Existence and Uniqueness of the Optimal Control in Hilbert Spaces for a Class of Linear Systems

Author(s)
M. Popescu

ABSTRACT

We analyze the existence and uniqueness of the optimal control for a class of exactly controllable linear systems. We are interested in the minimization of time, energy and final manifold in transfer problems. The state variables space X and, respectively, the control variables space U, are considered to be Hilbert spaces. The linear operator T(t) which defines the solution of the linear control system is a strong semigroup. Our analysis is based on some results from the theory of linear operators and functional analysis. The results obtained in this paper are based on the properties of linear operators and on some theorems from functional analysis.

We analyze the existence and uniqueness of the optimal control for a class of exactly controllable linear systems. We are interested in the minimization of time, energy and final manifold in transfer problems. The state variables space X and, respectively, the control variables space U, are considered to be Hilbert spaces. The linear operator T(t) which defines the solution of the linear control system is a strong semigroup. Our analysis is based on some results from the theory of linear operators and functional analysis. The results obtained in this paper are based on the properties of linear operators and on some theorems from functional analysis.

Cite this paper

nullM. Popescu, "Existence and Uniqueness of the Optimal Control in Hilbert Spaces for a Class of Linear Systems,"*Intelligent Information Management*, Vol. 2 No. 2, 2010, pp. 134-142. doi: 10.4236/iim.2010.22016.

nullM. Popescu, "Existence and Uniqueness of the Optimal Control in Hilbert Spaces for a Class of Linear Systems,"

References

[1] Q. W. Olbrot and L. Pandolfi, “Null controllability of a class of functional differential systems,” International Journal of Control, Vol. 47, pp. 193–208, 1988.

[2] H. J. Sussmann, “Nonlinear controlability and optimal control,” Marcel Dekker, New York, 1990.

[3] M. Popescu, “Variational transitory processes and nonlinear analysis in optimal control,” Technical Education Bucharest, 2007.

[4] M. Popescu, “Sweep method in analysis optimal control for rendezvous problems,” Journal of Applied Mathematics and Computing, Vol. 23, 1–2, pp.249–256, 2007.

[5] M. Popescu, “Optimal control in Hilbert space applied in orbital rendezvous problems,” Advances in Mathematical Problems in Engineering Aerospace and Science; (ed.Sivasundaram), Cambridge Scientific Publishers 2, pp. 135– 143, 2008.

[6] S. Chen and I. Lasiecka, “Feedback exact null controllability for unbounded control problems in Hilbert space,” Journal of Optimization Theory and Application, Vol. 74, pp. 191–219, 1992.

[7] M. Popescu, “On minimum quadratic functional control of affine nonlinear control,” Nonlinear Analysis, Vol. 56, pp. 1165–1173, 2004.

[8] M. Popescu, “Linear and nonlinear analysis for optimal pursuit in space,” Advances in Mathematical Problems in Engineering Aerospace and Science; (ed. Sivasundaram), Cambridge Scientific Publishers 2, pp. 107–126, 2008.

[9] M. Popescu, “Optimal control in Hilbert space applied in orbital rendezvous problems,” Advances in Mathematical Problems in Engineering Aerospace and Science; (ed. Sivasundaram), Cambridge Scientific Publishers 2, pp. 135– 143, 2008.

[10] M. Popescu, “Minimum energy for controllable nonlinear and linear systems,” Semineirre Theorie and Control Universite Savoie, France, 2009.

[11] M. Popescu, “Stability and stabilization dynamical systems,” Technical Education Bucharest, 2009.

[12] G. Da Prato, A. J. Pritchard, and J. Zabczyk, “On minimum energy problems,” SIAM J. Control and Optimization, Vol. 29, pp. 209–221, 1991.

[13] E. Priola and J. Zabczyk, “Null controllability with vanishing energy,” SIAM Journal on Control and Optimization, Vol. 42, pp. 1013–1032, 2003.

[14] F. Gozzi and P. Loreti, “Regularity of the minimum time function and minimum energy problems: The linear case,” SIAM Journal on Control and Optimization, Vol. 37, pp. 1195–1221, 1999.

[15] M. Popescu, “Fundamental solution for linear two-point boundary value problem,” Journal of Applied Mathematics and Computing, Vol. 31, pp. 385–394, 2009.

