
-
Previous Article
Optimal control of fish-feeding in a three-dimensional calm freshwater pond considering environmental concern
- JIMO Home
- This Issue
-
Next Article
Green cross-dock based supply chain network design under demand uncertainty using new metaheuristic algorithms
Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.
Readers can access Online First articles via the “Online First” tab for the selected journal.
Numerical solution to optimal control problems of oscillatory processes
The Institute of Control Systems, Azerbaijan National Academy of Sciences, 9, B. Vahabzadeh str., AZ1141, Baku, Azerbaijan |
A numerical approach to the investigation of optimal control problems of oscillatory processes with boundary and intermediate concentrated (lumped) control actions has been proposed in this paper. The corresponding analytical formulas for the components of the target functional gradient with respect to control actions considered on the class of piecewise continuous functions are obtained. The results of numerical experiments on the examples of solving model problems of optimal control of oscillatory processes with boundary and intermediate concentrated controls are provided. These results illustrate the dependence of the minimum settling time of oscillatory processes on the number and locations of concentrated control actions, on the process parameters, on the resistance coefficient of the medium (dissipation factor), and other factors.
References:
[1] |
K. R. Aida-Zade and J. A. Asadova,
Study of transients in oil pipelines, Automation and Remote Control, 72 (2011), 2563-2577.
doi: 10.1134/S0005117911120113. |
[2] |
K. R. Aida-Zade and J. A. Asadova, Optimal Control of Transients in Pipelines, Palmarium Academic Publishing, Saarbrücken, 2014, (in Russian). |
[3] |
A. G. Butkovsky, Optimal Control Theory for Systems with Distributed Parameters, Moscow, 1985, (in Russian). |
[4] |
S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5808-1.![]() ![]() ![]() |
[5] |
G. F. Guliyev, The problem of the point control for a hyperbolic equation, Avtomatika i Telemexanika, 1993 (1993), 80–84 (in Russian). |
[6] |
M. Gugat, G Leugering and G. Sklyar,
$L^p-$optimal boundary control for the wave equation, SIAM Journal Control and Optimization, 44 (2005), 49-74.
doi: 10.1137/S0363012903419212. |
[7] |
V. A. Il'in,
Two-endpoint boundary control of vibrations described by a finite-energy generalized solution of the wave equation, Differential Equations, 36 (2000), 1659-1675.
doi: 10.1007/BF02757368. |
[8] |
V. A. Il'in and V. V. Tikhomirov,
The wave equation with boundary control at two ends and the problem of the complete damping of a vibration process, Differential Equations, 35 (1999), 697-708.
|
[9] |
V. A. Il'in and E. I. Moiseyev, Optimal boundary control at one end with the free second end and a distribution of the string's total energy corresponding to this control, Dokladi RAN, 400 (2005), 585–591 (in Russian). |
[10] |
K. Kunisch and D. Wachsmuth,
On time optimal control of the wave equation and its numerical realization as parametric optimization problem, SIAM Journal Control and Optimization, 51 (2013), 1232-1262.
doi: 10.1137/120877520. |
[11] |
J.-L. Lions,
Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.
doi: 10.1137/1030001. |
[12] |
J. Loheac and E. Zuazua,
Norm saturating property of time optimal controls for wave-type equations, IFAC – Papers OnLine, 49 (2016), 37-42.
|
[13] |
F. P. Vasilyev, Optimization Methods, Moscow, Factorial Press, 2002.
![]() |
[14] |
L. N. Znamenskaya,
Control of string vibrations in the class of generalized solutions in $L_{2}$, Differential Equations, 38 (2002), 702-709.
doi: 10.1023/A:1020218926341. |
show all references
References:
[1] |
K. R. Aida-Zade and J. A. Asadova,
Study of transients in oil pipelines, Automation and Remote Control, 72 (2011), 2563-2577.
