Controllability of non-autonomous nonlinear differential system with non-instantaneous impulses

Approximation theorems for controllability problem governed by fractional differential equation

“…where Suppose that the functional J γ has a unique minimizer in the space Y, which is unique critical point of functional J γ , then the problem (21) has an optimal pair (u m , z m ), which converges to (u, z) with u m converges weakly to u and z m → z, where (u, z) is optimal pair of the problem (19).…”

“…Normally, the goal of a controllability problem is to steers or transfers the state variable from any initial state to any given target state by selecting a suitable control among available options. For more recent works on controllability, one may see [6,10,14,17,18,19,20,25,29,31]. Klamka [14] presented a survey paper on the controllability of dynamical systems.…”

Optimal controls for second‐order stochastic differential equations driven by mixed‐fractional Brownian motion with impulses

“…The second type is non‐instantaneous impulsive differential equations, ie, the impulsive action starts at fixed points and remains active on a finite time interval. For more recent works on dynamic systems with impulses, we refer to previous works . Hernández and O'Regan studied mild and classical solutions for a new class of non‐instantaneous impulsive differential equation of the form …”

“…For more recent works on dynamic systems with impulses, we refer to previous works. [11][12][13][14][15][16][17][18][19] Hernández and O'Regan 11 studied mild and classical solutions for a new class of non-instantaneous impulsive differential equation of the form ⎧ ⎪ ⎨ ⎪ ⎩ ′ (t) =  (t) + (t, (t)), t ∈ (e , t +1 ], = 0, 1, … , , (t) =  (t, (t)), t ∈ (t , e ], = 1, 2, … , , (0) = 0 .…”

Existence of solutions of non-autonomous fractional differential equations with integral impulse condition

“…In the same year, Chen et al [9] explored the existence of mild solutions for the initial value problem to a new class of abstract evolution equations with non-instantaneous impulses on ordered Banach spaces by using a perturbation technique and by dropping the compactness condition on the semigroup. Malik et al [28] used the Rothe's fixed point theorem to study the controllability of non-autonomous nonlinear differential system with non-instantaneous impulses in the space R n . By using Krasnoselskii's fixed point theorem, Wang et al [43] formed a set of sufficient conditions for the existence and stability for a class of impulsive non-autonomous differential equations.…”