Stability and solvability for a class of optimal control problems described by non-instantaneous impulsive differential equations

Abstract

In this paper, we investigate the existence and stability of solutions for a class of optimal control problems with 1-mean equicontinuous controls, and the corresponding state equation is described by non-instantaneous impulsive differential equations. The existence theorem is obtained by the method of minimizing sequence, and the stability results are established by using the related conclusions of set-valued mappings in a suitable metric space. An example with the measurable admissible control set, in which the controls are not continuous, is given in the end.

Asymptotic stability of non‐instantaneous impulsive systems and T‐S fuzzy non‐instantaneous impulsive control for nonlinear systems

“…where i = 1, 2, … , p, k ∈ ℕ + , 𝜐(t ) ∶  2 → ℝ + , 𝜐(t ) is continuous and bounded in (t k , s k ], 𝜐(t + k ) = 1. Therefore, making the original system (14) stable is equivalent to designing a noninstantaneous impulsive control scheme to stabilize the system (15). Remark 10.…”

“…Let a i be the largest eigenvalue of M T i + M i , a = max{a i }, 𝜉 i be the largest eigenvalue of N T i N i , 𝜉 = max{𝜉 i }, i = 1, 2, … , p, 𝜐(t ) is nonnegative and piecewise continuous in (t k , s k ], k ∈ ℕ + . The system (15) is semi-globally asymptotically stable if there exists a constant ℏ > 1 satisfies (i) a ≥ 0, (ii) 𝜐 2 (s k )𝜉e a(t k −s k−1 ) ≤ 1 ℏ . Remark 14.…”

“…Since then, the qualitative theory of non-instantaneous impulsive systems has developed. For instance, the existence and uniqueness of solutions for non-instantaneous impulsive systems via fixed point theorem can be found in [9][10][11]; achievements on periodic solutions via Poincáre operator, see [12][13][14]; the problem of controllability and time optimal control for various types of non-instantaneous impulsive systems have been widely studied in [15][16][17]. It is noticed that achievements on non-instantaneous impulsive differential systems are mainly reflected in theory and still relatively scarce in applications.…”