doi: 10.1134/S0005117911120113. |
[2] |
K. R. Aida-Zade and J. A. Asadova, Optimal Control of Transients in Pipelines, Palmarium Academic Publishing, Saarbrücken, 2014, (in Russian). |
[3] |
A. G. Butkovsky, Optimal Control Theory for Systems with Distributed Parameters, Moscow, 1985, (in Russian). |
[4] |
S. Ervedoza and E. Zuazua, Numerical Approximation of Exact Controls for Waves, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5808-1.![]() ![]() ![]() |
[5] |
G. F. Guliyev, The problem of the point control for a hyperbolic equation, Avtomatika i Telemexanika, 1993 (1993), 80–84 (in Russian). |
[6] |
M. Gugat, G Leugering and G. Sklyar,
$L^p-$optimal boundary control for the wave equation, SIAM Journal Control and Optimization, 44 (2005), 49-74.
doi: 10.1137/S0363012903419212. |
[7] |
V. A. Il'in,
Two-endpoint boundary control of vibrations described by a finite-energy generalized solution of the wave equation, Differential Equations, 36 (2000), 1659-1675.
doi: 10.1007/BF02757368. |
[8] |
V. A. Il'in and V. V. Tikhomirov,
The wave equation with boundary control at two ends and the problem of the complete damping of a vibration process, Differential Equations, 35 (1999), 697-708.
|
[9] |
V. A. Il'in and E. I. Moiseyev, Optimal boundary control at one end with the free second end and a distribution of the string's total energy corresponding to this control, Dokladi RAN, 400 (2005), 585–591 (in Russian). |
[10] |
K. Kunisch and D. Wachsmuth,
On time optimal control of the wave equation and its numerical realization as parametric optimization problem, SIAM Journal Control and Optimization, 51 (2013), 1232-1262.
doi: 10.1137/120877520. |
[11] |
J.-L. Lions,
Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68.
doi: 10.1137/1030001. |
[12] |
J. Loheac and E. Zuazua,
Norm saturating property of time optimal controls for wave-type equations, IFAC – Papers OnLine, 49 (2016), 37-42.
|
[13] |
F. P. Vasilyev, Optimization Methods, Moscow, Factorial Press, 2002.
![]() |
[14] |
L. N. Znamenskaya,
Control of string vibrations in the class of generalized solutions in $L_{2}$, Differential Equations, 38 (2002), 702-709.
doi: 10.1023/A:1020218926341. |



Problem No. | |||||
2,112 | 1 | 1.5 | 3, 72 | 5, 34 | |
3,168 | 1 | 1.5 | 3, 72 | 5, 34 | |
2,112 | 1 | 2 | 3, 1 | 5, 17 | |
2,112 | 1 | 3 | 3, 87 | 7, 05 | |
1,408 | 1 | 2 | 3, 1 | 5, 17 | |
2,112 | 1 | 4 | 3, 23 | 9, 38 |
Problem No. | |||||
2,112 | 1 | 1.5 | 3, 72 | 5, 34 | |
3,168 | 1 | 1.5 | 3, 72 | 5, 34 | |
2,112 | 1 | 2 | 3, 1 | 5, 17 | |
2,112 | 1 | 3 | 3, 87 | 7, 05 | |
