Synthesis of optimal boundary control of parabolic systems with delay and distributed parameters on the graph

V. V. Provotorov ; E. N. Provotorova

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 13 (2017) no. 2, pp. 209-224

Résumé

The problem of optimal boundary control of evolutionary systems with constant delay and distributed parameters on the graph. System status is determined by a weak solution of the boundary value problem for a parabolic equation in the space of Sobolev type whose elements are functions satisfying the conditions in a certain way matching all the internal nodes of the graph. The control action on the system and monitoring its state is made in the boundary nodes of the graph on the entire time interval. The dual status of the system is defined as a weak solution of the boundary value problem with delay and distributed parameters on the graph with the final condition. The conditions of weak unique solvability of the original and the dual challenges of weak continuous dependence of solutions on initial data. We present necessary and sufficient conditions for the existence of optimal control using the dual system state solved the problem of optimal control synthesis for the case of absence of restrictions on the control action and an analogue-known finite-dimensional case the Kalman results. The method used is applicable to many optimization problems of differential systems whose state is determined by weak solutions of evolution equations on networks. These results are fundamental in the study of problems of boundary control the dynamics of laminar flows of multiphase media. All techniques and methods can be used for the numerical solution of optimal control problem under consideration. Refs 20.

Détail

Keywords: information networks, differential equations, probability, spreading rumors.

Bibliographie

[1] Provotorov V. V., “Optimal control of parabolic systems with distributed parameters on the graph”, Vestnik of Saint Petersburg State University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2014, no. 3, 154–163 (In Russian)

[2] Provotorov V. V., “Start control of parabolic systems with distributed parameters on the graph”, Vestnik of Saint Petersburg State University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2015, no. 3, 126–142 (In Russian)

[3] Provotorov V. V., “Boundary control of a parabolic system with delay and distributed parameters on the graph”, Intern. conference “Stability and Control Processes” in memory of V. I. Zubov (SCP), Saint Petersburg State University, Saint Petersburg, 2015, 126–128

[4] Podvalny S. L., Provotorov V. V., “The questions of controllability of a parabolic systems with distributed parameters on the graph”, Intern. conference “Stability and Control Processes” in memory of V. I. Zubov (SCP), Saint Petersburg State University, Saint Petersburg, 2015, 117–119

[5] Provotorov V. V., Volkova A. S., Initial-boundary value problems with distributed parameters on the graph, Nauchnaya kniga Publ., Voronezh, 2014, 188 pp. (In Russian)

[6] Provotorov V. V., “Boundary control of a parabolic system with distributed parameters on a graph in the class of summable functions”, Automation and Remote Control, 76:2 (2015), 318–322 | DOI | MR | Zbl

[7] Lions J. L., Controle optimal de sistemes gouvernes par des eqations aux derivees partielles, Dunod Gauthier-Villars, Paris, 1968, 402 pp. | MR

[8] Volkova A. S., Provotorov V. V., “Generalized solutions and generalized eigenfunctions of the boundary value problems on a geometric graph”, Proceedings of Higher Educational Institutions. Mathematics, 2014, no. 3, 3–18 (In Russian)

[9] Provotorov V. V., “The expansion in eigenfunctions of the Sturm–Liouville problem on a graph bundle”, Proceedings of Higher Educational Institutions. Mathematics, 2008, no. 3, 50–62 (In Russian)

[10] Ladyzhenskaya I. A., Boundary-value problems of mathematical physics, Nauka Publ., M., 1973, 407 pp. (In Russian)

[11] Podval'ny S. L., Provotorov V. V., “Optimal problems for evolution system with distributed parameters on the graph”, Modern methods of Applied Mathematics, Control Theory and Computer Technology, PMTUKT-2014, VII Intern. scientific conference, Nauchnaia kniga Publ., Voronezh, 2015, 282–286 (In Russian) | Zbl

[12] Podval'ny S. L., Provotorov V. V., “Optimization for starting conditions of the parabolic system with distributed parameters on the graph”, Control systems and information technologies, 58:4 (2014), 70–74 (In Russian) | Zbl

[13] Volkova A. S., Gnilitskaya Yu. A., Provotorov V. V., “On the solvability of Boundary-Value problems for parabolic and hyperbolic equations on geometrical graphs”, Automation and Remote Control, 75:2 (2014), 405–412 | DOI | MR | Zbl

[14] Aleksandrov A. Yu., Zhabko A. P., “On asymptotic stability of solutions of nonlinear systems with delay”, Automation and Remote Control, 54:4 (2013), 4–7 (In Russian)

[15] Aleksandrov A. Yu., Zhabko A. P., “On the stability of solutions of a class nonlinear systems with delay”, Siberian Mathematical Journal, 44:6 (2003), 1217–1228 (In Russian)

[16] Veremey A. I., Korchanov V. I., “A multipurpose stabilization of dynamical systems of a class”, Automation and Remote Control, 1988, no. 9, 126–137 (In Russian) | Zbl

[17] Veremey A. I., Sotnikova I. V., “Plasma stabilization on the basis of the forecast with steady linear approach”, Vestnik of Saint Petersburg State University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2011, no. 1, 116–133 (In Russian)

[18] Karelin V. V., “Penal functions in a problem of management of supervision process”, Vestnik of Saint Petersburg State University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2010, no. 4, 109–114 (In Russian)

[19] Potapov D. E., “Optimal control of distributed high order systems of elliptic type with a spectral parameter and a discontinuous nonlinearity”, Proceedings of Russian Academy of Sciences. Series TeSU, 2013, no. 2, 19–24 (In Russian)

[20] Kamachkin A. M., Yevstafyeva V. V., “Oscillations in a relay control system at an external disturbance”, Control Applications of Optimization–2000, Proceedings of the 11th IFAC Workshop, v. 2, 2000, 459–462