Asymptotic expansion of a solution for one singularly perturbed optimal control problem with a convex integral quality index depends on slow variables and smooth control constraints

The paper deals with the problem of optimal control with a convex integral quality index depends on slow variables for a linear steady-state control system with a fast and slow variables in the class of piecewise continuous controls with a smooth control constraints x ε = A 11 x ε + A 12 y ε + B 1 u, εy ε = A 21 x ε + A 22 y ε + B 2 u, J ε u ≔φ x ε T + 0 T u(t) 2 dt→ min, t∈ 0, T , x ε0 = x 0 ,u ≤1, y ε0 = y 0 , where x ε ∈Rn , y ε ∈Rm , u∈Rr ; A ij , B i , i, j =1,2, - are constant matrices of the corresponding dimension, and φ(·) - is the strictly convex and cofinite function that is continuously differentiable in Rn in the sense of convex analysis. In the general case, Pontryagin’s maximum principle is a necessary and sufficient optimum condition for the optimization of a such a problem. The initial vector of the conjugate state l ε is the unique vector, thus determining the optimal control. It is proven that in the case of a finite number of control switching points, the asymptotics of the vector l ε has the character of a power series.