Stability Analysis of Optimal Trajectory for Nonlinear Optimal Control Problems

Due to the need for numerical calculation and mathematical modelling, this paper focuses on the stability of optimal trajectories for optimal control problems. )e basic ideas and techniques are based on the compactness of the optimal trajectory set and setvalued mapping theorem. )rough 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 righthand side functions of the state equations of optimal control can be approximated by other functions.


Introduction
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]. However, 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 then, the compactness of the optimal trajectories is discussed. Second, 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(·)) ∈ p ad [0, T] such that the cost functional J(y(·), u(·)) � h(y(T)) + T 0 f 0 (t, y(t))dt (1) reaches its minimum at (y(·), u(·)) for all (y(·), u(·)) ∈ P ad [0, T], where the admissible pair (y(·), u(·)) are solutions of the following nonlinear controlled system: Here, u(t) ∈ U[0, T] and P ad [0, T] is the set of all admissible pair, and the set of admissible controls is defined by where U is a metric space. 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.
ere exist L, C > 0 such that (H3) e function f: [0, T] × R n × U ⟼ R n satisfies the following Filippov-Roxin condition: e set f(t, y, U) � f(t, y, u) | u ∈ U is convex and closed for almost all t ∈ [0, T] and all y ∈ R n (H4) e function f 0 : [0, T] × R n ⟼ R is Borel measurable concerning t ∈ [0, T] and continuous concerning y ∈ R n , and h: R n ⟼ R is continuous.
Under the assumptions and from [8], equation (2) has a unique solution. Also, the solution of equation (2) depends continuously on the right-hand side function and is differentiable concerning initial data.
is paper is organized as follows. In part one, we construct the complete metric space of f satisfied the conditions of (H2-H3) and discuss the compactness of the feasible trajectories set. In part two, we consider the stability of optimal trajectory in sense of Baire category. In the last section, we give an example and conclusion.

Compactness of Optimal Trajectory
For the aim to consider the stability of optimal trajectory, first, we need some notions. Let Y � f | f satisfies the conditions of (H2) and (H3) . (5) then we know that space (Y, ρ) is a complete metric space easily. Set S(f) � y | y are optimal trajectories of optimal control problems · P) for each f ∈ Y .
en, the correspondence f ⟼ S(f) is a set-valued mapping; for convenience to discuss with the next, we ex- is the set of all feasible trajectories of optimal control problems (P) for each f ∈ Y, and we denote the set-valued mapping by S: Now, we consider the properties of feasible trajectories set Y S [0, T] ⊆ C([0, T]; R n ); we have the following theorem.

Theorem 1. Suppose assumptions (H1) and (H2) hold, then the feasible trajectories set Y S [0, T] is relatively compact in C([0, T]; R n ).
Proof. By the assumptions of (H1) and (H2), equation (2) has a unique solution for each f ∈ Y, and the solution is denoted by anks to the assumption (H2), it follows that Namely, is means that the feasible trajectories set Y S [0, T] is uniformly bounded.
Furthermore, we obtain Hence, at is, the feasible trajectories set Y S [0, T] is equicontinuous. From (7)-(11), by the Arzela-Ascoli theorem, we prove this theorem. Proof. Let y k (·) ⊆Y S [0, T] and y k (·) be the solution of the following equations: under the assumptions (H1)-(H4) and f k ∈ Y. Let y k (·) � y f k (·, y 0 , u k (·)) and assume that 2 Journal of Mathematics Next, we need to show that y(·) ∈ Y S [0, T]. Assume that where p > 1, and we have f ∈ Y. By the Mazur theorem, there exist convex combinations with α k By (H2) and (13), we have α k i f s, y k+i (s), u k+i (s) ds α k i f s, y(s), u k+i (s) ds. (16) Consider By the Filippov's Lemma, there exists u ∈ U[0, T] such that (17) is satisfied, namely, holds.

Stability of Optimal Trajectory
In this section, we discuss the stability of optimal trajectory based Baire category.
By assumption of (H4), we have

Theorem 4. Let (H1)-(H4) hold, and S(f) is closed for each
and by (H4), we see that For any fixed u ∈ U[0, T] in controlled system, we obtain From the abovementioned equations, for all u ∈ U[0, T], the inequality holds. Namely, y ∈ S(f) and S(f) is closed.
Since S(f) ⊂ Y S [0, T] is closed and Y S [0, T] is a compact set, we have the following theorem.  [5], and eorem 4.7 of [7], and therefore, we can obtain the theorem of stability. □ Theorem 7.
ere exists a dense residual subset W of Y, such that for any f ∈ W, S(f) is stable in the sense of Hausdorff metric. Remark 1. By eorem 7, if f ∈ Y, the optimal control problem can be approximated in sense of Baire category.

Conclusions
In this paper, under the condition of the lack of good properties of optimal control, we discuss the compactness of the feasible trajectories set when the right-hand side function of the controlled system is disturbed and give a suitable metric for the space of the right-hand side function. Combined with the given assumptions, the space is a complete metric space.
en, according to the compact upper semicontinuous set-valued mapping, the continuous dependence of the optimal trajectory on the right-hand function and the stability of the cost functional on the righthand function are obtained in the sense of Baire category.