Asymptotics of Solutions of Linear Singularly Perturbed Optimal Control Problems with a Convex Integral Performance Index and a Cheap Control

Abstract

We consider an optimal control problem for a linear system with constant coefficients with an integral convex performance index containing a small parameter multiplying the integral term in the class of piecewise continuous controls with smooth geometric constraints. Such problems are called cheap control problems. It is shown that the limit problem will be a problem with a terminal performance index. It is established that if the terminal term of the performance index is a convex (strictly convex) and continuously differentiable function, then the performance functional in the limit problem has similar properties. It is proved that, in the general case, convergence with respect to the performance functional is valid, and under the condition of strict convexity of the terminal term of the performance index in the original problem, convergence to the minimum point of the terminal summand of the performance index in the limit problem is valid. The limit of the defining vector in the original problem is found as the small parameter tends to zero. In particular, it is shown that the first component of the defining vector in the original problem converges to the defining vector in the limit problem. The problems of controlling a point of low mass in a medium with and without resistance with a terminal part depending on both slow and fast variables are considered in detail, and complete asymptotic expansions of the defining vectors in these problems are constructed.