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2163-2480

## Evolution Equations & Control Theory

September 2018 , Volume 7 , Issue 3

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**Abstract:**

We study the wave equation on an unbounded network of

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**Abstract:**

In a Hilbert space

where

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**Abstract:**

We consider the regularized Tikhonov-like dynamical equilibrium problem: find

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**Abstract:**

In this paper we study the exact boundary controllability for the following Boussinesq equation with variable physical parameters:

where

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**Abstract:**

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models and their existence theory along with numerical simulations have been studied earlier. One of the most suitable shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in [

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**Abstract:**

In this paper, we consider the Stokes equations in a two-dimensional channel with periodic conditions in the direction of the channel. We establish null controllability of this system using a boundary control which acts on the normal component of the velocity only. We show null controllability of the system, subject to a constraint of zero average, by proving an observability inequality with the help of a Müntz-Szász Theorem.

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**Abstract:**

In this paper, we establish the Carleman estimates for forward and backward stochastic fourth order Schrödinger equations, on basis of which, we can obtain the observability, unique continuation property and the exact controllability for the forward and backward stochastic fourth order Schrödinger equations.

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**Abstract:**

The present paper studies a new class of problems of optimal control theory with state constraints and second order delay-discrete (DSIs) and delay-differential inclusions (DFIs). The basic approach to solving this problem is based on the discretization method. Thus under the "regularity condition the necessary and sufficient conditions of optimality for problems with second order delay-discrete and delay-approximate DSIs are investigated. Then by using discrete approximations as a vehicle, in the forms of Euler-Lagrange and Hamiltonian type inclusions the sufficient conditions of optimality for delay-DFIs, including the peculiar transversality ones, are proved. Here our main idea is the use of equivalence relations for subdifferentials of Hamiltonian functions and locally adjoint mappings (LAMs), which allow us to make a bridge between the basic optimality conditions of second order delay-DSIs and delay-discrete-approximate problems. In particular, applications of these results to the second order semilinear optimal control problem are illustrated as well as the optimality conditions for non-delayed problems are derived.

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