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April 2018, 14(2): 473-496. doi: 10.3934/jimo.2017056

## Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method

 Mathematics Department, Faculty of Science, Assiut University, Assiut, 71516, Egypt

* Corresponding author: Kareem T. Elgindy

Received  June 2016 Revised  October 2016 Published  June 2017

In this paper, we introduce a novel fully exponentially convergent direct integral pseudospectral method for the numerical solution of optimal control problems governed by a parabolic distributed parameter system. The proposed method combines the superior advantages possessed by the family of pseudospectral methods with the well-conditioning of integral operators through the use of the integral formulation of the distributed parameter system equations, and the spectral accuracy provided by the latest technology of Gegenbauer barycentric quadratures in a fashion that allows us to take advantage of the strengths of these three methodologies. A rigorous error analysis of the method is presented, and a numerical test example is given to show the accuracy and efficiency of the proposed integral pseudospectral method.

Citation: Kareem T. Elgindy. Optimal control of a parabolic distributed parameter system using a fully exponentially convergent barycentric shifted gegenbauer integral pseudospectral method. Journal of Industrial & Management Optimization, 2018, 14 (2) : 473-496. doi: 10.3934/jimo.2017056
##### References:
 [1] H. Alzer, Sharp upper and lower bounds for the gamma function, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 139 (2009), 709-718. doi: 10.1017/S0308210508000644. [2] J. A. Burns, J. Borggaard, E. Cliff and L. Zietsman, An optimal control approach to sensor/actuator placement for optimal control of high performance buildings, In: International High Performance Buildings Conference, 2012. [3] S. P. Chakrabarty and F. B. Hanson, Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method, Mathematical Biosciences, 219 (2009), 129-141. doi: 10.1016/j.mbs.2009.03.005. [4] R.-Y. Chang and S.-Y. Yang, Solution of two-point-boundary-value problems by generalized orthogonal polynomials and application to optimal control of lumped and distributed parameter systems, International Journal of Control, 43 (1986), 1785-1802. doi: 10.1080/00207178608933572. [5] C. -P. Chen and F. Qi, The best lower and upper bounds of harmonic sequence, RGMIA research report collection, 2003. [6] B. Cushman-Roisin and J. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, 2nd Edition. Vol. 101 of International Geophysics Series. Academic Press, 2011. [7] K. Elgindy, Gegenbauer collocation integration methods: Advances in computational optimal control theory, Bull. Aust. Math. Soc., 89 (2014), 168-170. doi: 10.1017/S0004972713001044. [8] K. T. Elgindy, High-order adaptive Gegenbauer integral spectral element method for solving non-linear optimal control problems, Optimization, (2017), 811-836. doi: 10.1080/02331934.2017.1298597. [9] K. T. Elgindy, High-order numerical solution of second-order one-dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method, Numerical Methods for Partial Differential Equations, 32 (2016), 307-349. doi: 10.1002/num.21996. [10] K. T. Elgindy, High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures, Applied Numerical Mathematics, 113 (2017), 1-25. doi: 10.1016/j.apnum.2016.10.014. [11] K. T. Elgindy and K. A. Smith-Miles, Fast, accurate, and small-scale direct trajectory optimization using a Gegenbauer transcription method, Journal of Computational and Applied Mathematics, 251 (2013), 93-116. doi: 10.1016/j.cam.2013.03.032. [12] K. T. Elgindy and K. A. Smith-Miles, Optimal Gegenbauer quadrature over arbitrary integration nodes, Journal of Computational and Applied Mathematics, 242 (2013), 82-106. doi: 10.1016/j.cam.2012.10.020. [13] D. R. Gardner, S. A. Trogdon and R. W. Douglass, A modified tau spectral method that eliminates spurious eigenvalues, Journal of Computational Physics, 80 (1989), 137-167. [14] I.-R. Horng and J.-H. Chou, Application of shifted Chebyshev series to the optimal control of linear distributed-parameter systems, International Journal of Control, 42 (1985), 233-241. doi: 10.1080/00207178508933359. [15] G. Mahapatra, Solution of optimal control problem of linear diffusion equations via Walsh functions, IEEE Transactions on Automatic Control, 25 (1980), 319-321. doi: 10.1109/TAC.1980.1102278. [16] R. Padhi and S. Balakrishnan, Proper orthogonal decomposition based optimal neurocontrol synthesis of a chemical reactor process using approximate dynamic programming, Neural Networks, 16 (2003), 719-728, Advances in Neural Networks Research: IJCNN'03. [17] M. A. Patterson and A. V. Rao, GPOPS-Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming ACM Transactions on Mathematical Software (TOMS) 41 (2014), Art. 1, 37 pp. doi: 10.1145/2558904. [18] J. Rad, S. Kazem and K. Parand, Optimal control of a parabolic distributed parameter system via radial basis functions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 2559-2567. doi: 10.1016/j.cnsns.2013.01.007. [19] W. F. Ramirez, Application of Optimal Control Theory to Enhanced Oil Recovery, Vol. 21. Elsevier, 1987. [20] A. P. Sage and C. C. White, Optimum Systems Control, Prentice Hall, 1977. [21] L. N. Trefethen, Spectral Methods in MATLAB SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. [22] M.-L. Wang and R.-Y. Chang, Optimal control of linear distributed parameter systems by shifted Legendre polynomial functions, Journal of Dynamic Systems, Measurement, and Control, 105 (1983), 222-226. [23] J.-M. Zhu and Y.-Z. Lu, Application of single-step method of block-pulse functions to the optimal control of linear distributed-parameter systems, International Journal of Systems Science, 19 (1988), 2459-2472. doi: 10.1080/00207728808547126.

