Analysis of attainability sets of bilinear control systems

Ekimov, A. V.; Balykina, Yu. Е.; Svirkin, Michail V.

doi:10.1063/1.4992357

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“…The third inequality S nχ[ X ] is obvious. This follows from Property 1 of the characteristic number of a matrix (see (2)). In the same way, inequalities (14) can be proved using (16).…”

Section: Property 4 (The Relation Between the Characteristic Number Omentioning

confidence: 99%

“…From this, we get two left-hand side inequalities in (14). The last inequality S nχ[ X ] follows from Property 1 on the characteristic number of a matrix (see (2)). This proves Lemma 1.…”

Section: Property 4 (The Relation Between the Characteristic Number Omentioning

confidence: 99%

“…In [1] A. M. Lyapunov presented the fundamental concepts of stability theory and outlined two approaches to the problem of the stability of motion. In practice, the second method of Lyapunov is of considerable current use [2][3][4][5][6][7]. Lyapunov's first method is also widely used for the stability analysis of both linear and nonlinear systems that may be time-invariant or time-varying [8].…”

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Lyapunov’s first method: estimates of characteristic numbers of functional matrices

Ermolin¹,

Vlasova²

2019

Vestnik SPbSU. Applied Mathematics. Computer Science. Control P

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This paper contains the development of theoretical fundamentals of the first method of Lyapunov. We analyze the relations between characteristic numbers of functional matrices, their rows, and columns. We consider Lyapunov's results obtained to evaluate and calculate characteristic numbers for products of scalar functions and prove a theorem on the generalization of these results to the products of matrices. This theorem states necessary and sufficient conditions for the existence of rigorous estimates for characteristic numbers of matrix products. Also, we prove a theorem that establishes a relationship between the characteristic number of a square non-singular matrix and the characteristic number of its inverse matrix, and the determinant. The stated relations and properties of the characteristic numbers of square matrices we reformulate in terms of the Lyapunov exponents. Examples of matrices illustrate the proved theorems.

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“…The third inequality S nχ[ X ] is obvious. This follows from Property 1 of the characteristic number of a matrix (see (2)). In the same way, inequalities (14) can be proved using (16).…”

Section: Property 4 (The Relation Between the Characteristic Number Omentioning

confidence: 99%

“…From this, we get two left-hand side inequalities in (14). The last inequality S nχ[ X ] follows from Property 1 on the characteristic number of a matrix (see (2)). This proves Lemma 1.…”

Section: Property 4 (The Relation Between the Characteristic Number Omentioning

confidence: 99%

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