Trajectory Controllability of Nonlinear Integro-Differential System—An Analytical and a Numerical Estimations

Affiliation(s)

Mallory Hall, Virginia Military Institute, Lexington, USA.

Roop Hall, James Madison University, Harrisonburg, USA.

Mallory Hall, Virginia Military Institute, Lexington, USA.

Roop Hall, James Madison University, Harrisonburg, USA.

ABSTRACT

A stronger concept of complete (exact) controllability which we call Trajectory Controllability is introduced in this paper. We study the Trajectory Controllability of an abstract nonlinear integro-differential system in the finite and infinite dimensional space setting. We will then discuss how approximations to these problems can be found computationally using finite difference methods and optimization. Examples will be presented in one, two and three dimensions.

A stronger concept of complete (exact) controllability which we call Trajectory Controllability is introduced in this paper. We study the Trajectory Controllability of an abstract nonlinear integro-differential system in the finite and infinite dimensional space setting. We will then discuss how approximations to these problems can be found computationally using finite difference methods and optimization. Examples will be presented in one, two and three dimensions.

Cite this paper

D. Chalishajar, H. Chalishajar and J. David, "Trajectory Controllability of Nonlinear Integro-Differential System—An Analytical and a Numerical Estimations,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1729-1738. doi: 10.4236/am.2012.311239.

D. Chalishajar, H. Chalishajar and J. David, "Trajectory Controllability of Nonlinear Integro-Differential System—An Analytical and a Numerical Estimations,"

References

[1] E. J. Davison and E. C. Kunze, “Controllability of Inte-gro-Differential Systems in Banach Space,” SIAM Journal on Control and Optimization, Vol. 8, No. 1, 1970, pp. 489-497.

[2] R. K. George, “Approximate Controllability of Nonautonomous Semilinear Systems,” Nonlinear Analysis—TMA, Vol. 24, No. 1, 1995, pp. 1377-1393.

[3] R. K. George, D. N. Chalishajar and A. K. Nandakumaran, “Exact Controllability of Generalised Hammerstein Type Equations,” Electronic Journal of Differential Equation, Vol. 142, No. 1, 2006, pp. 1-15.

[4] J. L. Lions, “Exact Controllability, Stabilization and Perturbations for Distributed Systems,” SIAM Review, Vol. 30, No. 1, 1998, pp. 1-68. doi:10.1137/1030001

[5] D. N. Chalishajar, “Control-lability of Nonlinear IntegroDifferential Third Order Dispersion Equation,” Journal of Mathematical Analysis and Applications, Vol. 348, No. 1, 2008, pp. 480-486. doi:10.1016/j.jmaa.2008.07.047

[6] R. K. George, D. N. Chalishajar and A. K. Nandakunaran, “Exact Controlla-bility of Nonlinear Third Order Dispersion Equation,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007, pp. 1028-1044. doi:10.1016/j.jmaa.2006.10.084

[7] D. N. Chalishajar and F. S. Acharya, “Controllability of Neutral Impulsive Differential Inclusion with Nonlocal Conditions,” Applied Mathematics, Vol. 2, No. 1, 2011, pp. 1486-1496. doi:10.4236/am.2011.212211

[8] A. K. Nandakumaran and R. K. George, “Approximate Controllability of Non-autonomous Semilinear Systems,” Revista Mathematica UCM, Vol. 8, No. 1, 1995, pp. 181-196.

[9] S. Micu and E. Zuazua, “On the Null Controllability of the Heat Equation in Unbounded Domains,” Bulletin des Sciences Mathématiques, Vol. 129, No. 2, 2005, pp. 175-185. doi:10.1016/j.bulsci.2004.04.003

[10] F. Cardetti and M. Gordina, “A Note on Local Controllability on Li Groups,” System and Control Letters, Vol. 52, No. 12, 1990, pp. 979-987.

[11] J. Klamka, “Constrained Controllability of Semilinear Systems with Delayed Controls,” Bulletin of the Polish Academy of Sciences, Vol. 56, No. 4, 2008, pp. 333-337.

