Singularly perturbed control problems in the case of the stability of the spectrum of the matrix of an optimal system

Abstract

The paper considers a singularly perturbed control problem with a quadratic quality functional. Such problems in their standard formulation under known spectrum restrictions (the points of the spectrum of the optimal system are not purely imaginary and are located symmetrically with respect to the imaginary axis) were previously considered using the Vasilyeva - Butuzov method of boundary functions. If at least one of the points of the spectrum for some values of the independent variable falls on the imaginary axis, the boundary functions method does not work. It is precisely this situation with the assumption of purely imaginary points of the spectrum that is investigated in this paper. In this case, you have to develop a different approach based on the ideas of the regularization method S.A. Lomov. It should also be noted that in the control problems considered earlier, the cost functional either did not depend on a small parameter at all, or allowed a smooth dependence on the parameter. In this paper, an irregular dependence on a small parameter is allowed, in particular, the presence in them of a rapidly changing damping function in the form of an exponential factor under the integral sign. In this case, the spectrum behavior of the optimal system depends on the damping coefficient, which (under certain conditions) can shift the spectrum in one direction or another in the complex plane. In this case, a situation may arise when some points of the spectrum for individual values (or even on a certain continuum set) of an independent variable can become purely imaginary. This situation is not amenable to investigation by the previously mentioned Vasilieva - Butuzov method of boundary functions. However, it can be fully studied using the regularization method S.A. Lomov, the algorithm of which is applied to the considered control problem in the present paper. The presentation of this method begins with a brief description of the maximum principle of L.S. Pontryagin for the classical optimal control problem, which then, along with other ideas, is used to justify the results in the considered control problem.