Yu.F. Dolgii. Exact solutions of an inverse optimal stabilization problem for systems with aftereffect of neutral type ... P. 35-44

Vol. 25, no. 1, 2019

An optimal stabilization problem is considered for systems of differential equations with aftereffect of neutral type. To simplify the representation of a continuous quadratic functional, an isomorphism of functional spaces is used. The optimal stabilization problem is formulated in a functional space of states with a special metric. A statement of the inverse optimal stabilization problem is presented; this statement is related to the recovery of a system with a given representation of an optimal stabilizing control. Sufficient conditions for the solvability of the inverse problem are obtained, and conditions under which the inverse problem admits analytical solutions are specified. A method for finding exact solutions to this problem is proposed. For systems of differential equations with delay-type aftereffect, exact solutions of the inverse problem were obtained earlier. An example of the exact solution of the inverse problem is given for a system of differential equations with aftereffect of neutral type.

Keywords: differential equations with aftereffect of neutral type, optimal stabilization, Riccati equation

Received November 6, 2018

Revised February 8, 2019

Accepted February 11, 2019

Yuriy Filippovich Dolgii, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekateriburg, 620002, Russia, e-mail: yurii.dolgii@imm.uran.ru

Cite this article as:  Yu.F. Dolgii. Exact solutions of an inverse optimal stabilization problem for systems with aftereffect of neutral type, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp. 35–44 . 

REFERENCES

1.   Krasovskii N.N. On the analytic construction of an optimal control in a system with time lags. PMM, J. Appl. Math. Mech., 1962, vol. 26, no. 1, pp. 50–67. doi: 10.1016/0021-8928(62)90101-6 

2.   Gibson J.S. Linear-quadratic optimal control of hereditary differential systems: infinite dimensional Riccati equations and numerical approximations. SIAM J. Control Optim., 1983, vol. 21, no. 1, pp. 95–139. doi: 10.1137/0321006 

3.   Delfour M.C., McCalla C., Mitter S.K. Stability and the infinite-time quadratic cost problem for linear hereditary differential systems. SIAM J. Control, 1975, vol. 13, no. 1, pp. 48–88. doi: 10.1137/0313004 

4.   Andreeva E.A., Kolmanovskii V.B., Shaikhet L.E. Upravlenie sistemami s posledeistviem (Control of systems with aftereffect). Moscow: Nauka Publ., 1992, 336 p. ISBN: 5-02-014875-Х .

5.   Dolgii Yu.F. Stabilization of linear autonomous systems of differential equations with distributed delay. Autom. Remote Control, 2007, vol. 68, no. 10, pp. 1813–1825. doi: 10.1134/S0005117907100098 

6.   Zhelonkina N.I., Lozhnikov A.B., Sesekin A.N. On pulse optimal control of linear systems with aftereffect. Autom. Remote Control, 2013, vol. 74, no. 11, pp. 1802–1809. doi: 10.1134/S0005117913110039 

7.   Yanushevsky R.T. Optimal control of linear differential-difference systems of neutral type. Int. J. Control, 1989, vol. 49, no. 6, pp. 1835–1850.

8.   Dolgii Yu.F. Optimal stabilization of systems of differential equations with aftereffect of neutral type. Proc. Systems Dynamics and Control Processes — SDCP-2014, 2015, pp. 155–162 (in Russian). ISBN: 978-5-8295-0364-2 .

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