Controllability of partially prescribed matrices

• Autores: Gloria Cravo
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 60, Fasc. 3, 2009, págs. 335-348
• Idioma: español
• DOI: 10.1007/bf03191375
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• Resumen

  • LetF be an infinite field and letn,p 1,p 2,p 3 be positive integers such thatn =p 1 +p 2 +p 3. Let C_{1,2} \in F^{p_1 \times p_2 }, C_{1,3} \in F^{p_1 \times p_3 } and C_{2,1} \in F^{p_2 \times p_1 }. In this paper we show that appart from an exception, there always exist C_{1,1} \in F^{p_1 \times p_1 }, C_{2,2} \in F^{p_2 \times p_2 } and C_{2,3} \in F^{p_2 \times p_3 } such that the pair (A_1 , A_2 ) = \left( {\left[ {\begin{array}{*{20}c} {C_{1,1} } \\ {C_{2,1} } \\ \end{array} \begin{array}{*{20}c} {C_{1,2} } \\ {C_{2,2} } \\ \end{array} } \right],\left[ {\begin{array}{*{20}c} {C_{1,3} } \\ {C_{2,3} } \\ \end{array} } \right]} \right) is completely controllable. In other words, we study the possibility of the linear system _χ (t) =A _1χ(t) +A _2ζ(t) being completely controllable, whenC_1,2,C_1,3 andC_2,1 are prescribed and the other blocks are unknown. We also describe the possible characteristic polynomials of a partitioned matrix of the form C = \left[ {\begin{array}{*{20}c} {C_{1,1} } \\ {C_{2,1} } \\ {C_{3,1} } \\ \end{array} \begin{array}{*{20}c} {C_{1,2} } \\ {C_{2,2} } \\ {C_{3,2} } \\ \end{array} \begin{array}{*{20}c} {C_{1,3} } \\ {C_{2,3} } \\ {C_{3,3} } \\ \end{array} } \right] \in F^{n \times n} , whereC_1,1,C_2,2,C_3,3 are square submatrices (not necessarily with the same size), whenC_1,2,C_1,3 andC_2,1 are fixed and the other blocks vary.