Geodesic
On the boundedness and unboundedness of external polyhedral estimates for reachable sets of linear differential systems
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 134-145
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The boundedness and unboundedness properties of external polyhedral (parallelepiped-valued) estimates are investigated for reachable sets of linear differential systems with a stable matrix. Boundedness and unboundedness criteria on an infinite time interval are presented for two types of estimates (“touching” estimates, which were introduced earlier, and estimates with constant orientation matrix). Conditions for the system matrix and bounding sets are given under which there are bounded estimates among the mentioned estimates, under which there are unbounded estimates, and under which all the estimates are bounded or all the estimates are unbounded. In terms of the exponents of the estimates, the possible degree of their growth is described. For two-dimensional systems, the classification and comparison of possible situations of the boundedness or unboundedness for estimates of both types are given and boundedness criteria for estimates with special (orthogonal and “quasi-orthogonal”) constant orientation matrices are found. Results of numerical modeling are presented.
Keywords: reachable sets, linear systems, polyhedral estimates, parallelepipeds, interval analysis. 
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E. K. Kostousova. On the boundedness and unboundedness of external polyhedral estimates for reachable sets of linear differential systems. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 134-145. http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a11/

[1] Kurzhanskii A. B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 392 pp. | MR | Zbl

[2] Kurzhanski A. B., Vályi I., Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, 1997, 321 pp. | MR | Zbl

[3] Kalmykov S. A., Shokin Yu. I., Yuldashev Z. Kh., Metody intervalnogo analiza, Nauka, Novosibirsk, 1986, 222 pp. | MR | Zbl

[4] Gorban A. N., Shokin Yu. I., Verbitskii V. I., “Simultaneously dissipative operators and the infinitesimal wrapping effect in interval spaces”, Vychisl. tekhnologii, 2:4 (1997), 16–48 | MR

[5] Chernousko F. L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem. Metod ellipsoidov, Nauka, M., 1988, 319 pp. | MR

[6] Kostousova E. K., Kurzhanskii A. B., “Garantirovannye otsenki tochnosti vychislenii v zadachakh upravleniya i otsenivaniya”, Vychisl. tekhnologii, 2:1 (1997), 19–27 | MR

[7] Kurzhanski A. B., Varaiya P., “On ellipsoidal techniques for reachability analysis. Part I: External approximations”, Optimiz. Methods Software, 17:2 (2002), 177–206 | DOI | MR | Zbl

[8] Kirilin M. N., Kurzhanski A. B., “Ellipsoidal techniques for reachability problems under nonellipsoidal constraints”, Nonlinear Control Systems 2001, (Nolcos 2001), A Proceedings Volume from the 5th IFAC Symposium (St. Petersburg, Russia, 4–6 July 2001), eds. Kurzhanski A. B., Fradkov A. L., Elsevier Science Ltd., 2002, 735–740

[9] Kostousova E. K., “O vneshnikh poliedralnykh otsenkakh dlya mnozhestv dostizhimosti sistem s bilineinoi neopredelennostyu”, Prikl. matematika i mekhanika, 66:4 (2002), 559–571 | MR | Zbl

[10] Kostousova E. K., “State estimation for dynamic systems via parallelotopes: optimization and parallel computations”, Optimiz. Methods Software, 9:4 (1998), 269–306 | DOI | MR | Zbl

[11] Kostousova E. K., “Ob ogranichennosti vneshnikh poliedralnykh otsenok mnozhestv dostizhimosti lineinykh sistem”, Zhurn. vychisl. matematiki i mat. fiziki, 48:6 (2008), 974–989 | Zbl

[12] Demidovich B. P., Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967, 472 pp. | MR

[13] Lankaster P., Teoriya matrits, Nauka, M., 1978, 280 pp. | MR

[14] Shilov G. E., Matematicheskii analiz. Konechnomernye lineinye prostranstva, Nauka, M., 1969, 432 pp. | MR

[15] Gantmakher F. R., Teoriya matrits, Nauka, M., 1988, 552 pp. | MR | Zbl