Construction of an attainability set for the Brockett integrator. (Russian, English) Zbl 1106.49035

Prikl. Mat. Mekh. 68, No. 5, 707-724 (2004); translation in J. Appl. Math. Mech. 68, No. 5, 631-646 (2004).
The Brockett integrator [R. W. Brockett, “Differential Geometric Control Theory”, R. W. Brockett, et al. (eds.), Boston: Birkhäuser, 181–191 (1983; Zbl 0503.00014)] is one of the first classical examples of the system for which the control problem solution requires the nonlinear (discontinuous) control law to be introduced. The authors apply some results of the optimal control theory [L. S. Pontryagin, V. G. Boltyanskij, R. V. Gamkrelidze and E. F. Mischenko, “Mathematical theory of optimal processes”, Nauka, Moscow (1969; Zbl 0179.14001)] and solve the problem on attainability sets construction for nonlinear dynamical system known as the nonholonomous Brockett integrator. It is proved that the attainability set boundary is characterized by the optimal trajectory points constructed for the control problem with integral quality exponent prescribing the figure area limited by the motion trajectory of controlled system.


49K15 Optimality conditions for problems involving ordinary differential equations
49K35 Optimality conditions for minimax problems
Full Text: DOI


[1] Krasovskii, A. N.; Krasovskii, N. N., Control under Lack of Information (1994), Birkhauser · Zbl 0827.93001
[2] Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; Mishchenko, Ye. F., (Mathematical Theory of Optimal Processes (1976), Nauka: Nauka Basel)
[3] Brockett, R. W., Asymptotic stability and feedback stabilization, (Brockett, R. W.; etal., Differential Geometric Control Theory (1983), Birkhauser: Birkhauser Moscow), 181-191 · Zbl 0528.93051
[4] Astolfi, A.; Rapaport, A., Robust stabilization of the angular velocity of a rigid body, Systems and Control Letters, 34, 257-264 (1998) · Zbl 0909.93062
[5] Clarke, F. H.; Ledyayev, Yu. S.; Stern, R. J., Proximal analysis and feedback construction, Trudy Inst. Mat. Mekh., Ural. Otd. Ross. Akad. Nauk, 6, 1, 91-109 (2000) · Zbl 1116.93031
[6] Nikol’Slkii, M. S., Interior estimation of the attainability set of a non-linear Brockett integrator, Diff. Uravn., 96, 11, 1501-1505 (2000)
[7] Chernous’Ko, F. L.; Melikyan, A. A., Game Problems of Control and Search (1978), Nauka: Nauka Boston · Zbl 0398.49010
[8] Chernous’Ko, F. L.; Bolotnik, N. N.; Gradetskii, V. G., Manipulator Robots: Dynamics, Control, Optimization (1989), Nauka: Nauka Moscow · Zbl 0693.70001
[9] Blagodatskikh, V. I., (Introduction to Optimal Control (Linear Theory) (2001), Vysshaya Shkola: Vysshaya Shkola Moscow)
[10] Krasovskii, N. N.; Subbotin, A. I., Positional Differential Games (1974), Nauka: Nauka Moscow · Zbl 0298.90067
[11] Subbotin, A. I., Minimax Inequalities and the Hamilton-Jacobi Equations (1991), Nauka: Nauka Moscow · Zbl 0733.70014
[12] Subbotin, A. I.; Chentsov, A. G., Optimization of Guarantee in Control Problems (1981), Nauka: Nauka Moscow · Zbl 0542.90106
[13] Subbotin, A. I., Generalized Solutions of First-Order PDEs: The Dynamical Optimization Perspective (1995), Birkhauser: Birkhauser Moscow · Zbl 0820.35003
[14] Subbotin, A. I., The method of Cauchy characteristics and generalized solutions of the Hamilton-Jacobi-Bellman equations, Dokl. Akad. Nauk SSSR, 320, 3, 556-561 (1991)
[15] Subbotin, A. I.; Taras’Vev, A. M.; Ushakov, V. N., Generalized characteristics of the Hamilton-Jacobi equations, Izv. Akad. Nauk. Tekhn. Kibemetika, 1, 190-197 (1993)
[16] Kartashev, A. P.; Rozhdestvenskii, B. L., Ordinary Differential Equations and the Elements of the Variational Calculus (1976), Nauka: Nauka Boston
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