# American Institute of Mathematical Sciences

March  2019, 9(1): 101-112. doi: 10.3934/naco.2019008

## Solving optimal control problem using Hermite wavelet

 1 Department of Mathematics, Faculty of Mathematical Science and Statistics, University of Birjand, Birjand, Iran 2 Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

Received  May 2018 Revised  July 2018 Published  October 2018

In this paper, we derive the operational matrices of integration, derivative and production of Hermite wavelets and use a direct numerical method based on Hermite wavelet, for solving optimal control problems. The properties of Hermite polynomials are used for finding these matrices. First, we approximate the state and control variables by Hermite wavelets basis; then, the operational matrices is used to transfer the given problem into a linear system of algebraic equations. In fact, operational matrices of Hermite wavelet are employed to achieve a linear algebraic equation, in place of the dynamical system in terms of the unknown coefficients. The solution of this system gives us the solution of the original problem. Numerical examples with time varying and time invariant coefficient are given to demonstrate the applicability of these matrices.

Citation: Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati. Solving optimal control problem using Hermite wavelet. Numerical Algebra, Control & Optimization, 2019, 9 (1) : 101-112. doi: 10.3934/naco.2019008
##### References:

show all references

##### References:
Approximate (linestyle is -) and exact (linestyle is :) solution for x(t)
Approximate (linestyle is -) and exact (linestyle :) solution for u(t)
Approximate (linestyle -) and exact (linestyle :) solution for x(t)
Approximate (linestyle -) and exact (linestyle :) solution for u(t)
Comparison of the optimal values of J (Example 4.1)
 Exact value of J Kafash et al. [17] Saberi Nik et al. [24] Approximated solution via HW 0.1929092981 0.192914197 0.193415452 0.1929092981
 Exact value of J Kafash et al. [17] Saberi Nik et al. [24] Approximated solution via HW 0.1929092981 0.192914197 0.193415452 0.1929092981
The exact and approximated values of x(t) and u(t) for Example 4.1
 x(t) u(t) Time Approximated solution via HW Exact solution Approximated solution via HW Exact solution 0.0 1.0000 1.0000 -0.3859 -0.3858 0.2 0.7594 0.7594 -0.2769 -0.2769 0.4 0.5799 0.5799 -0.1902 -0.1902 0.6 0.4472 0.4472 -0.1189 -0.1189 0.8 0.3505 0.3505 -0.0571 -0.0571 1 0.2820 0.2820 0.0000 0.0000
 x(t) u(t) Time Approximated solution via HW Exact solution Approximated solution via HW Exact solution 0.0 1.0000 1.0000 -0.3859 -0.3858 0.2 0.7594 0.7594 -0.2769 -0.2769 0.4 0.5799 0.5799 -0.1902 -0.1902 0.6 0.4472 0.4472 -0.1189 -0.1189 0.8 0.3505 0.3505 -0.0571 -0.0571 1 0.2820 0.2820 0.0000 0.0000
Comparison of the optimal values of J (Example 4.2)
 Exact solution [19] Hashemi Mehne and Hashemi Borzabadi[12] Approximated solution via HW 6.1586 6.1748 6.1495
 Exact solution [19] Hashemi Mehne and Hashemi Borzabadi[12] Approximated solution via HW 6.1586 6.1748 6.1495
The exact and approximated values of x(t) and u(t) for Example 4.2
 x(t) u(t) Time Approximated solution via HW Exact solution Approximated solution via HW Exact solution 0.0 0.0000 0.0000 1.1028 1.1029 0.2 0.2264 0.2265 1.4185 1.4188 0.4 0.4896 04897 1.9646 1.9648 0.6 0.8321 0.8324 2.8293 2.8293 0.8 1.3097 1.3100 4.1515 4.1526 1 2.0000 2.0000 6.1300 6.1493
 x(t) u(t) Time Approximated solution via HW Exact solution Approximated solution via HW Exact solution 0.0 0.0000 0.0000 1.1028 1.1029 0.2 0.2264 0.2265 1.4185 1.4188 0.4 0.4896 04897 1.9646 1.9648 0.6 0.8321 0.8324 2.8293 2.8293 0.8 1.3097 1.3100 4.1515 4.1526 1 2.0000 2.0000 6.1300 6.1493
