Time-Optimal Control Problem for *n*× *n* Co-Operative Parabolic Systems with Control in Initial Conditions

ABSTRACT

In this paper, time-optimal control problem for a liner *n*× *n* co-operative parabolic system involving Laplace operator is considered. This problem is, steering an initial state y(0)=u , with control u so that an observation y(t) hitting a given target set in minimum time. First, the existence and uniqueness of solutions of such system under conditions on the coefficients are proved. Afterwards necessary and sufficient conditions of optimality are obtained. Finally a scaler case is given.

Cite this paper

M. Shehata, "Time-Optimal Control Problem for*n*× *n* Co-Operative Parabolic Systems with Control in Initial Conditions," *Advances in Pure Mathematics*, Vol. 3 No. 9, 2013, pp. 38-43. doi: 10.4236/apm.2013.39A1006.

M. Shehata, "Time-Optimal Control Problem for

References

[1] J. L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer-Verlag, New York, 1971. http://dx.doi.org/10.1007/978-3-642-65024-6

[2] J. L. Lions and E. Magenes, “Non-Homogeneous Boundary Value Problem and Applications, I, II,” Spring-Verlage, New York, 1972.

[3] H. O. Fattorini, “Infinite Dimensional Optimization Theory and Opotimal Control,” Cambridge University Press, Cambridge, 1998.

[4] K. D. Phung, G. Wang and X. Zhang, “On the Existence of Time Optimal Controls for Linear Evolution Equations,” Discrete and Continuous Dynamical Systems, Series B, Vol. 8, No. 4, 2007, pp. 925-941.

http://dx.doi.org/10.3934/dcdsb.2007.8.925

[5] G. Knowles, “Time Optimal Control of Parabolic Systems with Boundary Condition Involving Time Delays,” Journal of Optimization Theory and Applications, Vol. 25, No. 1, 1978, pp. 563-574.

http://dx.doi.org/10.1007/BF00933521

[6] H. O. Fattorini, “Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems,” North-Holland Mathematics Studies, Vol. 201, Elsevier, Amsterdam, 2005.

[7] X. Li and J. Yong, “Optimal Control Theory for Infinite Dimensional Systems,” Birkhauser, Boston, 1995.

http://dx.doi.org/10.1007/978-1-4612-4260-4

[8] H. A. El-Saify, H. M. Serag and M. A. Shehata, “Time-Optimal Control for Co-Operative Hyperbolic Systems Involving Laplace Operator,” Journal of Dynamical and Control Systems, Vol. 15, No. 3, 2009, pp. 405-423.

http://dx.doi.org/10.1007/s10883-009-9067-y

[9] M. A. Shehata, “Some Time-Optimal Control Problems for n × n Co-Operative Hyperbolic Systems with Distributed or Boundary Controls,” Journal of Mathematical Sciences: Advances and Applications, Vol. 18, No. 1-2, 2012, pp. 63-83.

[10] A. Kowalewski, “Optimal Control via Initial State of an Infinite Order Time Delay Hyperbolic System,” 15th International Conference on Methods and Models in Automation and Robotics (MMAR), 23-26 August 2010, pp. 275-277.

[11] R. A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975.

[12] J. Fleckinger, J. Hernández and F. de Thélin, “On the Existence of Multiple Principal Eigenvalues for Some Indefinite Linear Eigenvalue Problems,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas, Vol. 97, No. 2, 2003, pp. 461-466.

[1] J. L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Springer-Verlag, New York, 1971. http://dx.doi.org/10.1007/978-3-642-65024-6

[2] J. L. Lions and E. Magenes, “Non-Homogeneous Boundary Value Problem and Applications, I, II,” Spring-Verlage, New York, 1972.

[3] H. O. Fattorini, “Infinite Dimensional Optimization Theory and Opotimal Control,” Cambridge University Press, Cambridge, 1998.

[4] K. D. Phung, G. Wang and X. Zhang, “On the Existence of Time Optimal Controls for Linear Evolution Equations,” Discrete and Continuous Dynamical Systems, Series B, Vol. 8, No. 4, 2007, pp. 925-941.

http://dx.doi.org/10.3934/dcdsb.2007.8.925

[5] G. Knowles, “Time Optimal Control of Parabolic Systems with Boundary Condition Involving Time Delays,” Journal of Optimization Theory and Applications, Vol. 25, No. 1, 1978, pp. 563-574.

http://dx.doi.org/10.1007/BF00933521

[6] H. O. Fattorini, “Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems,” North-Holland Mathematics Studies, Vol. 201, Elsevier, Amsterdam, 2005.

[7] X. Li and J. Yong, “Optimal Control Theory for Infinite Dimensional Systems,” Birkhauser, Boston, 1995.

http://dx.doi.org/10.1007/978-1-4612-4260-4

[8] H. A. El-Saify, H. M. Serag and M. A. Shehata, “Time-Optimal Control for Co-Operative Hyperbolic Systems Involving Laplace Operator,” Journal of Dynamical and Control Systems, Vol. 15, No. 3, 2009, pp. 405-423.

http://dx.doi.org/10.1007/s10883-009-9067-y

[9] M. A. Shehata, “Some Time-Optimal Control Problems for n × n Co-Operative Hyperbolic Systems with Distributed or Boundary Controls,” Journal of Mathematical Sciences: Advances and Applications, Vol. 18, No. 1-2, 2012, pp. 63-83.

[10] A. Kowalewski, “Optimal Control via Initial State of an Infinite Order Time Delay Hyperbolic System,” 15th International Conference on Methods and Models in Automation and Robotics (MMAR), 23-26 August 2010, pp. 275-277.

[11] R. A. Adams, “Sobolev Spaces,” Academic Press, New York, 1975.

[12] J. Fleckinger, J. Hernández and F. de Thélin, “On the Existence of Multiple Principal Eigenvalues for Some Indefinite Linear Eigenvalue Problems,” Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas, Vol. 97, No. 2, 2003, pp. 461-466.