Nonconvex Differential Variational Inequality and State-Dependent Sweeping Process

Reduction of state dependent sweeping process to unconstrained differential inclusion

“…Other demonstrations have been proposed. Haddad [17] used a semi-implicit discritization and Azzam et al [3] employed a semi-reduction method. For the perturbed differential inclusion (1.2) when C(t, u) ≡ K (t) is independent of the state, we refer to [4,5,7,15].…”

Nonsmooth dynamics of generalized Nash games

“…In this section, we show that there exists a solution to the perturbed state-dependent sweeping process (3.6) connected with the GNEP. Our first existence result relies on recent progress on nonsmooth dynamical system [29,43,58]. Proof.…”

Nonconvex Sweeping Process with a Moving Set Depending on the State

“…It results from (14) and (15) that for any n ∈ N, any t ∈ [0, T ]\S, and for any y ∈ H , y, −u n (t) + z n (t) ≤ σ y, γ ∂d C(θ n (t),u n (θ n (t))) (u n (θ n (t))) (17) and y, z n (t) ≤ σ (y, G (δ n (t), u n (δ n (t)))) .…”

“…Further, for each n ∈ N and any t ∈ [0, T ]\S, from (16), we have y, ξ k (t) ≤ sup q≥n y, −u q (t) + z q (t) for all k ≥ n and y, ζ k (t) ≤ sup q≥n y, z q (t) for all k ≥ n, and taking the limit in both inequalities as k → +∞ gives through (17) and (18) y, −u(t) + z(t) ≤ sup q≥n y, −u q (t) + z q (t) ≤ sup q≥n σ y, γ ∂d C(θ q (t),u q (θ q (t))) u q θ q (t) -For each t ∈ [0, T ] and each u ∈ H , the sets C(t, u) are nonempty closed in H and r-prox-regular for some constant r > 0; -There are real constants L 1 > 0, L 2 ∈]0, 1[ such that, for all t, s ∈ [0, T ] and…”

“…Azzam et al [1] obtained, in finite dimensions, a solution for (II ), with a multimapping G, via a reduction to an unconstrained differential inclusion. In [17], assuming that the prox-regular sets C(t, x) are contained in a fixed compact set of H and using (without a fixed point theorem) the scheme u n 0 = u 0 , u n i+1 = Proj C(t n i+1 ,u n i ) (u n i − T 2 n g n i ) with g n i ∈ G(t n i , u n i ) (where t n i := i T 2 n , i = 0, . .…”