[12] J. Klamka, “Constrained Controllability of Semilinear Systems with Delay,” Nonlinear Dynamics, Vol. 56, No. 1-2, 2009, pp. 169-177. doi:10.1007/s11071-008-9389-4

[13] P. Linz, “A Survey of Methods for the Solution of Volterra Integral Equations of the First Kind in the Applications and Numerical Solution of Integral Equations,” Nonlinear Analysis—TMA, 1980, pp. 189-194.

[14] K. Deimling, “Nonlinear Volterra Integral Equations of the First Kind,” Nonlinear Analysis—TMA, Vol. 25, No. 1, 1995, pp. 951-957.

[15] K. Deimling, “Multivalued Differential Equations,” Walter De Gruyter, The Netherlands, 1992. doi:10.1515/9783110874228

[16] D. N. Chalishajar, “Controllability of Damped SecondOrder Initial Value Problem for a Class of Differential Inclusions with Non-local Conditions on Noncompact Intervals,” Nonlinear Functional Analysis and Applications (Korea), Vol. 14, No. 1, 2009, pp. 25-44.

[17] M. C. Joshi and R. K. Bose, “Some Topics in Nonlinear Functional Analysis,” Hasted Press, New York, 1985.

[18] M. D. Gunzburger, “Pers-pectives in Flow Control and Optimization. Advances in Design and Control,” SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 2003.

[19] E. Polak, “Computational Methods in Optimization,” Academic Press, Cambridge, 1971.

[20] J. A. David, H. T. Tran and H. T. Banks, “HIV Model Analysis and Estimation Im-plementation under Optimal Control Based Treatment Strategies,” International Journal of Pure and Applied Mathematics, Vol. 57, No. 3, 2009, pp. 357-392.

[21] C. T. Kelley, “Iterative Methods for Optimization,” SIAM: Society for Industrial and Applied Mathematics, Phila-delphia, 1999.

[22] L. F. Shampine and M. W. Reichelt, “The MATLAB ODE Suite,” SIAM Journal on Scientific Computing, Vol. 18, No. 1, 1997, pp. 1-22. doi:10.1137/S1064827594276424

[23] L. F. Reichelt, M. W. Shampine and J. A. Kierzenka. “Solving Index-1 DAEs in MATLAB and Simulink,” SIAM Review, Vol. 41, No. 3, 1999, pp. 538-552. doi:10.1137/S003614459933425X

[24] R. D. Skeel and M. Berzins, “A Method for the Spatial Discretization of Parabolic Equations in One Space Variable,” SIAM Journal on Scientific and Statistical Computing, Vol. 11, No. 1, 1990, pp. 1-32. doi:10.1137/0911001

[25] R. L. Burden and J. D. Faires, “Numberical Analysis,” Brookes/Cole Publisher, Salt Lake City, 2011.

[26] C. T. Kelley, “Iterative Methods for Linear and Nonlinear Equations,” SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 1995. doi:10.1137/1.9781611970944

[1] E. J. Davison and E. C. Kunze, “Controllability of Inte-gro-Differential Systems in Banach Space,” SIAM Journal on Control and Optimization, Vol. 8, No. 1, 1970, pp. 489-497.

[2] R. K. George, “Approximate Controllability of Nonautonomous Semilinear Systems,” Nonlinear Analysis—TMA, Vol. 24, No. 1, 1995, pp. 1377-1393.

[3] R. K. George, D. N. Chalishajar and A. K. Nandakumaran, “Exact Controllability of Generalised Hammerstein Type Equations,” Electronic Journal of Differential Equation, Vol. 142, No. 1, 2006, pp. 1-15.

[4] J. L. Lions, “Exact Controllability, Stabilization and Perturbations for Distributed Systems,” SIAM Review, Vol. 30, No. 1, 1998, pp. 1-68. doi:10.1137/1030001

[5] D. N. Chalishajar, “Control-lability of Nonlinear IntegroDifferential Third Order Dispersion Equation,” Journal of Mathematical Analysis and Applications, Vol. 348, No. 1, 2008, pp. 480-486. doi:10.1016/j.jmaa.2008.07.047

[6] R. K. George, D. N. Chalishajar and A. K. Nandakunaran, “Exact Controlla-bility of Nonlinear Third Order Dispersion Equation,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007, pp. 1028-1044. doi:10.1016/j.jmaa.2006.10.084

[7] D. N. Chalishajar and F. S. Acharya, “Controllability of Neutral Impulsive Differential Inclusion with Nonlocal Conditions,” Applied Mathematics, Vol. 2, No. 1, 2011, pp. 1486-1496. doi:10.4236/am.2011.212211

[8] A. K. Nandakumaran and R. K. George, “Approximate Controllability of Non-autonomous Semilinear Systems,” Revista Mathematica UCM, Vol. 8, No. 1, 1995, pp. 181-196.

