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Topics from Linear Algebra
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
Obviously, χ(λ) $ \chi (\lambda ) $ has the leading term λn $ \lambda^{n} $ . Any solution of the characteristic equation χ(λ)=0 $ \chi (\lambda ) = 0 $ is said to be an eigenvalue of the matrix A. The set of all eigenvalues is called the spectrum of the matrix A, denoted by σ(A) .
Controlling consensus in networks with symmetries
Published in International Journal of Control, 2022
Francesco Lo Iudice, Anna Di Meglio, Fabio Della Rossa, Francesco Sorrentino
Having dealt with controlling the dynamics along the group consensus subspace, we can now turn to stabilising the dynamics transverse to this subspace. To this aim, note that the spectrum of the matrix in (22) is composed of the following set of eigenvalues with the geometric multiplicity of the null eigenvalue being equal to 3, and the other two eigenvalues being simple. Hence, in order to apply Algorithm 1, we must first consider that , with , and . Then, setting i = 1, and as D is initialised as the empty matrix, then as Δ is the empty set and we can enter the inner while loop. The three vectors spanning are the last three rows of the matrix that brings the system in the IRR-coordinates, namely
Adjoint-based SQP method with block-wise quasi-Newton Jacobian updates for nonlinear optimal control
Published in Optimization Methods and Software, 2021
Pedro Hespanhol, Rien Quirynen
A proof for Theorem 3.7 can be found in [16,38], based on nonlinear systems theory. Using this result, let us define the linear contraction rate for a Gauss-Newton method with exact Jacobian information at the local solution point of the KKT conditions. In what follows, we show that the local contraction rate for the block-TR1 Gauss-Newton SQP method in the limit for , coincides with the exact Jacobian based linear convergence rate in (39). The following result states that the spectrum of iteration matrix at the solution point coincides with the spectrum of iteration matrix , using the notation to denote the spectrum, i.e. the set of eigenvalues for a matrix P.
Energy estimates and model order reduction for stochastic bilinear systems
Published in International Journal of Control, 2020
We introduce a reachability Gramian P as a positive definite solution to An inequality is considered in (2), since the existence of a positive definite solution is not ensured when having an equality. The existence of a solution to (2) goes back to Damm and Benner (2014), Redmann (2018b) and is given if which we assume to hold throughout the remainder of the paper. Here, denotes the spectrum of a matrix. Condition (3) is called mean square asymptotic stability (Damm, 2004; Khasminskii, 1980; Redmann, 2018b). It means that if the control components in the bilinear term of (1a) would truly be white noise, then the second moment of the solution would tend to zero in the uncontrolled setting (B=0) if .