Abstract
The optimal speed problem for the class of linear discrete-time systems with the infinite-dimensional state vector and degenerate operator is solved. Some properties of convex sets are formulated and proved. Necessary and sufficient conditions under which this problem has a solution in the case of the zero point located on the boundary of the reachability set are established. The optimality conditions are written as a discrete-time maximum principle. For the inner points, the degenerate character of the maximum principle is demonstrated. For an inner point, the optimal speed problem is solved by developing an algorithm with reduction to the admissible case of the boundary point. Some examples are given.
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Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, B.F., Matematicheskaya teoriya optimal’nykh protsessov, Moscow: Nauka, 1969. Translated under the title The Mathematical Theory of Optimal Processes, New York: Wile, 1962.
Boltyanskii, V.G., Matematicheskie metody optimal’nogo upravleniya, Moscow: Nauka, 1969. Translated under the title Mathematical Methods of Optimal Control (Balskrishnan–Neustadt Series), New York: Holt, Rinehart and Winston, 1971.
Moiseev, N.N., Elementy teorii optimal’nykh sistem (Elements of Theory of Optimal Systems), Moscow: Nauka, 1975.
Bellman, R., Dynamic Programming, Princeton: Princeton Univ. Press, 1957. Translated under the title Dinamicheskoe programmirovanie, Moscow: Inostrannaya Literatura, 1960.
Sirotin, A.N. and Formal’skii, A.M., Reachability and Controllability of Discrete-Time Systems under Control Actions Bounded in Magnitude and Norm, Autom. Remote Control, 2003, vol. 64, no. 12, pp. 1844–1857.
Fisher, M.E. and Gayek, J.E., Estimating Reachable Sets for Two-Dimensional Linear Discrete Systems, J. Optim. Theory Appl., 1988, vol. 56, no. 1, pp. 67–88.
Kostousova, E.K., On External Polyhedral Estimation of Reachability Sets in the Extended Space for Linear Multistep Systems with Integral-Type Control Constraints, Vychisl. Tekhnol., 2004, vol. 9, no. 4, pp. 54–72.
Agrachev, A.A. and Sachkov, Yu.L., Geometricheskaya teoriya upravleniya (Geometrical Theory of Control), Moscow: Nauka, 2005.
Evtushenko, Yu.G., Metody resheniya ekstremal’nykh zadach i ikh prilozheniya v sistemakh optimizatsii (Methods for Solving Extremum Problems and Their Applications in Optimization Problems), Moscow: Nauka, 1982.
Boltyanskii, V.G., Optimal’noe upravlenie diskretnymi sistemami (Optimal Control of Discrete-Time Systems), Moscow: Nauka, 1973.
Propoi, A.I., Elementy teorii optimal’nykh diskretnykh protsessov (Elements of the Theory of Optimal Discrete Processes), Moscow: Nauka, 1973.
Holtzman, J.M. and Halkin, H., Directional Convexity and the Maximum Principle for Discrete Systems, J. SIAM Control, 1966, vol. 4, no. 2, pp. 263–275.
Desoer, C.A. and Wing, J., The Minimal Time Regulator Problem for Linear Sampled-Data Systems: General Theory, J. Franklin Inst., 1961, vol. 272, no. 3, pp. 208–228.
Lin, W.-S., Time-Optimal Control Strategy for Saturating Linear Discrete Systems, Int. J. Control, 1986, vol. 43, no. 5, pp. 1343–1351.
Moroz, A.I., Synthesis of an Optimally Rapid Control for a Linear Discrete Third-Order Object. I, Autom. Remote Control, 1965, vol. 26, no. 2, pp. 193–206.
Ibragimov, D.N. and Sirotin, A.N., On the Problem of Operation Speed for the Class of Linear Infinite- Dimensional Discrete-Time Systems with Bounded Control, Autom. Remote Control, 2017, vol. 78, no. 10, pp. 1731–1756.
Rockafellar, R.T., Convex Analysis, Princeton: Princeton Univ. Press, 1970. Translated under the title Vypuklyi analiz, Moscow: Mir, 1973.
Berger, M., Géométrie II, Paris: CEDIC and Fernand Nathan, 1977. Translated under the title Geometry II, Berlin–Heidelberg: Springer-Verlag, 1987.
Ibragimov, D.N. and Sirotin, A.N., On the Problem of Optimal Speed for the Discrete Linear System with Bounded Scalar Control on the Basis of 0-controllability Sets, Autom. Remote Control, 2015, vol. 76, no. 9, pp. 1517–1540.
Ibragimov, D.N., Optimal Speed Control for Aerostat’s Motion, Tr. Mosk. Aviats. Inst., 2015, no. 83. http://trudymai.ru/published.php
Kolmogorov, A.N. and Fomin, S.V., Elementy teorii funktsii i funktsional’nogo analiza (Elements of the Theory of Functions and Functional Analysis), Moscow: Fizmatlit, 2012.
Dunford, N. and Schwartz, J.T., Linear Operators, part 2: Spectral Theory, Self Adjoint Operators in Hilbert Space, New York: Interscience, 1963. Translated under the title Lineinye operatory, tom 2: Spektral’naya teoriya. Samosopryazhennye operatory v gil’bertovom prostranstve, Moscow: Mir, 1966.
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Russian Text © D.N. Ibragimov, 2019, published in Avtomatika i Telemekhanika, 2019, No. 3, pp. 3–25.
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Ibragimov, D.N. On the Optimal Speed Problem for the Class of Linear Autonomous Infinite-Dimensional Discrete-Time Systems with Bounded Control and Degenerate Operator. Autom Remote Control 80, 393–412 (2019). https://doi.org/10.1134/S0005117919030019
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DOI: https://doi.org/10.1134/S0005117919030019