[9] S. Micu and E. Zuazua, “On the Null Controllability of the Heat Equation in Unbounded Domains,” Bulletin des Sciences Mathématiques, Vol. 129, No. 2, 2005, pp. 175-185. doi:10.1016/j.bulsci.2004.04.003

[10] F. Cardetti and M. Gordina, “A Note on Local Controllability on Li Groups,” System and Control Letters, Vol. 52, No. 12, 1990, pp. 979-987.

[11] J. Klamka, “Constrained Controllability of Semilinear Systems with Delayed Controls,” Bulletin of the Polish Academy of Sciences, Vol. 56, No. 4, 2008, pp. 333-337.

[12] J. Klamka, “Constrained Controllability of Semilinear Systems with Delay,” Nonlinear Dynamics, Vol. 56, No. 1-2, 2009, pp. 169-177. doi:10.1007/s11071-008-9389-4

[13] P. Linz, “A Survey of Methods for the Solution of Volterra Integral Equations of the First Kind in the Applications and Numerical Solution of Integral Equations,” Nonlinear Analysis—TMA, 1980, pp. 189-194.

[14] K. Deimling, “Nonlinear Volterra Integral Equations of the First Kind,” Nonlinear Analysis—TMA, Vol. 25, No. 1, 1995, pp. 951-957.

[15] K. Deimling, “Multivalued Differential Equations,” Walter De Gruyter, The Netherlands, 1992. doi:10.1515/9783110874228

[16] D. N. Chalishajar, “Controllability of Damped SecondOrder Initial Value Problem for a Class of Differential Inclusions with Non-local Conditions on Noncompact Intervals,” Nonlinear Functional Analysis and Applications (Korea), Vol. 14, No. 1, 2009, pp. 25-44.

[17] M. C. Joshi and R. K. Bose, “Some Topics in Nonlinear Functional Analysis,” Hasted Press, New York, 1985.

[18] M. D. Gunzburger, “Pers-pectives in Flow Control and Optimization. Advances in Design and Control,” SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 2003.

[19] E. Polak, “Computational Methods in Optimization,” Academic Press, Cambridge, 1971.

[20] J. A. David, H. T. Tran and H. T. Banks, “HIV Model Analysis and Estimation Im-plementation under Optimal Control Based Treatment Strategies,” International Journal of Pure and Applied Mathematics, Vol. 57, No. 3, 2009, pp. 357-392.

[21] C. T. Kelley, “Iterative Methods for Optimization,” SIAM: Society for Industrial and Applied Mathematics, Phila-delphia, 1999.

[22] L. F. Shampine and M. W. Reichelt, “The MATLAB ODE Suite,” SIAM Journal on Scientific Computing, Vol. 18, No. 1, 1997, pp. 1-22. doi:10.1137/S1064827594276424

[23] L. F. Reichelt, M. W. Shampine and J. A. Kierzenka. “Solving Index-1 DAEs in MATLAB and Simulink,” SIAM Review, Vol. 41, No. 3, 1999, pp. 538-552. doi:10.1137/S003614459933425X

[24] R. D. Skeel and M. Berzins, “A Method for the Spatial Discretization of Parabolic Equations in One Space Variable,” SIAM Journal on Scientific and Statistical Computing, Vol. 11, No. 1, 1990, pp. 1-32. doi:10.1137/0911001

[25] R. L. Burden and J. D. Faires, “Numberical Analysis,” Brookes/Cole Publisher, Salt Lake City, 2011.

[26] C. T. Kelley, “Iterative Methods for Linear and Nonlinear Equations,” SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 1995. doi:10.1137/1.9781611970944