show all references

##### References:
 [1] H. Alzer, Sharp upper and lower bounds for the gamma function, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 139 (2009), 709-718. doi: 10.1017/S0308210508000644. [2] J. A. Burns, J. Borggaard, E. Cliff and L. Zietsman, An optimal control approach to sensor/actuator placement for optimal control of high performance buildings, In: International High Performance Buildings Conference, 2012. [3] S. P. Chakrabarty and F. B. Hanson, Distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method, Mathematical Biosciences, 219 (2009), 129-141. doi: 10.1016/j.mbs.2009.03.005. [4] R.-Y. Chang and S.-Y. Yang, Solution of two-point-boundary-value problems by generalized orthogonal polynomials and application to optimal control of lumped and distributed parameter systems, International Journal of Control, 43 (1986), 1785-1802. doi: 10.1080/00207178608933572. [5] C. -P. Chen and F. Qi, The best lower and upper bounds of harmonic sequence, RGMIA research report collection, 2003. [6] B. Cushman-Roisin and J. Beckers, Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects, 2nd Edition. Vol. 101 of International Geophysics Series. Academic Press, 2011. [7] K. Elgindy, Gegenbauer collocation integration methods: Advances in computational optimal control theory, Bull. Aust. Math. Soc., 89 (2014), 168-170. doi: 10.1017/S0004972713001044. [8] K. T. Elgindy, High-order adaptive Gegenbauer integral spectral element method for solving non-linear optimal control problems, Optimization, (2017), 811-836. doi: 10.1080/02331934.2017.1298597. [9] K. T. Elgindy, High-order numerical solution of second-order one-dimensional hyperbolic telegraph equation using a shifted Gegenbauer pseudospectral method, Numerical Methods for Partial Differential Equations, 32 (2016), 307-349. doi: 10.1002/num.21996. [10] K. T. Elgindy, High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures, Applied Numerical Mathematics, 113 (2017), 1-25. doi: 10.1016/j.apnum.2016.10.014. [11] K. T. Elgindy and K. A. Smith-Miles, Fast, accurate, and small-scale direct trajectory optimization using a Gegenbauer transcription method, Journal of Computational and Applied Mathematics, 251 (2013), 93-116. doi: 10.1016/j.cam.2013.03.032. [12] K. T. Elgindy and K. A. Smith-Miles, Optimal Gegenbauer quadrature over arbitrary integration nodes, Journal of Computational and Applied Mathematics, 242 (2013), 82-106. doi: 10.1016/j.cam.2012.10.020. [13] D. R. Gardner, S. A. Trogdon and R. W. Douglass, A modified tau spectral method that eliminates spurious eigenvalues, Journal of Computational Physics, 80 (1989), 137-167. [14] I.-R. Horng and J.-H. Chou, Application of shifted Chebyshev series to the optimal control of linear distributed-parameter systems, International Journal of Control, 42 (1985), 233-241. doi: 10.1080/00207178508933359. [15] G. Mahapatra, Solution of optimal control problem of linear diffusion equations via Walsh functions, IEEE Transactions on Automatic Control, 25 (1980), 319-321. doi: 10.1109/TAC.1980.1102278. [16] R. Padhi and S. Balakrishnan, Proper orthogonal decomposition based optimal neurocontrol synthesis of a chemical reactor process using approximate dynamic programming, Neural Networks, 16 (2003), 719-728, Advances in Neural Networks Research: IJCNN'03. [17] M. A. Patterson and A. V. Rao, GPOPS-Ⅱ: A MATLAB software for solving multiple-phase optimal control problems using hp-adaptive Gaussian quadrature collocation methods and sparse nonlinear programming ACM Transactions on Mathematical Software (TOMS) 41 (2014), Art. 1, 37 pp. doi: 10.1145/2558904. [18] J. Rad, S. Kazem and K. Parand, Optimal control of a parabolic distributed parameter system via radial basis functions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 2559-2567. doi: 10.1016/j.cnsns.2013.01.007. [19] W. F. Ramirez, Application of Optimal Control Theory to Enhanced Oil Recovery, Vol. 21. Elsevier, 1987. [20] A. P. Sage and C. C. White, Optimum Systems Control, Prentice Hall, 1977. [21] L. N. Trefethen, Spectral Methods in MATLAB SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719598. [22] M.-L. Wang and R.-Y. Chang, Optimal control of linear distributed parameter systems by shifted Legendre polynomial functions, Journal of Dynamic Systems, Measurement, and Control, 105 (1983), 222-226. [23] J.-M. Zhu and Y.-Z. Lu, Application of single-step method of block-pulse functions to the optimal control of linear distributed-parameter systems, International Journal of Systems Science, 19 (1988), 2459-2472. doi: 10.1080/00207728808547126.
The figure shows the plots of the approximate optimal cost functional $J_{N,N}^*$ (upper left), the feasibility of the optimal solution $\boldsymbol{Z}^*$ as reported by the solver (upper right), the maximum error in the initial condition (3), ${\psi}_1$, at the $101$ linearly spaced nodes $(x_i, y_i)$ in the $y$-and $t$-directions from $0$ to $4$, and $0$ to $1$, respectively in a semi-logarithmic scale (lower left), and the maximum error in the boundary condition (11), ${\psi}_2$ (lower right). All of the plots were generated using the same $101$ points $(x_i, y_i)$
The figure shows the state and control profiles on $D_{4,1}^2$ using $N = 4, 12$ and $\alpha = -0.2$
The figure shows the state and control profiles at the midpoint $y = 2$ using $N = 4, 12$ and $\alpha = -0.2$
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