Approximate Controllability of Infinite Dimensional Systems of the n-th Order

Abstract

The objective of the article is to obtain general conditions for several types of controllability at once for an abstract differential equation of arbitrary order, instead of conditions for a fixed order equation. This innovative approach was possible owing to analyzing the n-th order linear system in the Frobenius form which generates a Jordan transition matrix of the Vandermonde form. We extensively used the fact that the knowledge of the inverse of a Jordan transition matrix enables us to directly verify the controllability by Chen's theorem. We used the explicit analytical form of the inverse Vandermonde matrix. This enabled us to obtain more general conditions for different types of controllability for infinite dimensional systems than the conditions existing in the literature so far. The methods introduced can be easily adapted to the analysis of other dynamic properties of the systems considered.

Recursive Matrix Calculation Paradigm by the Example of Structured Matrix

“…86-95). Control issues: investigating the so-called controllability [9] of the higher-order systems leads to the issue of inverting the classic Vandermonde matrix [10] (in the case of distinct zeros of the system characteristic polynomial) and the generalized Vandermonde matrix [11] (for systems with multiple characteristic polynomial zeros). As the examples of the higher-order models of the physical objects, we can mention Timoshenko's elastic beam equation [12] (fourth order) and Korteweg-de Vries's equation of waves on shallow water surfaces [13,14] (third, fifth, and seventh order).…”

Approximate controllability of infinite dimensional system with internal damping dependent on fractional powers of system operator

“…Controllability condition is derived using duality principle. An interesting approach to approximate controllability investigation may be found in [21], where a higher order infinite dimensional system is investigated. The investigation of controllability conditions is based on Frobenius form which generates a Jordan transition matrix of the Vandermonde form.…”

The $(P,Q)$ generalized anti-reflexive extremal rank solutions to a system of matrix equations

“…Obviously, if let P D Q D J n in Definition 1, then matrix A is the well-known centrosymmetric (or anti-centrosymmetric) matrix, which plays an important role in many areas (see, e.g., [6,13,[16][17][18][19]), and has been widely and extensively studied (see, e.g., [1,25,28]). Moreover, let P D Q, then matrix A is called generalized centrosymmetric (or anti-centrosymmetric) matrix [12,22].…”