Approximation of the set of trajectories of the nonlinear control system with limited control resources
Abstract
In this paper the control system described by a Urysohn type integral equation is studied. It is assumed that the control functions have integral constraint. Approximation of the set of trajectories generated by all admissible control functions is considered. Step by step way, the set of admissible control functions is replaced by a set consisting of a finite number of control functions which generates a finite number of trajectories. An evaluation of the Hausdorff distance between the set of trajectories of the system and the set, consisting of a finite number of trajectories is obtained.
Keyword : control system, nonlinear integral equation, integral constraint, set of trajectories, approximation
How to Cite
Huseyin, N., Huseyin, A., & Guseinov, K. (2018). Approximation of the set of trajectories of the nonlinear control system with limited control resources. Mathematical Modelling and Analysis, 23(1), 152-166. https://doi.org/10.3846/mma.2018.010
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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A. Huseyin, N. Huseyin and Kh. G. Guseinov. Approximation of the sections of the set of trajectories of the control system described by a nonlinear Volterra integral equation. Math. Model. Anal., 20:502-515, 2015.
N. Huseyin. Compactness of the set of trajectories of the control system described by a Urysohn type integral equation. Int. J. Optim. Control Theory Appl. (IJOCTA), 7:59-65, 2017.
N. Huseyin, Kh. G. Guseinov and V. N. Ushakov. Approximate construction of the set of trajectories of the control system described by a Volterra integral equation. Math. Nachr., 288:1891-1899, 2015.
G. Ibragimov, A. Rakhmanov, I. A. Alias and A. M. J. Mai Zurwatul. The soft landing problem for an infinite system of second order differential equations. Numerical Algebra, Control and Optimization, 7:89-94, 2017.
M. A. Krasnoselskii and S. G. Krein. On the principle of averaging in nonlinear mechanics. Uspekhi Mat. Nauk., 10:147-153, 1955.
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N. N. Subbotina and A. I. Subbotin. Alternative for the encounter-evasion differential game with constraints on the momenta of the players controls. J. Appl. Math. Mech., 39:376-385, 1975.
P. S. Urysohn. On a type of nonlinear integral equation. Mat. Sb., 31:236-255, 1923.
E. Vainikko and G. Vainikko. Product quasi-interpolation in logarithmically singular integral equations. Math. Model. Anal., 17(5):696-714, 2012.
J. Appell, A. Carbone and P. P. Zabrejko. Kantorovich majorants for nonlinear operators and applications to Urysohn integral equations. Rend. Mat. Appl., 12:675-688, 1992.
E. J. Balder. On existence problems for the optimal control of certain nonlinear integral equations of Urysohn type. J. Optim. Theory Appl., 42:447-465, 1984.
M. L. Bennati. An existence theorem for optimal controls of systems defined by Urysohn integral equations. Ann. Mat. Pura Appl., 121:187-197, 1979.
F. Brauer. On a nonlinear integral equation for population growth problems. SIAM J. Math. Anal., 69:312-317, 1975.
D. A. Carlson. An elementary proof of the maximum principle for optimal control problems governed by a Volterra integral equation. J. Optim. Theory Appl., 54:43-61, 1987.
A. G. Chentsov. Approximative realization of integral constraints and generalized constructions in the class of vector finitely additive measures. Proc. Steklov Inst. Math., Suppl.2:S10-S60, 2002.
R. Conti. Problemi di Controllo e di Controllo Ottimale. UTET, Torino, 1974.
C. Corduneanu. Integral Equations and Applications. Cambridge University Press, Cambridge, 1991.
K. G. Guseinov, A. A. Neznakhin and V. N. Ushakov. Approximate construction of reachable sets of control systems with integral constraints on the control. J. Appl. Math. Mech., 63:557-567, 1999.
M. I. Gusev and I.V. Zykov. On extremal properties of the boundary points of reachable sets for control systems with integral constraints. Tr. Inst. Math. Mekh. UrO RAN, 23(1):103-115, 2017.
W. Heisenberg. Physics and Philosophy. The Revolution in Modern Science. George Allen and Unwin, London, 1958.
D. Hilbert. Grundzuge Einer Allgemeinen Theorie der Linearen Integralgleichungen. Druck und Verlag von B.G.Teubner, Leipzig und Berlin, 1912.
A. Huseyin and N. Huseyin. Dependence on the parameters of the set of trajectories of the control system described by a nonlinear Volterra integral equation. Appl. Math., 59:303-317, 2014.
A. Huseyin, N. Huseyin and Kh. G. Guseinov. Approximation of the sections of the set of trajectories of the control system described by a nonlinear Volterra integral equation. Math. Model. Anal., 20:502-515, 2015.
N. Huseyin. Compactness of the set of trajectories of the control system described by a Urysohn type integral equation. Int. J. Optim. Control Theory Appl. (IJOCTA), 7:59-65, 2017.
N. Huseyin, Kh. G. Guseinov and V. N. Ushakov. Approximate construction of the set of trajectories of the control system described by a Volterra integral equation. Math. Nachr., 288:1891-1899, 2015.
G. Ibragimov, A. Rakhmanov, I. A. Alias and A. M. J. Mai Zurwatul. The soft landing problem for an infinite system of second order differential equations. Numerical Algebra, Control and Optimization, 7:89-94, 2017.
M. A. Krasnoselskii and S. G. Krein. On the principle of averaging in nonlinear mechanics. Uspekhi Mat. Nauk., 10:147-153, 1955.
N. N. Krasovskii. Theory of Control of Motion: Linear Systems. Nauka, Moscow, 1968.
A. R. Matviichuk and V. N. Ushakov. On the construction of resolving controls in control problems with phase constraints. J. Comput. Syst. Sci. Int., 45(1):1-16, 2006.
B. T. Polyak. Convexity of the reachable set of nonlinear systems under L2 bounded controls. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 11(2-3):255-267, 2004.
N. N. Subbotina and A. I. Subbotin. Alternative for the encounter-evasion differential game with constraints on the momenta of the players controls. J. Appl. Math. Mech., 39:376-385, 1975.
P. S. Urysohn. On a type of nonlinear integral equation. Mat. Sb., 31:236-255, 1923.
E. Vainikko and G. Vainikko. Product quasi-interpolation in logarithmically singular integral equations. Math. Model. Anal., 17(5):696-714, 2012.