Reconstruction of Distributed Controls in Hyperbolic Systems by Dynamic Method

In the paper an inverse dynamic problem is considered. It consists of reconstructing a priori unknown distributed controls in dynamical systems described by boundary value problems for partial differential equations of hyperbolic type. The source information for solving the inverse problem is the results of approximate measurements of the states (velocities) of the observed system's motion. The problem is solved in the dynamic case, i.e. to solve the problem we can use only the approximate measurements accumulated by this moment. Unknown controls must be reconstructed in dynamics (during the process, during the motion of the system). The problem under consideration is ill-posed. We propose the method of dynamic regularization to solve the problem. This method was elaborated by Yu.S. Osipov and his school. New modifications of dynamic regularizing solution algorithms are devised in this paper. Using these algorithms in contrast to tradition approach we can obtain stronger convergence of regularized approximations, in particular the piecewise uniform convergence. We also demonstrate a finite-dimensional approximation of the problem and the present results of numerical modelling. These results enable us to assess the ability of modified algorithms to reconstruct the subtle structure of desired controls.

Keywords

dynamical system, control, reconstruction, method of dynamic regularization, piecewise uniform convergence.

References

1. Krasovskii N.N., Subbotin A.I. Game-Theoretical Control Problems. N.Y., Springer-Verlag, 1988.
2. Kryazhimskii A.V., Osipov Yu.S. Modeling of a Control in a Dynamic System. Engineering Cybernetics, 1983, vol. 21, no. 2, pp. 38-47.
3. Osipov Yu.S., Vasil'ev F.P., Potapov M.M. Osnovy metoda dinamicheskoy regulyarizatsii [Foundations of the Dynamical Regularization Method]. Moscow, Moscow Univ., 1999.
4. Osipov Yu.S., Kryazhimskii A.V., Maksimov V.I. Metody dinamicheskogo vosstanovleniya vkhodov upravlyaemykh sistem [Methods of Dynamic Reconstruction of Inputs of Controlled Systems]. Yekaterinburg, UrB RAS Pub., 2011.
5. Korotkii A.I. Inverse Problems of the Dynamics of Controllable Systems with Distributed Parameters. Russian Mathematics (Izvestiya VUZ. Matematika), 1995, vol. 39, no. 11, pp. 94-115.
6. Korotkii A.I. Pryamye i obratnye zadachi upravlyaemykh sistem s raspredelennymi parametrami [Direct and Inverse Problems of Controlled Systems with Distributed Parameters]. Doctoral Dissertation in Physics and Mathematics, Yekaterinburg, 1993.
7. Korotkii M.A. The Reconstruction of Controls by Regularization Methods with Uneven Stabilizers. J. of Applied Mathematics and Mechanics, 2009, vol. 73, issue 1, pp. 26-35.
8. Tikhonov A.N., Arsenin V.Y. Solution of Ill-Posed Problems. N.Y., Wiley, 1977.
9. Ivanov V.K., Vasin V.V., Tanana V.P. Theory of Linear Ill-Posed Problems and its Applications. Utrecht, VSP, 2002.
10. Lavrent'ev M.M., Romanov V.G., Shishatskii S.P. Ill-Posed Problems of Mathematical Physics and Analysis. Providence, AMS, 1980.
11. Ladyzhenskaya O.A. The Boundary Value Problems of Mathematical Physics. Berlin, Heidelberg, N.Y., Springer-Verlag, 1985.
12. Ladyzhenskaya O.A., Uraltseva N.N. Linear and Quasilinear Elliptic Equations. N.Y., London, Academic Press, 1968.
13. Tikhonov A.N., Leonov A.S., Yagola A.G. Nonlinear Ill-Posed Problems. Vols 1 and 2. London, Chapman and Hall, 1998.