Forward Euler Solutions and Weakly Invariant Time‐Delayed Systems

Abstract

This paper presents a necessary and sufficient condition for the weak invariance property of a time-delayed system parametrized by a differential inclusion. The aforementioned condition generalizes the well-known Hamilton-Jacobi inequality that characterizes weakly invariant systems in the nondelay setting. The forward Euler approximation scheme used in the theory of discontinuous differential equations is extended to the time-delayed context by incorporating the delay and tail functions featuring the dynamics. Accordingly, an existence theorem of weakly invariant trajectories is established under the extended forward Euler approach.

Criteria on Contraction and Incremental Stability of Dynamical Systems with Time Delay

“…In Assumption 2, 'Upper semicontinuity' for the set-valued mapping F is defined as follows. holds for any (x, y 1 , ..., y m , r) ∈ R n×(m+1) × R. With Gronwall inequality [26], it can be easily seem equivalent to: there exists Θ > 0 such that sup{|v| : v ∈ F(x, y 1 , ..., y m , r)} ≤ Θ holds for any (x, y 1 , ..., y m , r) ∈ R n×(m+1) × R.…”

“…Here we take an example of Hopfield system with time delay for illustration. The system is formulated as follows, ẋ(t) = D(t)x(t) + T(t)g(x(t)) + σ(t)S(t)u(x τ (t))) + J(t) (26) where x = [x 1 , x 2 ] is the state vector, σ(t) is the switched function with respect to time t, takes value between 0 and 1. The parameter matrix are…”

“…Here define the norm with subscript σ(t) as |x| 0 = 0. , where N(t) = j, t ∈ [t j−1 , t j ), α j = max{1/4, (1/2) − (1/2) j }, t j = (T 0 /2) • j, j = 1, 2, ..., such that in the continuous region of the right-hand of system (26), it holds that α N(t) + ν 0 (−D(t) + T(t)) < 0 α N(t) + ν 1 (−D(t) + T(t)) + exp α N(t) t exp α N(t−τ(t)) (t − τ(t)) S(t) 1 < 0.5 − 1 + 0.25 < 0…”

“…Together with (24), where K = 5, M = 2, conditon (24) holds for σ(t) = 0, 1. And for t > 10, we have Figure 3 shows the dynamical trajectories of two of the solutions with initial function defined as x 0 (•) and y 0 (•) for system (26). And Figure 4 shows the errors between the two dynamical trajectories of their segments.…”

Criteria on Exponential Incremental Stability of Dynamical Systems with Time Delay

“…. , y m , r) ∈ R n×(m+1) × R. With Gronwall inequality [41], it can easily be seen as equivalent to: there exists Θ > 0 such that sup{|v| : v ∈ F(x, y 1 , . .…”