1,408 | 1 | 2 | 3, 1 | 5, 17 | |
2,112 | 1 | 4 | 3, 23 | 9, 38 |
0, 1 | 0, 2112 | 0, 95 | 1 | 1, 05 |
0.2 | 0, 4224 | 0, 85 | 0, 85 | 0, 9 |
0, 3 | 0, 6336 | 0, 75 | 0, 75 | 0, 8 |
0, 4 | 0, 8448 | 0, 85 | 0, 85 | 0, 95 |
0, 5 | 1,056 | 1, 05 | 1, 05 | 1, 1 |
0, 6 | 1, 2672 | 0, 85 | 0, 85 | 0, 95 |
0, 7 | 1, 4784 | 0, 75 | 0, 75 | 0, 8 |
0, 8 | 1, 6896 | 0.85 | 0.85 | 0, 9 |
0, 9 | 1, 9008 | 0.95 | 0, 95 | 1, 05 |
0, 1 | 0, 2112 | 0, 95 | 1 | 1, 05 |
0.2 | 0, 4224 | 0, 85 | 0, 85 | 0, 9 |
0, 3 | 0, 6336 | 0, 75 | 0, 75 | 0, 8 |
0, 4 | 0, 8448 | 0, 85 | 0, 85 | 0, 95 |
0, 5 | 1,056 | 1, 05 | 1, 05 | 1, 1 |
0, 6 | 1, 2672 | 0, 85 | 0, 85 | 0, 95 |
0, 7 | 1, 4784 | 0, 75 | 0, 75 | 0, 8 |
0, 8 | 1, 6896 | 0.85 | 0.85 | 0, 9 |
0, 9 | 1, 9008 | 0.95 | 0, 95 | 1, 05 |
(0, 1; 0, 9) | (0, 2112; 1, 6896) | 0, 85 | 0, 87 | 0, 94 |
(0, 2; 0, 8) | (0, 4224; 1, 2672) | 0, 65 | 0, 69 | 0, 75 |
(0, 3; 0, 7) | (0, 6336; 0, 8448) | 0, 66 | 0, 69 | 0, 74 |
(0,332; 0,668) | (0, 7012; 0, 7096) | 0, 72 | 0, 75 | 0, 84 |
(0, 4; 0, 6) | (0, 8448; 0.4224) | 0, 63 | 0, 69 | 0, 76 |
(0, 1; 0, 2) | (0.2112; 0, 2112) | 0, 82 | 0, 85 | 0, 94 |
(0, 2; 0, 3) | (0, 4224; 0, 2112) | 0, 75 | 0, 78 | 0, 85 |
(0, 3; 0, 4) | (0, 6336; 0, 2112) | 0.65 | 0.68 | 0, 75 |
(0, 4; 0, 5) | (0, 8448; 0, 2112) | 0.64 | 0, 67 | 0, 79 |
(0, 6; 0, 7) | (1, 2672; 0, 2112) | 0, 66 | 0, 68 | 0, 75 |
(0, 2; 0, 6) | (0, 4224; 0, 8448) | 0, 88 | 0, 9 | 0, 92 |
(0, 1; 0, 9) | (0, 2112; 1, 6896) | 0, 85 | 0, 87 | 0, 94 |
(0, 2; 0, 8) | (0, 4224; 1, 2672) | 0, 65 | 0, 69 | 0, 75 |
(0, 3; 0, 7) | (0, 6336; 0, 8448) | 0, 66 | 0, 69 | 0, 74 |
(0,332; 0,668) | (0, 7012; 0, 7096) | 0, 72 | 0, 75 | 0, 84 |
(0, 4; 0, 6) | (0, 8448; 0.4224) | 0, 63 | 0, 69 | 0, 76 |
(0, 1; 0, 2) | (0.2112; 0, 2112) | 0, 82 | 0, 85 | 0, 94 |
(0, 2; 0, 3) | (0, 4224; 0, 2112) | 0, 75 | 0, 78 | 0, 85 |
(0, 3; 0, 4) | (0, 6336; 0, 2112) | 0.65 | 0.68 | 0, 75 |
(0, 4; 0, 5) | (0, 8448; 0, 2112) | 0.64 | 0, 67 | 0, 79 |
(0, 6; 0, 7) | (1, 2672; 0, 2112) | 0, 66 | 0, 68 | 0, 75 |
(0, 2; 0, 6) | (0, 4224; 0, 8448) | 0, 88 | 0, 9 | 0, 92 |
(0,252; 0, 5; 0,748) | (0, 5322; 0, 5237; 0, 5237) | 0, 61 | 0, 63 | 0, 65 |
(0, 1; 0, 5; 0, 9) | (0, 2112; 0, 8448; 0, 8448) | 0, 89 | 0, 91 | 0, 94 |
(0, 1; 0, 2; 0, 3) | (0, 2112; 0, 2112; 0, 2112) | 0, 82 | 0, 83 | 0, 87 |
(0, 1; 0, 3; 0, 5) | (0, 2112; 0, 4224; 0, 4224) | 0, 6 | 0, 62 | 0, 64 |
(0, 1; 0, 4; 0, 7) | (0, 2112; 0, 6336; 0, 6336) | 0, 7 | 0, 72 | 0, 76 |
(0, 4; 0, 5; 0, 6) | (0, 8448; 0, 2112; 0, 2112) | 0, 72 | 0, 74 | 0, 75 |
(0, 1; 0, 2; 0, 9) | (0, 2112; 0, 2112; 1, 4784) | 0, 83 | 0, 85 | 0, 86 |
