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Convergence analysis of a parallel projection algorithm for solving convex feasibility problems
2D system analysis via dual problems and polynomial matrix inequalities
1.  Arzamas Polytechnic Institute, Alekseev Nizhny Novgorod State Technical University, 607220, Arzamas, Russian Federation 
References:
[1] 
P. Agathoklis, E. Jury and M. Mansour, Algebraic necessary and sufficient conditions for the stability of 2D discrete systems,, Circuits and Systems II: Analog and Digital Signal Processing, 40 (1993), 251. 
[2] 
B. Anderson, P. Agathoklis, E. Jury and M. Mansour, Stability and the matrix Lyapunov equation for discrete 2dimensional systems,, Circuits and Systems, 33 (1986), 261. doi: 10.1109/TCS.1986.1085912. 
[3] 
S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Studies in Applied Mathematics, (1994). doi: 10.1137/1.9781611970777. 
[4] 
G. Chesi, LMI techniques for optimization over polynomials in control: A survey,, Automatic Control, 55 (2010), 2500. doi: 10.1109/TAC.2010.2046926. 
[5] 
G. Chesi and R. Middleton, Necessary and sufficient LMI conditions for stability and performance analysis of 2D mixed continuousdiscretetime systems,, Automatic Control, 59 (2014), 996. doi: 10.1109/TAC.2014.2299353. 
[6] 
B. Cichy, P. Augusta, E. Rogers, K. Galkowski et al., On the control of distributed parameter systems using a multidimensional systems setting,, Mechanical Syst. and Signal Processing, 22 (2008), 1566. 
[7] 
G. Dullerud and R. D'Andrea, Distributed control of heterogeneous systems,, Automatic Control, 49 (2004), 2113. doi: 10.1109/TAC.2004.838499. 
[8] 
D. Henrion and J.B. Lasserre, Convergent relaxations of polynomial matrix inequalities and static output feedback,, Automatic Control, 51 (2006), 192. doi: 10.1109/TAC.2005.863494. 
[9] 
D. Henrion and J.B. Lasserre, Detecting global optimality and extracting solutions in GloptiPoly,, in Positive Polynomials in Control (eds. D. Henrion and A. Garulli), (2005), 293. doi: 10.1007/10997703_15. 
[10] 
T. Hinamoto, 2D Lyapunov equation and filter design based on the FornasiniMarchesini second model,, Circuits and Systems I: Fundamental Theory and Applications, 40 (1993), 102. 
[11] 
V. Kamenetskiy and Y. Pyatnitskiy, An iterative method of Lyapunov function construction for differential inclusions,, Systems & Control Letters, 8 (1987), 445. doi: 10.1016/01676911(87)900855. 
[12] 
S. Knorn and R. Middleton, Stability of twodimensional linear systems with singularities on the stability boundary using LMIs,, Automatic Control, 58 (2013), 2579. doi: 10.1109/TAC.2013.2264852. 
[13] 
J.B. Lasserre, Global optimization with polynomials and the problem of moments,, SIAM Journal on Optimization, 11 (2001), 796. doi: 10.1137/S1052623400366802. 
[14] 
Y. Li, M. Cantoni and E. Weyer, On waterlevel error propagation in controlled irrigation channels,, in Proc. IEEE Conf. Decision Control and Eur. Control Conf., (2005), 2101. 
[15] 
M. C. D. Oliveira, J. C. Geromel and J. Bernussou, Extended H_{2} and H_{∞} norm characterizations and controller parametrizations for discretetime systems,, International Journal of Control, 75 (2002), 666. doi: 10.1080/00207170210140212. 
[16] 
W. Paszke, E. Rogers and K. Galkowski, H_{2}/H_{∞} output informationbased disturbance attenuation for differential linear repetitive processes,, International Journal of Robust and Nonlinear Control, 21 (2011), 1981. doi: 10.1002/rnc.1672. 
[17] 
V. Pozdyayev, Atomic optimization. I. Search space transformation and onedimensional problems,, Automation and Remote Control, 74 (2013), 2069. doi: 10.1134/S0005117913120096. 
[18] 
V. Pozdyayev, Atomic optimization, II, Multidimensional problems and polynomial matrix inequalities,, Automation and Remote Control, 75 (2014), 1155. doi: 10.1134/S0005117914060150. 
[19] 
V. Pozdyayev, Necessary conditions for 2D systems' stability,, in Preprints, (2015), 800. 
[20] 
R. Rabenstein and L. Trautmann, Towards a framework for continuous and discrete multidimensional systems,, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 73. 
[21] 
R. P. Roesser, A discrete statespace model for linear image processing,, Automatic Control, 20 (1975), 1. 
[22] 
E. Rogers, K. Galkowski and D. Owens, Control systems theory and applications for linear repetitive processes,, in Lecture Notes in Control and Information Sciences, (2007). 
[23] 
E. Rogers and D. Owens, Stability analysis for linear repetitive processes,, in Lecture Notes in Control and Information Sciences, (1992). doi: 10.1007/BFb0007165. 
[24] 
E. Rogers and D. Owens, Kronecker product based stability tests and performance bounds for a class of 2D continuousdiscrete linear systems,, Linear Algebra and its Applications, 353 (2002), 33. doi: 10.1016/S00243795(02)002872. 
show all references
References:
[1] 
P. Agathoklis, E. Jury and M. Mansour, Algebraic necessary and sufficient conditions for the stability of 2D discrete systems,, Circuits and Systems II: Analog and Digital Signal Processing, 40 (1993), 251. 
[2] 
B. Anderson, P. Agathoklis, E. Jury and M. Mansour, Stability and the matrix Lyapunov equation for discrete 2dimensional systems,, Circuits and Systems, 33 (1986), 261. doi: 10.1109/TCS.1986.1085912. 
[3] 
S. Boyd, L. Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory,, Studies in Applied Mathematics, (1994). doi: 10.1137/1.9781611970777. 
[4] 
G. Chesi, LMI techniques for optimization over polynomials in control: A survey,, Automatic Control, 55 (2010), 2500. doi: 10.1109/TAC.2010.2046926. 
[5] 
G. Chesi and R. Middleton, Necessary and sufficient LMI conditions for stability and performance analysis of 2D mixed continuousdiscretetime systems,, Automatic Control, 59 (2014), 996. doi: 10.1109/TAC.2014.2299353. 
[6] 
B. Cichy, P. Augusta, E. Rogers, K. Galkowski et al., On the control of distributed parameter systems using a multidimensional systems setting,, Mechanical Syst. and Signal Processing, 22 (2008), 1566. 
[7] 
G. Dullerud and R. D'Andrea, Distributed control of heterogeneous systems,, Automatic Control, 49 (2004), 2113. doi: 10.1109/TAC.2004.838499. 
[8] 
D. Henrion and J.B. Lasserre, Convergent relaxations of polynomial matrix inequalities and static output feedback,, Automatic Control, 51 (2006), 192. doi: 10.1109/TAC.2005.863494. 
[9] 
D. Henrion and J.B. Lasserre, Detecting global optimality and extracting solutions in GloptiPoly,, in Positive Polynomials in Control (eds. D. Henrion and A. Garulli), (2005), 293. doi: 10.1007/10997703_15. 
[10] 
T. Hinamoto, 2D Lyapunov equation and filter design based on the FornasiniMarchesini second model,, Circuits and Systems I: Fundamental Theory and Applications, 40 (1993), 102. 
[11] 
V. Kamenetskiy and Y. Pyatnitskiy, An iterative method of Lyapunov function construction for differential inclusions,, Systems & Control Letters, 8 (1987), 445. doi: 10.1016/01676911(87)900855. 
[12] 
S. Knorn and R. Middleton, Stability of twodimensional linear systems with singularities on the stability boundary using LMIs,, Automatic Control, 58 (2013), 2579. doi: 10.1109/TAC.2013.2264852. 
[13] 
J.B. Lasserre, Global optimization with polynomials and the problem of moments,, SIAM Journal on Optimization, 11 (2001), 796. doi: 10.1137/S1052623400366802. 
[14] 
Y. Li, M. Cantoni and E. Weyer, On waterlevel error propagation in controlled irrigation channels,, in Proc. IEEE Conf. Decision Control and Eur. Control Conf., (2005), 2101. 
[15] 
M. C. D. Oliveira, J. C. Geromel and J. Bernussou, Extended H_{2} and H_{∞} norm characterizations and controller parametrizations for discretetime systems,, International Journal of Control, 75 (2002), 666. doi: 10.1080/00207170210140212. 
[16] 
W. Paszke, E. Rogers and K. Galkowski, H_{2}/H_{∞} output informationbased disturbance attenuation for differential linear repetitive processes,, International Journal of Robust and Nonlinear Control, 21 (2011), 1981. doi: 10.1002/rnc.1672. 
[17] 
V. Pozdyayev, Atomic optimization. I. Search space transformation and onedimensional problems,, Automation and Remote Control, 74 (2013), 2069. doi: 10.1134/S0005117913120096. 
[18] 
V. Pozdyayev, Atomic optimization, II, Multidimensional problems and polynomial matrix inequalities,, Automation and Remote Control, 75 (2014), 1155. doi: 10.1134/S0005117914060150. 
[19] 
V. Pozdyayev, Necessary conditions for 2D systems' stability,, in Preprints, (2015), 800. 
[20] 
R. Rabenstein and L. Trautmann, Towards a framework for continuous and discrete multidimensional systems,, Int. J. of Applied Mathematics and Computer Science, 13 (2003), 73. 
[21] 
R. P. Roesser, A discrete statespace model for linear image processing,, Automatic Control, 20 (1975), 1. 
[22] 
E. Rogers, K. Galkowski and D. Owens, Control systems theory and applications for linear repetitive processes,, in Lecture Notes in Control and Information Sciences, (2007). 
[23] 
E. Rogers and D. Owens, Stability analysis for linear repetitive processes,, in Lecture Notes in Control and Information Sciences, (1992). doi: 10.1007/BFb0007165. 
[24] 
E. Rogers and D. Owens, Kronecker product based stability tests and performance bounds for a class of 2D continuousdiscrete linear systems,, Linear Algebra and its Applications, 353 (2002), 33. doi: 10.1016/S00243795(02)002872. 
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