Abstract:In this paper we constructed an expansion of an asymptotic solution for singularly perturbed linear-quadratic optimal control with quadratic parameter problem. This formulation is based on direct substituting a postulated asymptotic expansion of boundary function type for the solution into the problem condition and on defining optimal control problems for finding asymptotic terms. The solutions of which form ( ) for the asymptotic solution, is proven. In physical mathematic problem we say the singularly perturbed linearly quadratic problems is "fast and slow " velocity , in this work we discussed the concept " The slowest velocity" by taken quadratic small parameter in the singularly perturbed linearly quadratic problems.