(0, 1; 0, 2; 0, 8) | (0, 2112; 0, 2112; 1, 2672) | 0, 71 | 0, 73 | 0, 74 |
(0, 1; 0, 2; 0, 7) | (0, 2112; 0, 2112; 1,056) | 0, 74 | 0, 74 | 0, 77 |
(0,252; 0, 5; 0,748) | (0, 5322; 0, 5237; 0, 5237) | 0, 61 | 0, 63 | 0, 65 |
(0, 1; 0, 5; 0, 9) | (0, 2112; 0, 8448; 0, 8448) | 0, 89 | 0, 91 | 0, 94 |
(0, 1; 0, 2; 0, 3) | (0, 2112; 0, 2112; 0, 2112) | 0, 82 | 0, 83 | 0, 87 |
(0, 1; 0, 3; 0, 5) | (0, 2112; 0, 4224; 0, 4224) | 0, 6 | 0, 62 | 0, 64 |
(0, 1; 0, 4; 0, 7) | (0, 2112; 0, 6336; 0, 6336) | 0, 7 | 0, 72 | 0, 76 |
(0, 4; 0, 5; 0, 6) | (0, 8448; 0, 2112; 0, 2112) | 0, 72 | 0, 74 | 0, 75 |
(0, 1; 0, 2; 0, 9) | (0, 2112; 0, 2112; 1, 4784) | 0, 83 | 0, 85 | 0, 86 |
(0, 1; 0, 2; 0, 8) | (0, 2112; 0, 2112; 1, 2672) | 0, 71 | 0, 73 | 0, 74 |
(0, 1; 0, 2; 0, 7) | (0, 2112; 0, 2112; 1,056) | 0, 74 | 0, 74 | 0, 77 |
(0, 2; 0, 4; 0, 6; 0, 8) | (0, 4224; 0, 4224; 0, 4224; 0, 4224) | 0, 51 | 0, 54 | 0, 57 |
(0, 1; 0, 2; 0, 8; 0, 9) | (0, 2112; 0, 2112; 1, 2672; 0, 2112) | 0, 7 | 0, 72 | 0, 76 |
(0, 1; 0, 3; 0, 7; 0, 9) | (0, 2112; 0, 4224; 0, 8448; 0, 4224) | 0, 51 | 0, 53 | 0, 59 |
(0, 1; 0, 2; 0, 3; 0, 4) | (0, 2112; 0, 2112; 0, 2112; 0, 2112) | 0, 72 | 0, 73 | 0, 74 |
(0, 1; 0, 2; 0, 3; 0, 9) | (0, 2112; 0, 2112; 0, 2112; 1, 2672) | 0, 71 | 0, 73 | 0, 77 |
(0, 1; 0, 2; 0, 3; 0, 8) | (0, 2112; 0, 2112; 0, 2112; 1,056) | 0, 62 | 0, 63 | 0, 66 |
(0, 1; 0, 2; 0, 3; 0, 7) | (0, 2112; 0, 2112; 0, 2112; 0, 8448) | 0, 7 | 0, 72 | 0, 75 |
(0, 2; 0, 4; 0, 6; 0, 8) | (0, 4224; 0, 4224; 0, 4224; 0, 4224) | 0, 51 | 0, 54 | 0, 57 |
(0, 1; 0, 2; 0, 8; 0, 9) | (0, 2112; 0, 2112; 1, 2672; 0, 2112) | 0, 7 | 0, 72 | 0, 76 |
(0, 1; 0, 3; 0, 7; 0, 9) | (0, 2112; 0, 4224; 0, 8448; 0, 4224) | 0, 51 | 0, 53 | 0, 59 |
(0, 1; 0, 2; 0, 3; 0, 4) | (0, 2112; 0, 2112; 0, 2112; 0, 2112) | 0, 72 | 0, 73 | 0, 74 |
(0, 1; 0, 2; 0, 3; 0, 9) | (0, 2112; 0, 2112; 0, 2112; 1, 2672) | 0, 71 | 0, 73 | 0, 77 |
(0, 1; 0, 2; 0, 3; 0, 8) | (0, 2112; 0, 2112; 0, 2112; 1,056) | 0, 62 | 0, 63 | 0, 66 |
(0, 1; 0, 2; 0, 3; 0, 7) | (0, 2112; 0, 2112; 0, 2112; 0, 8448) | 0, 7 | 0, 72 | 0, 75 |
[1] |
J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341 |
[2] |
Nicolás Matte Bon. Topological full groups of minimal subshifts with subgroups of intermediate growth. Journal of Modern Dynamics, 2015, 9: 67-80. doi: 10.3934/jmd.2015.9.67 |
[3] |
Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021076 |
[4] |
Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571 |
[5] |
