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2163-2480

## Evolution Equations & Control Theory

December 2018 , Volume 7 , Issue 4

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**Abstract:**

We prove exact rate of decay for solutions to a class of second order ordinary differential equations with degenerate potentials, in particular, for potential functions that grow as different powers in different directions in a neigborhood of zero. As a tool we derive some decay estimates for scalar second order equations with non-autonomous damping.

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**Abstract:**

The main purpose of this work is to prove a new variant of Mehrenberger's inequality. Subsequently, we apply it to establish several observability estimates for the wave equation subject to Ventcel dynamic condition.

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**Abstract:**

In the present paper we derive the existence and uniqueness of the solution for the optimal control problem governed by the stochastic FitzHugh-Nagumo equation with recovery variable. Since the drift coefficient is characterized by a cubic non-linearity, standard techniques cannot be applied, instead we exploit the Ekeland's variational principle.

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**Abstract:**

Some multi-agent stochastic differential games described by a stochastic linear system driven by a Brownian motion and having an exponential quadratic payoff for the agents are formulated and solved. The agents have either complete observations or partial observations of the system state. The agents act independently of one another and the explicit optimal feedback control strategies form a Nash equilibrium. In the partially observed problem the observations are the same for all agents which occurs in broadcast situations. The optimal control strategies and optimal payoffs are given explicitly. The method of solution for both problems does not require solving either Hamilton-Jacobi-Isaacs equations or backward stochastic differential equations.

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**Abstract:**

In this paper, we study the effect of an internal or boundary time-delay on the stabilization of a moving string. The models adopted here are nonlinear and of "Kirchhoff" type. The well-posedness of the systems is proven by means of the Faedo-Galerkin method. In both cases, we prove that the solution of the system approaches the equilibrium in an exponential manner in the energy norm. To this end we request that the delayed term be dominated by the damping term. This is established through the multiplier technique.

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**Abstract:**

This paper analyzes the time-dependence and backward controllability of pullback attractors for the trajectory space generated by a non-autonomous evolution equation without uniqueness. A pullback trajectory attractor is called *backward controllable* if the norm of its union over the past is controlled by a continuous function, and *backward compact* if it is backward controllable and pre-compact in the past on the underlying space. We then establish two existence theorems for such a backward compact trajectory attractor, which leads to the existence of a pullback attractor with the backward compactness and backward boundedness in two original phase spaces respectively. An essential criterion is the existence of an increasing, compact and absorbing brochette. Applying to the non-autonomous Jeffreys-Oldroyd equations with a backward controllable force, we obtain a backward compact trajectory attractor, and also a pullback attractor with backward compactness in the negative-exponent Sobolev space and backward boundedness in the Lebesgue space.

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**Abstract:**

A cantilevered piezoelectric smart composite beam, consisting of perfectly bonded elastic, viscoelastic and piezoelectric layers, is considered. The piezoelectric layer is actuated by a voltage source. Both fully dynamic and electrostatic approaches, based on Maxwell's equations, are used to model the piezoelectric layer. We obtain (ⅰ) fully-dynamic and electrostatic Rao-Nakra type models by assuming that the viscoelastic layer has a negligible weight and stiffness, (ⅱ) fully-dynamic and electrostatic Mead-Marcus type models by neglecting the in-plane and rotational inertia terms. Each model is a perturbation of the corresponding classical smart composite beam model. These models are written in the state-space form, the existence and uniqueness of solutions are obtained in appropriate Hilbert spaces. Next, the stabilization problem for each closed-loop system, with a thorough analysis, is investigated for the natural

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**Abstract:**

In this paper, we propose a modified quasi-boundary value method to solve an inverse source problem for a time fractional diffusion equation. Under some boundedness assumption, the corresponding convergence rate estimates are derived by using an a priori and an a posteriori regularization parameter choice rules, respectively. Based on the superposition principle, we propose a direct inversion algorithm in a parallel manner.

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