ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2014. Vol. 8

Identity of Optimality Conditions for Elastic Vibrations in Different Auxiliary Interpretations of Wave Problem

N. V. Kurganova, E. A. Lutkovskaya, V. A. Terletsky

In this paper an optimal control problem is considered, where the controlled process is described by a non-linear wave equation. The state vector consists of the solution for the non-linear wave equation and its first-order partial derivatives with respect to the independent variables. The vector of controls includes distributed and boundary controls. The statement of the problem allows any combinations of boundary conditions of the first, second, and third kinds on both left and right borders of the domain. For the initial optimal control problem two types of equivalent auxiliary optimal control problems are constructed. They differ by the ways of describing the controlled process. The first approach describes the controlled process by a hyperbolic system of four differential equations of the first order. The second approach uses one differential equation of the second order and two DE of the first order, identical to the ones used at the first equivalent problem. It is necessary to reduce the initial wave equation to the corresponding equivalent systems in order not only to construct a convenient concept of a generalized solution, but to obtain the necessary optimality conditions as well. In spite of formally different Pontryagin’s functions used for the two equivalent control problems, it was proved that the corresponding conjugate problems take such forms that for the same set of controls the values of the Pontryagin’s functions coincide throughout the domain. This property justifies identity of optimality conditions in forms of variational and Pontryagin’s maximum principles, obtained for the both equivalent optimal control problems.

optimal control, variational and Pontryagin’s maximum principles, wave equation, conjugate problem
517.977.56 MSC 49K20

1. Petrovsky I. G. Lectures on Partial Differential Equations. Dover Publ., New York, 1991.

2. Rozhdestvenskii B. L. and Yanenko N. N. Systems of Quasilinear Equations and Their Applications to Gas Dynamics. Amer. Math. Soc., New York, 1983.

3. Terletsky V.A., Lutkovskaya E.A. Variational Maximum Principle in the Problem of Optimal Control of Nonlinear Wave Processes (in Russian) Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya: Matematika [The Bulletin of Irkutsk State University. Series: Mathematics], 2010, vol. 3, no. 3, pp.105-117.

4. Terletskii V. A. and Lutkovskaya E. A. Generalized Solution of a Nonlinear Wave Equation with Nonlinear Boundary Conditions of the First, Second, and Third Kinds. Differential Equations, 2009, Vol. 45, No. 3, pp. 416–428. PleiadesPublishing, Ltd., 2009.

5. Tikhonov A.N., Samarskii A.A., Equations of Mathematical Physics. Courier Dover Publications, New York, 1990.

Full text (russian)