscript. They should be numbered in ascending order and the numbers should be inserted in square brackets. References should be organized as follows. First initial(s), surname(s) of the author(s) and title of article or book. Then, for papers: title of periodical or collective work, volume number (year of issue), issue number, and numbers of the first and the last page; for books: publisher's name(s), place and year of issue. Example:

- R. E. Kalman: Mathematical description of linear dynamical system. SIAM J. Control. 1(2), (1963), 152-192.
- F. C. Shweppe: Uncertain dynamic systems. Prentice-Hall, Englewood Cliffs, N.J. 1970.

Please, give full titles of journals; only common words like Journal, Proceedings, Conference, etc. may be abbreviated ( to J., Proc., Conf., ... respectively). References to publications in the body of the manuscript should be indicated by the numbers of the adequate references in square brackets. When the paper is set in TeX the preferable form of preparing references is Bib TeX bib database.

**Footnotes** should be placed in the manuscript, beginning with "Received..." (date to be filled in by Editor), the author's institutional affiliation and acknowledgement of financial support,