Comparison between different methods for optimal value of J (Example 4.3)
 Exact value Hsieh [13] Jaddu [16] Majdalawi [18] Our proposed method 0.06936094 0.0702 0.0693689 0.0693668896 0.0693688962
 Exact value Hsieh [13] Jaddu [16] Majdalawi [18] Our proposed method 0.06936094 0.0702 0.0693689 0.0693668896 0.0693688962
The approximate and exact values of J (Example 4.4)
 Exact value Approximated value via HW Error 0.16666666666 0.1666666666 0.4×10−14
 Exact value Approximated value via HW Error 0.16666666666 0.1666666666 0.4×10−14
Comparison between different methods for optimal value of J (Example 4.5)
 Elnagar [7] Jaddu [16] Abu Haya [1] Rafiei [20] Our method via HW 0.48427022 0.4842676003 0.4842678105 0.4842677529 0.4842676962
 Elnagar [7] Jaddu [16] Abu Haya [1] Rafiei [20] Our method via HW 0.48427022 0.4842676003 0.4842678105 0.4842677529 0.4842676962
 [1] Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487 [2] Bin Han, Qun Mo. Analysis of optimal bivariate symmetric refinable Hermite interpolants. Communications on Pure & Applied Analysis, 2007, 6 (3) : 689-718. doi: 10.3934/cpaa.2007.6.689 [3] Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255 [4] Meng Zhao, Aijie Cheng, Hong Wang. A preconditioned fast Hermite finite element method for space-fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3529-3545. doi: 10.3934/dcdsb.2017178 [5] R. Wong, L. Zhang. Global asymptotics of Hermite polynomials via Riemann-Hilbert approach. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 661-682. doi: 10.3934/dcdsb.2007.7.661 [6] Shu-Lin Lyu. On the Hermite--Hadamard inequality for convex functions of two variables. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 1-8. doi: 10.3934/naco.2014.4.1 [7] Irene I. Bouw, Sabine Kampf. Syndrome decoding for Hermite codes with a Sugiyama-type algorithm. Advances in Mathematics of Communications, 2012, 6 (4) : 419-442. doi: 10.3934/amc.2012.6.419 [8] Tijana Levajković, Hermann Mena, Amjad Tuffaha. The stochastic linear quadratic optimal control problem in Hilbert spaces: A polynomial chaos approach. Evolution Equations & Control Theory, 2016, 5 (1) : 105-134. doi: 10.3934/eect.2016.5.105 [9] Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367 [10] Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967 [11] Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist, Amit K. Sanyal. Embedded geodesic problems and optimal control for matrix Lie groups. Journal of Geometric Mechanics, 2011, 3 (2) : 197-223. doi: 10.3934/jgm.2011.3.197 [12] S. S. Dragomir, I. Gomm. Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 271-278. doi: 10.3934/naco.2012.2.271 [13] Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275 [14] Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028 [15] N. U. Ahmed. Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach. Evolution Equations & Control Theory, 2012, 1 (2) : 235-250. doi: 10.3934/eect.2012.1.235 [16] Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311 [17] Tobias Breiten, Karl Kunisch, Laurent Pfeiffer. Numerical study of polynomial feedback laws for a bilinear control problem. Mathematical Control & Related Fields, 2018, 8 (3&4) : 557-582. doi: 10.3934/mcrf.2018023 [18] Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027 [19] Zhiming Chen, Shaofeng Fang, Guanghui Huang. A direct imaging method for the half-space inverse scattering problem with phaseless data. Inverse Problems & Imaging, 2017, 11 (5) : 901-916. doi: 10.3934/ipi.2017042 [20] Karl Kunisch, Markus Müller. Uniform convergence of the POD method and applications to optimal control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4477-4501. doi: 10.3934/dcds.2015.35.4477

Impact Factor: