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Application of 2D systems to investigation of a process of gas filtration. (English) Zbl 1256.49051

Summary: In this paper, a process of gas filtration described by a 2D system with controls is considered. Sufficient conditions for the existence of an optimal process are proved.

MSC:

49N90 Applications of optimal control and differential games
35Q93 PDEs in connection with control and optimization
Full Text: DOI

References:

[1] Berkson E., Gillespie T. A. (1984) Absolutely continuous functions of two variables and well-bounded operators. Journal of the London Mathematical Society 2(30): 305–321 · doi:10.1112/jlms/s2-30.2.305
[2] Cesari L. (1969) Optimization with partial differential equations in Dieudonne-Rashevsky form and conjugate problems. Archive for Rational Mechanics and Analysis 33: 339–357 · Zbl 0182.20302 · doi:10.1007/BF00247693
[3] Cesari L. (1983) Optimization theory and application. Springer, New York · Zbl 0506.49001
[4] Dymkov M., Gaishun I., Gałkowski K., Owens D. H., Rogers E. (2004) Control theory for a class of 2D continuous-discrete linear systems. International Journal of Control 77(9): 847–860 · Zbl 1060.93055 · doi:10.1080/00207170410001726796
[5] Egorov A. I. (1966) Necessary optimality conditions for systems with distributed parameters. Matematicheskii Sbornik 69(3): 371–421 (Russian) · Zbl 0163.11305
[6] Fornasini E., Marchesini G. (1976) State-space realization theory of two-dimensional filters. IEEE Transactions on Automatic Control AC-21(4): 484–491 · Zbl 0332.93072 · doi:10.1109/TAC.1976.1101305
[7] Fornasini E., Marchesini G. (1978) Doubly indexed dynamical systems; state space models and structural properties. Mathematical Systems Theory 12(1): 59–72 · Zbl 0392.93034 · doi:10.1007/BF01776566
[8] Gałkowski K., Vinnikov V. (2004) Multidimensional control systems: Theory with a view to applications. International Journal of Control 77(9): 801 · doi:10.1080/00207170412331294770
[9] Hildebrandt T. H. (1963) Introduction to the theory of integration. Academic Press, New York, London · Zbl 0112.28302
[10] Idczak D., Walczak S. (2000) On the existence of a solution for some distributed optimal control hyperbolic system. International Journal of Mathematics and Mathematical Science 23(5): 297–311 · Zbl 0959.49004 · doi:10.1155/S0161171200000880
[11] Ioffe A. D., Tikhomirov V. M. (1979) Theory of extremal problems. Academic Press, New York
[12] Kaczorek, T. (1985). Two-dimensional linear systems. Lecture Notes in Control and Information Sciences, Springer. · Zbl 0593.93031
[13] Kaczorek T. (2000) Reachability and controllability of 2D positive linear systems with state feedbacks. Control and Cybernetics 29(1): 141–151 · Zbl 1012.93030
[14] Kaczorek T. (2001) Perfect observers for singular 2-D Fornasini-Marchesini models. IEEE Transactions on Automatic Control 46(10): 1671–1675 · Zbl 1009.93044 · doi:10.1109/9.956070
[15] Lam J., Xu S., Zou Y., Lin Z., Gałkowski K. (2004) Robust output feedback stabilization for two-dimensional continuous systems in Roesser form. Applied Mathematics Letters 17(12): 1331–1341 · Zbl 1122.93405 · doi:10.1016/j.am1.2004.04.004
[16] Lomadze V., Rogers E., Wood J. (2009) Singular 2D behaviors: Fornasini-Marchesini and Givone-Roesser models. Georgian Mathematical Journal 16(1): 105–130 · Zbl 1163.93013
[17] Łojasiewicz S. (1988) An introduction to the theory of real functions. Wiley, Chichester
[18] Pontriagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F. (1960) The mathematical theory of optimal processes. Interscience, New York
[19] Rehbock V., Wang S., Teo K. L. (1998) Computing optimal control with a hyperbolic partial differential equation. Journal of the Australian Mathematical Society Series B 40(2): 266–287 · Zbl 0917.65060 · doi:10.1017/S0334270000012510
[20] Surryyanarayana M. B. (1973) Necessary conditions for optimization problems with hyperbolic partial differential equations. SIAM Journal of Control 11(4): 130–147 · Zbl 0253.35061 · doi:10.1137/0311011
[21] Tikhonov, A. N., & Samarski, A. A. (1978). Equations of Mathematical Physics. Moscow.
[22] Walczak S. (1984) Some properties of cones in normed spaces and their application to investigating extremal problems. Journal of Optimization Theory and Applications 42(4): 561–582 · Zbl 0504.52001 · doi:10.1007/BF00934567
[23] Walczak S. (1987) Absolutely continuous functions of several variables and their applications to differential equations. Bulletin of the Polish Academy of Science and Mathematics 35(11–12): 733–744
[24] Xu S., Lam J., Gałkowski K., Lin Z. (2006) An LMI approach to the computations of lower bounds for stability margins of 2D discrete systems. Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms 13(2): 221–236 · Zbl 1162.93374
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