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

We consider an optimal control problem for a linear system with constant coefficients withan integral convex performance index containing a small parameter multiplying the integral termin the class of piecewise continuous controls with smooth geometric constraints. Such problemsare called cheap control problems. It is shown that the limit problem will be a problem with aterminal performance index. It is established that if the terminal term of the performance index isa convex (strictly convex) and continuously differentiable function, then the performancefunctional 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 strictconvexity of the terminal term of the performance index in the original problem, convergence tothe minimum point of the terminal summand of the performance index in the limit problem isvalid. The limit of the defining vector in the original problem is found as the small parametertends to zero. In particular, it is shown that the first component of the defining vector in theoriginal problem converges to the defining vector in the limit problem. The problems ofcontrolling a point of low mass in a medium with and without resistance with a terminal partdepending on both slow and fast variables are considered in detail, and complete asymptoticexpansions of the defining vectors in these problems are constructed.