# American Institute of Mathematical Sciences

September  2018, 7(3): 373-401. doi: 10.3934/eect.2018019

## Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems

 Laboratory LIBMA Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, 40000 Marrakech, Morroco

Received  February 2017 Revised  March 2018 Published  July 2018

We consider the regularized Tikhonov-like dynamical equilibrium problem: find $u: [0, +∞ [\to\mathcal H$ such that for a.e. $t \ge 0$ and every $y∈K$, $\langle \dot{u}(t), y-u(t)\rangle +F(u(t), y)+\varepsilon(t) \langle u(t), y-u(t)\rangle \ge 0$, where $F:K×K \to \mathbb{R}$ is a monotone bifunction, $K$ is a closed convex set in Hilbert space $\mathcal H$ and the control function $\varepsilon(t)$ is assumed to tend to 0 as $t \to +∞$. We first establish that the corresponding Cauchy problem admits a unique absolutely continuous solution. Under the hypothesis that $\int_{0}^{+∞} \varepsilon (t) dt <∞$, we obtain weak ergodic convergence of $u(t)$ to $x∈K$ solution of the following equilibrium problem $F(x, y) \ge 0, \;\forall y∈K$. If in addition the bifunction is assumed demipositive, we show weak convergence of $u(t)$ to the same solution. By using a slow control $\int_{0}^{+∞} \varepsilon (t) dt = ∞$ and assuming that the bifunction $F$ is 3-monotone, we show that the term $\varepsilon (t)u(t)$ asymptotically acts as a Tikhonov regularization, which forces all the trajectories to converge strongly towards the element of minimal norm of the closed convex set of equilibrium points of $F$. Also, in the case where $\varepsilon$ has a slow control property and $\int_{0}^{+∞}\vert \dot{\varepsilon} (t) \vert dt < +∞$, we show that the strong convergence property of $u(t)$ is satisfied. As applications, we propose a dynamical system to solve saddle-point problem and a neural dynamical model to handle a convex programming problem. In the last section, we propose two Tikhonov regularization methods for the proximal algorithm. We firstly use the prox-penalization algorithm $(ProxPA)$ by iteration $x_{n+1} = J^{F_n}_{λ_n}(x_n)$ where $F_n(x, y) = F(x, y)+\varepsilon_n \langle x, y-x\rangle$, and $\varepsilon_n$ is the Liapunov parameter; afterwards, we propose the descent-proximal (forward-backward) algorithm $(DProxA)$: $x_{n+1} = J^F_{λ_n} ((1 - λ_n\varepsilon_n)x_n)$. We provide low conditions that guarantee a strong convergence of these algorithms to least norm element of the set of equilibrium points.

Citation: Aicha Balhag, Zaki Chbani, Hassan Riahi. Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems. Evolution Equations & Control Theory, 2018, 7 (3) : 373-401. doi: 10.3934/eect.2018019
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