Differential Stability of a Class of Convex Optimal Control Problems

Abstract

A parametric constrained convex optimal control problem, where the initial state is perturbed and the linear state equation contains a noise, is considered in this paper. Formulas for computing the subdifferential and the singular subdifferential of the optimal value function at a given parameter are obtained by means of some recent results on differential stability in mathematical programming. The computation procedures and illustrative examples are presented.

Painlevé-Kuratowski convergences of solutions to nonlinear multiobjective optimal control problems

“…Let x (1) , x (2) , u, α 0 be the concentration of blood glucose, the net hormonal concentration, the insulin injection level and the desired constant glucose level, respectively. The goal of this model is to find the insulin injection level which minimizes the cost of the treatment and the difference between x 1 and α 0 .…”

“…Because φ is linear in the second and the third components and U is convex, we obtain that K(φ) is convex. Setting f (t, y, v) = f (1) (y), f (2) (v) , where f (1) (y) = (y (1) − α 0 ) 2 and f (2) (v) = v 2 for all (t, y, v) ∈ [t 0 , t 1 ] × ρB R 2 × Ω, y = (y (1) , y (2) ). Then, the functions f (1) and f (2) are strictly convex on convex subsets ρB R 2 and Ω, respectively.…”

“…Setting f (t, y, v) = f (1) (y), f (2) (v) , where f (1) (y) = (y (1) − α 0 ) 2 and f (2) (v) = v 2 for all (t, y, v) ∈ [t 0 , t 1 ] × ρB R 2 × Ω, y = (y (1) , y (2) ). Then, the functions f (1) and f (2) are strictly convex on convex subsets ρB R 2 and Ω, respectively. For all ẑ = (x, û), z = (x, ū) ∈ K(φ), ẑ = z, and for all λ ∈ (0, 1), we have…”

“…Fortunately, the assumption (H3) is still valid in this case. The interpretation of the assumption (H3) in this context can be understood as follows: The injection level of insulin u has an impact on the net hormonal concentration x (2) in the body. Subsequently, the concentration of blood glucose x (1) is generated based on the value of x (2) .…”

“…In [24], the authors reduced a parametric optimal control problem under linear state equations to a mathematical convex programming problem, then, by using the Fréchet upper subdifferential of the objective function and the existence of a local upper Lipschitzian selection of the solution map, the authors obtained formulas for computing the subdifferential of the optimal value function to such problem. Then, the techniques and results in this paper have been improved by [2]. For the convergence conditions, by using assumptions related to the uniform boundedness of the functions of state equations and equicontinuity of the control set, [29] formulated convergence conditions of solutions to nonlinear optimal control problems with perturbed state equations via the concept of essential solutions.…”