[16] L. Frisoli, A. Borelli, and Montagner, et al., “Arm rehabilitation with a robotic exoskeleleton in Virtual Reality,” Proceedings of IEEE ICORR’07, International Conference on Rehabilitation Robotics, 2007.

[17] P. Garrec, “Systemes mecaniques,” in: Coiffet. P et Kheddar A., Teleoperation et telerobotique, Ch 2., Hermes, Paris, France, 2002.

[18] P. Garrec, F. Geffard, Y. Perrot (CEA List), G. Piolain, and A. G. Freudenreich (AREVA/NC La Hague), “Evaluation tests of the telerobotic system MT200-TAO in AREVANC/ La Hague hot-cells,” ENC 2007, Brussels, Belgium, 2007.

[1] Q. W. Olbrot and L. Pandolfi, “Null controllability of a class of functional differential systems,” International Journal of Control, Vol. 47, pp. 193–208, 1988.

[2] H. J. Sussmann, “Nonlinear controlability and optimal control,” Marcel Dekker, New York, 1990.

[3] M. Popescu, “Variational transitory processes and nonlinear analysis in optimal control,” Technical Education Bucharest, 2007.

[4] M. Popescu, “Sweep method in analysis optimal control for rendezvous problems,” Journal of Applied Mathematics and Computing, Vol. 23, 1–2, pp.249–256, 2007.

[5] M. Popescu, “Optimal control in Hilbert space applied in orbital rendezvous problems,” Advances in Mathematical Problems in Engineering Aerospace and Science; (ed.Sivasundaram), Cambridge Scientific Publishers 2, pp. 135– 143, 2008.

[6] S. Chen and I. Lasiecka, “Feedback exact null controllability for unbounded control problems in Hilbert space,” Journal of Optimization Theory and Application, Vol. 74, pp. 191–219, 1992.

[7] M. Popescu, “On minimum quadratic functional control of affine nonlinear control,” Nonlinear Analysis, Vol. 56, pp. 1165–1173, 2004.

[8] M. Popescu, “Linear and nonlinear analysis for optimal pursuit in space,” Advances in Mathematical Problems in Engineering Aerospace and Science; (ed. Sivasundaram), Cambridge Scientific Publishers 2, pp. 107–126, 2008.

[9] M. Popescu, “Optimal control in Hilbert space applied in orbital rendezvous problems,” Advances in Mathematical Problems in Engineering Aerospace and Science; (ed. Sivasundaram), Cambridge Scientific Publishers 2, pp. 135– 143, 2008.

[10] M. Popescu, “Minimum energy for controllable nonlinear and linear systems,” Semineirre Theorie and Control Universite Savoie, France, 2009.

[11] M. Popescu, “Stability and stabilization dynamical systems,” Technical Education Bucharest, 2009.

[12] G. Da Prato, A. J. Pritchard, and J. Zabczyk, “On minimum energy problems,” SIAM J. Control and Optimization, Vol. 29, pp. 209–221, 1991.

[13] E. Priola and J. Zabczyk, “Null controllability with vanishing energy,” SIAM Journal on Control and Optimization, Vol. 42, pp. 1013–1032, 2003.

[14] F. Gozzi and P. Loreti, “Regularity of the minimum time function and minimum energy problems: The linear case,” SIAM Journal on Control and Optimization, Vol. 37, pp. 1195–1221, 1999.

[15] M. Popescu, “Fundamental solution for linear two-point boundary value problem,” Journal of Applied Mathematics and Computing, Vol. 31, pp. 385–394, 2009.

[16] L. Frisoli, A. Borelli, and Montagner, et al., “Arm rehabilitation with a robotic exoskeleleton in Virtual Reality,” Proceedings of IEEE ICORR’07, International Conference on Rehabilitation Robotics, 2007.

[17] P. Garrec, “Systemes mecaniques,” in: Coiffet. P et Kheddar A., Teleoperation et telerobotique, Ch 2., Hermes, Paris, France, 2002.

[18] P. Garrec, F. Geffard, Y. Perrot (CEA List), G. Piolain, and A. G. Freudenreich (AREVA/NC La Hague), “Evaluation tests of the telerobotic system MT200-TAO in AREVANC/ La Hague hot-cells,” ENC 2007, Brussels, Belgium, 2007.