Evolution Equations & Control Theory
December 2021 , Volume 10 , Issue 4
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In this paper, we study the blow – up of solutions to the semilinear Moore – Gibson – Thompson (MGT) equation with nonlinearity of derivative type
This paper deals with the solvability in the semilinear abstract evolution equation with countable time delays,
in a Banach space
We consider the control of semilinear stochastic partial differential equations (SPDEs) via deterministic controls. In the case of multiplicative noise, existence of optimal controls and necessary conditions for optimality are derived. In the case of additive noise, we obtain a representation for the gradient of the cost functional via adjoint calculus. The restriction to deterministic controls and additive noise avoids the necessity of introducing a backward SPDE. Based on this novel representation, we present a probabilistic nonlinear conjugate gradient descent method to approximate the optimal control, and apply our results to the stochastic Schlögl model. We also present some analysis in the case where the optimal control for the stochastic system differs from the optimal control for the deterministic system.
In this paper, an initial boundary value problem for a parabolic type Kirchhoff equation with time-dependent nonlinearity is considered. A new blow-up criterion for nonnegative initial energy is given and upper and lower bounds for the blow-up time are also derived. These results partially generalize some recent ones obtained by Han and Li in [Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(2018), 3283-3297].
In this article, we deal with the existence of S-asymptotically
In this paper, the rich feedback theory of regular linear systems in the Salamon-Weiss sense as well as some advanced tools in semigroup theory are used to formulate and solve control problems for network systems. In fact, we derive necessary and sufficient conditions for approximate controllability of such systems. These criteria, in some particular cases, are given by the well-known Kalman's controllability rank condition.
We will consider the full von Kármán thermoelastic system with free boundary conditions and dissipation imposed only on the in-plane displacement. It will be shown that the corresponding solutions are exponentially stable, though there is no mechanical dissipation on the vertical displacements. The main tools used are: (i) partial analyticity of the linearized semigroup and (ii) trace estimates which exploit the hidden regularity harvested from partial analyticity.
This article is concerned with Hadamard's well posedness of a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain
In this paper, we study an inverse problem for linear parabolic system with variable diffusion coefficients subject to dynamic boundary conditions. We prove a global Lipschitz stability for the inverse problem involving a simultaneous recovery of two source terms from a single measurement and interior observations, based on a recent Carleman estimate for such problems.
We study the backstepping stabilization of higher order linear and nonlinear Schrödinger equations on a finite interval, where the boundary feedback acts from the left Dirichlet boundary condition. The plant is stabilized with a prescribed rate of decay. The construction of the backstepping kernel is based on a challenging successive approximation analysis. This contrasts with the case of second order pdes. Second, we consider the case where the full state of the system cannot be measured at all times but some partial information such as measurements of a boundary trace are available. For this problem, we simultaneously construct an observer and the associated backstepping controller which is capable of stabilizing the original plant. Wellposedness and regularity results are provided for all pde models. Although the linear part of the model is similar to the KdV equation, the power type nonlinearity brings additional difficulties. We give two examples of boundary conditions and partial measurements. We also present numerical algorithms and simulations verifying our theoretical results to the fullest extent. Our numerical approach is novel in the sense that we solve the target systems first and obtain the solution to the feedback system by using the bounded invertibility of the backstepping transformation.
This article investigates the controllability for neutral stochastic delay functional integro-differential equations driven by a fractional Brownian motion, with Hurst parameter lesser than
The main objective of this paper is to study the optimal distributed control of the three dimensional non-autonomous primitive equations of large-scale ocean and atmosphere dynamics. We apply the well-posedness and regularity results of solutions for this system as well as some abstract results from the nonlinear functional analysis to establish the existence of optimal controls as well as the first-order necessary optimality condition for an associated optimal control problem in which a distributed control is applied to the temperature.
We consider a port-Hamiltonian system on an open spatial domain
We consider the 3D Navier-Stokes-Voigt equations in a bounded domain with unbounded variable delay. We study the stability of stationary solutions by the classical direct method, and by an appropriate Lyapunov functional. We also give a sufficient condition of parameters for the polynomial stability of the stationary solution in a special case of unbounded variable delay. Finally, when the condition for polynomial stability is not satisfied, we stabilize the stationary by using the finite Fourier modes and by internal feedback control with a support large enough.
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