Karel Kadlec, Bohdan Maslowski. Ergodic boundary and point control for linear stochastic PDEs driven by a cylindrical Lévy process. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 4039-4055. doi: 10.3934/dcdsb.2020137 |
[6] |
Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133 |
[7] |
Yuefen Chen, Yuanguo Zhu. Indefinite LQ optimal control with process state inequality constraints for discrete-time uncertain systems. Journal of Industrial and Management Optimization, 2018, 14 (3) : 913-930. doi: 10.3934/jimo.2017082 |
[8] |
Karine Beauchard, Morgan Morancey. Local controllability of 1D Schrödinger equations with bilinear control and minimal time. Mathematical Control and Related Fields, 2014, 4 (2) : 125-160. doi: 10.3934/mcrf.2014.4.125 |
[9] |
Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139 |
[10] |
Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141 |
[11] |
Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4461-4476. doi: 10.3934/dcds.2021043 |
[12] |
R.S. Dahiya, A. Zafer. Oscillatory theorems of n-th order functional differential equations. Conference Publications, 2001, 2001 (Special) : 435-443. doi: 10.3934/proc.2001.2001.435 |
[13] |
Roberto de A. Capistrano-Filho, Vilmos Komornik, Ademir F. Pazoto. Pointwise control of the linearized Gear-Grimshaw system. Evolution Equations and Control Theory, 2020, 9 (3) : 693-719. doi: 10.3934/eect.2020029 |
[14] |
Shijin Deng, Weike Wang, Shih-Hsien Yu. Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary. Networks and Heterogeneous Media, 2007, 2 (3) : 383-395. doi: 10.3934/nhm.2007.2.383 |
[15] |
Peter V. Gordon, Cyrill B. Muratov. Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source. Networks and Heterogeneous Media, 2012, 7 (4) : 767-780. doi: 10.3934/nhm.2012.7.767 |
[16] |
Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations and Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032 |
[17] |
Mourad Azi, Mohand Ouamer Bibi. Optimal control of a dynamical system with intermediate phase constraints and applications in cash management. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 279-291. doi: 10.3934/naco.2021005 |
[18] |
María Isabel Cortez, Samuel Petite. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2891-2901. doi: 10.3934/dcds.2020153 |
[19] |
Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial and Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1 |
[20] |
Volker Rehbock, Iztok Livk. Optimal control of a batch crystallization process. Journal of Industrial and Management Optimization, 2007, 3 (3) : 585-596. doi: 10.3934/jimo.2007.3.585 |
2020 Impact Factor: 1.801
Tools
Metrics
Other articles
by authors
[Back to Top]