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Evolution Equations & Control Theory

December 2018 , Volume 7 , Issue 4

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Exact rate of decay for solutions to damped second order ODE's with a degenerate potential
Tomáš Bárta
2018, 7(4): 531-543 doi: 10.3934/eect.2018025 +[Abstract](152) +[HTML](55) +[PDF](416.56KB)

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.

Observability of wave equation with Ventcel dynamic condition
Imen Benabbas and Djamel Eddine Teniou
2018, 7(4): 545-570 doi: 10.3934/eect.2018026 +[Abstract](161) +[HTML](106) +[PDF](451.6KB)

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.

Optimal control for the stochastic FitzHugh-Nagumo model with recovery variable
Francesco Cordoni and Luca Di Persio
2018, 7(4): 571-585 doi: 10.3934/eect.2018027 +[Abstract](126) +[HTML](61) +[PDF](451.85KB)

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.

Some partially observed multi-agent linear exponential quadratic stochastic differential games
Tyrone E. Duncan
2018, 7(4): 587-597 doi: 10.3934/eect.2018028 +[Abstract](105) +[HTML](49) +[PDF](333.49KB)

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.

Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback
Abdelkarim Kelleche and Nasser-Eddine Tatar
2018, 7(4): 599-616 doi: 10.3934/eect.2018029 +[Abstract](127) +[HTML](70) +[PDF](414.34KB)

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.

Backward controllability of pullback trajectory attractors with applications to multi-valued Jeffreys-Oldroyd equations
Yangrong Li, Renhai Wang and Lianbing She
2018, 7(4): 617-637 doi: 10.3934/eect.2018030 +[Abstract](104) +[HTML](62) +[PDF](493.26KB)

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.

Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results
Ahmet Özkan Özer
2018, 7(4): 639-668 doi: 10.3934/eect.2018031 +[Abstract](117) +[HTML](56) +[PDF](1859.49KB)

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 \begin{document}$B^*-$\end{document}type state feedback controllers. The fully dynamic Rao-Nakra model with four state feedback controllers is shown to be not asymptotically stable for certain choices of material parameters whereas the electrostatic model is exponentially stable with only three state feedback controllers (by the spectral multipliers method). Similarly, the fully dynamic Mead-Marcus model lacks of asymptotic stability for certain solutions whereas the electrostatic model is exponentially stable by only one state feedback controller.

Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method
Zhousheng Ruan, Sen Zhang and Sican Xiong
2018, 7(4): 669-682 doi: 10.3934/eect.2018032 +[Abstract](167) +[HTML](55) +[PDF](541.78KB)

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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