A gradient-based approach for identification of the zeroth-order coefficient 𝑝(𝑥) in the parabolic equation 𝑢𝑡 = (𝑘(𝑥)𝑢𝑥)𝑥 − 𝑝(𝑥)𝑢 from Dirichlet-type measured output

This paper is devoted to the inverse problem of identifying an unknown spacewise-dependent zeroth-order coefficient {p(x)} in the 1D diffusion equation {u_{t}=(k(x)u_{x})_{x}-p(x)u} from boundary Dirichlet measured output {f(t)\coloneq u(0,t)}, {t\in[0,T]}. Compactness and Lipschitz continuity of the input-output operator {\Phi[p]\coloneq u(x,t;p)|_{x=0^{+}}}, {\Phi[\,\cdot\,]\colon\mathcal{P}\subset H^{1}(0,l)\mapsto L^{2}(0,T)} are proved. Then an existence of a quasi-solution of the inverse problem is obtained. We prove Fréchet differentiability of the Tikhonov functional and derive an explicit gradient formula for the Fréchet gradient through the solutions of the direct and corresponding adjoint problems solutions. This allows to use gradient-type algorithms for the numerical solution of the considered inverse problem.

Feedback stabilization of one-dimensional parabolic systems related to formations

This paper is concerned with the problem of stabilizing one-dimensional parabolic systems related to formations by using finitedimensional controllers of a modal type. The parabolic system is described by a Sturm-Liouville operator, and the boundary condition is different from any of Dirichlet type, Neumann type, and Robin type, since it contains the time derivative of boundary values. In this paper, it is shown that the system is formulated as an evolution equation with unbounded output operator in a Hilbert space, and further that it is stabilized by using an RMF (residual mode filter)-based controller which is of finite-dimension. A numerical simulation result is also given to demonstrate the validity of the finite-dimensional controller

Stability of Switched Feedback Time-Varying Dynamic Systems Based on the Properties of the Gap Metric for Operators

The stabilization of dynamic switched control systems is focused on and based on an operator-based formulation. It is assumed that the controlled object and the controller are described by sequences of closed operator pairs(L,C)on a Hilbert spaceHof the input and output spaces and it is related to the existence of the inverse of the resulting input-output operator being admissible and bounded. The technical mechanism addressed to get the results is the appropriate use of the fact that closed operators being sufficiently close to bounded operators, in terms of the gap metric, are also bounded. That philosophy is followed for the operators describing the input-output relations in switched feedback control systems so as to guarantee the closed-loop stabilization.

Stochastic linearization of nonlinear point dissipative systems

Stochastic linearization produces a linear system with the same covariance kernel as the original nonlinear system. The method passes from factorization of finite-dimensional covariance kernels through convergence results to the final input/output operator representation of the linear system.

Brain function theories, EEG sources, and dynamic states

This commentary discusses three features of the general theoretical framework proposed by Nunez: (1) Functional concepts, such as computation and control, are not foundational. (2) A mismatch between the concept of subcortical input and EEG output is problematic for the input/output operator concept of cortical dynamics. (3) The concept of brain state is relatively static.