Abstract
Due to the need for numerical calculation and mathematical modelling, this paper focuses on the stability of optimal trajectories for optimal control problems. The basic ideas and techniques are based on the compactness of the optimal trajectory set and set-valued mapping theorem. Through lack of optimal control stability, the result of generic stability for optimal trajectories is obtained under the perturbations of the right-hand side functions of the state equations; in the sense of Baire category, the right-hand side functions of the state equations of optimal control can be approximated by other functions.
Highlights
Due to the need for numerical calculation and mathematical modelling, we will consider the influence of optimal trajectory with the changing right-hand side functions of the state equations on the approximate functions. ere are many experts who have done much work on the stability of the optimal control problem [1,2,3,4], and especially in the case of the disturbance of the right-hand side function, the stability of the optimal control is discussed [5,6,7]
In the actual problem, the real decisive factor for the optimal control problem is the stability of the optimal trajectory, and sometimes, the optimal control does not necessarily converge; that is, when the trajectory converges, the corresponding control does not necessarily converge. erefore, this paper discusses the stability in these cases, focusing on two problems: First, the compactness of the feasible trajectories corresponding to the perturbations of right-hand side functions of the state equations is discussed, and the compactness of the optimal trajectories is discussed
In order to discuss the optimal control problem (P), let us start with some basic assumptions: (H1) e terminal time T > 0 is fixed, and the metric space U is compact
Summary
Due to the need for numerical calculation and mathematical modelling, we will consider the influence of optimal trajectory with the changing right-hand side functions of the state equations on the approximate functions. ere are many experts who have done much work on the stability of the optimal control problem [1,2,3,4], and especially in the case of the disturbance of the right-hand side function, the stability of the optimal control is discussed [5,6,7]. Erefore, this paper discusses the stability in these cases, focusing on two problems: First, the compactness of the feasible trajectories corresponding to the perturbations of right-hand side functions of the state equations is discussed, and the compactness of the optimal trajectories is discussed. The stability of the optimal trajectories set corresponding to the perturbations of the right-hand side functions is discussed. Consider Bolza problem, Problem P, as follows: Find an optimal pair (y(·), u(·)) ∈ pad[0, T] such that the cost functional. The solution of equation (2) depends continuously on the right-hand side function and is differentiable concerning initial data. We construct the complete metric space of f satisfied the conditions of (H2-H3) and discuss the compactness of the feasible trajectories set.
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