In this paper the explicit (closed form) solutions to several application-motivated parabolic problems are presented. The boundary stabilization problem is converted to a problem of solving a specific linear hyperbolic partial differential equation (PDE). This PDE is then solved for several particular cases. Closed loop solutions to the original parabolic problem are also found explicitly. Outp...

A method is described for solving the inverse problem of determining the profile of an acoustic horn when time-domain reflectance (TDR) is known only at the entrance. The method involves recasting Webster's horn equation in terms of forward and backward propagating wave variables. An essential feature of this method is a requirement that the backward propagating wave be continuous at the wave-f...

This work is concerned with the numerical computation of null controls of minimal L∞-norm for the linear heat equation with a bounded potential. Both, the cases of internal and boundary (Dirichlet and Neumann) controls are considered. Dual arguments allow to reduce the search of controls to the unconstrained minimization of a conjugate function with respect to the initial condition of a backwar...

One of the most frequently used techniques to solve polynomial eigenvalue problems is linearization, in which the polynomial eigenvalue problem is turned into an equivalent linear eigenvalue problem with the same eigenvalues, and with easily recoverable eigenvectors. The eigenvalues and eigenvectors of the linearization are usually computed using a backward stable solver such as the QZ algorith...

This article is devoted to the explanation of the onset of localization and the formation of shear bands in high strain-rate plasticity of metals. We employ the Arrhenius constitutive model and show Hadamard instability for the linearized problem. For the nonlinear model, using an asymptotic procedure motivated by the theory of relaxation and the Chapman-Enskog expansion, we derive an effective...

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness of viscosity solutions growing at most like o(1+ |x|p) at infinity for such HJB equations and more generally for degenerate parabolic equations with a superli...

The long time numerical approximation of the parabolic p-Laplacian problem with a time-independent forcing term and sufficiently smooth initial data is studied. Convergence and stability results which are uniform for t ∈ [0,∞) are established in the L2, W 1,p norms for the backward Euler and the Crank–Nicholson schemes with the finite element method (FEM). This result extends the existing unifo...