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doi: 10.3934/jimo.2021166
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Numerical solution to optimal control problems of oscillatory processes

Kamil Aida-Zade and  Jamila Asadova ,

The Institute of Control Systems, Azerbaijan National Academy of Sciences, 9, B. Vahabzadeh str., AZ1141, Baku, Azerbaijan

* Corresponding author: asadova.jamilya64@gmail.com

Received  October 2020 Revised  July 2021 Early access September 2021

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.

Keywords: Boundary and intermediate concentrated (lumped) control actions, pointwise source, gradient of target functional, minimal settling time of oscillatory process.
Mathematics Subject Classification: Primary: 49J15, 49J35.
Citation: Kamil Aida-Zade, Jamila Asadova. Numerical solution to optimal control problems of oscillatory processes. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021166
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.

Figure 1.  Plots of optimal piecewise continuous control of the oscillatory process for Problem $ III $: a) control at the left end; b) control at the right end; c) intermediate pointwise control located at the point $ \bar{x}_{1} = 0,5 $
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Figure 2.  Plots of optimal piecewise-continuous control actions by intermediate concentrated sources in Problem $ VI $ under the following arrangement: a) $ \bar{x}_{1} = 0,4 $; b) $ \bar{x}_{2} = 0,8 $
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Figure 3.  Plots of optimal control actions of four ILS for problem $ III $ at the following arrangement of ILS: a) $ \bar{x}_{1} = 0,2 $; b) $ \bar{x}_{2} = 0,4 $; c) $ \bar{x}_{3} = 0,6 $; d) $ \bar{x}_{4} = 0,8 $
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Table 1.  Parameters of six model problems of oscillatory processes control with boundary and intermediate lumped controls
Problem No. $ \alpha $ $ \hat{{\varphi }}_{10} \, $ $ \, \hat{{\varphi }}_{1T} \, $ $ \eta _{0} (0) $ $ \eta _{T} (0) $
$ I $ 2,112 1 1.5 3, 72 5, 34
$ II $ 3,168 1 1.5 3, 72 5, 34
$ III $ 2,112 1 2 3, 1 5, 17
$ IV $ 2,112 1 3 3, 87 7, 05
$ V $ 1,408 1 2 3, 1 5, 17
$ VI $ 2,112 1 4 3, 23 9, 38
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Problem No. $ \alpha $ $ \hat{{\varphi }}_{10} \, $ $ \, \hat{{\varphi }}_{1T} \, $ $ \eta _{0} (0) $ $ \eta _{T} (0) $
$ I $ 2,112 1 1.5 3, 72 5, 34
$ II $ 3,168 1 1.5 3, 72 5, 34
$ III $ 2,112 1 2 3, 1 5, 17
$ IV $ 2,112 1 3 3, 87 7, 05
$ V $ 1,408 1 2 3, 1 5, 17
$ VI $ 2,112 1 4 3, 23 9, 38
Table 2.  Dependence of the optimal settling time for problems $ I $, $ III $, and $ VI $ on the arrangement of the single intermediate lumped control action
$ T_{opt}^{} $
$ \bar{x}_{1} $ $ f_{0}^{1} (0) $ $ I $ problem $ III $ problem $ VI $ problem
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
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$ T_{opt}^{} $
$ \bar{x}_{1} $ $ f_{0}^{1} (0) $ $ I $ problem $ III $ problem $ VI $ problem
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
Table 3.  The optimal settling time of oscillatory processes in the problems $ I $, $ III $ and $ VI $ in the presence of two ILS
$ T_{opt}^{} $
$ (\bar{x}_{1} ,\, \, \bar{x}_{2} ) $ $ (f_{0}^{1} (0),\, \, f_{0}^{2} (0)) $ $ I $ problem $ III $ problem $ VI $ problem
(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
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$ T_{opt}^{} $
$ (\bar{x}_{1} ,\, \, \bar{x}_{2} ) $ $ (f_{0}^{1} (0),\, \, f_{0}^{2} (0)) $ $ I $ problem $ III $ problem $ VI $ problem
(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
Table 4.  The optimal settling time in the presence of three ILS
$ T_{opt}^{} $
$ (\bar{x}_{1} ,\, \, \bar{x}_{2} ,\bar{x}_{3} ) $ $ (f_{0}^{1} (0),\, \, f_{0}^{2} (0),\, \, f_{0}^{3} (0)) $ $ I $ prob. $ III $ prob. $ VI $ prob.
(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
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$ T_{opt}^{} $
$ (\bar{x}_{1} ,\, \, \bar{x}_{2} ,\bar{x}_{3} ) $ $ (f_{0}^{1} (0),\, \, f_{0}^{2} (0),\, \, f_{0}^{3} (0)) $ $ I $ prob. $ III $ prob. $ VI $ prob.
(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
Table 5.  The minimal settling time of oscillating processes in the case of four ILS
$ T_{opt}^{} $
$ (\bar{x}_{1} ,\, \, \bar{x}_{2} ,\bar{x}_{3} ,\bar{x}_{4} ) $ $ (f_{0}^{1} (0),\, \, f_{0}^{2} (0),\, \, f_{0}^{3} (0),\, \, f_{0}^{4} (0)) $ $ I $ prob. $ III $ prob. $ VI $ prob.
(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
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$ T_{opt}^{} $
$ (\bar{x}_{1} ,\, \, \bar{x}_{2} ,\bar{x}_{3} ,\bar{x}_{4} ) $ $ (f_{0}^{1} (0),\, \, f_{0}^{2} (0),\, \, f_{0}^{3} (0),\, \, f_{0}^{4} (0)) $ $ I $ prob. $ III $ prob. $ VI $ prob.
(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
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