Hybrid discrete-continuous control systems: I. Representation of solutions

Abstract

We consider the Cauchy problem for linear control hybrid discrete-continuous systems in symmetric and normal forms. We obtain a solution of the problem in the form of definite integrals, where the solutions of special adjoint systems are used as the kernels of integral transform; this generalizes the representation of solutions by the Cauchy formula for ordinary linear dynamical systems to hybrid discrete-continuous systems. We discuss some other approaches to finding solutions of hybrid discrete-continuous systems. The results are used for deriving criteria for the relative controllability of considered systems and are refined in the time-independent case. We consider an example illustrating the obtained results.

On optimal fuzzy best proximity coincidence points of fuzzy order preserving proximal Ψ(σ, α)-lower-bounding asymptotically contractive mappings in non-Archimedean fuzzy metric spaces

“… be an ordered set and a non-Archimedean fuzzy metric space (Vetro and Salimi 2013 ; Chauan et al 2013b ; Abbas et al 2015 ) endowed with a fuzzy metric ; , , where is the metric of the Banach space , being also a complete metric space , the distance being identified with the norm, where: is a numerable set of switching laws for some with parameterizations (see Remark 22), is defined by where ; is a fuzzy order preserving strong proximal -lower-bounding mapping which, together with the non-contractive invertible mapping , describes the solution of the positive discrete n th dimensional linear system as follows: for any initial condition , v n = Q n x n ; quantifies the unmodeled dynamics (De la Sen et al 2010 ; De la Sen 1986 ; Marchenko 2015a , b ), and ; . This implies that: provided that G n is nonsingular; .…”

“… is a fuzzy order preserving strong proximal -lower-bounding mapping which, together with the non-contractive invertible mapping , describes the solution of the positive discrete n th dimensional linear system as follows: for any initial condition , v n = Q n x n ; quantifies the unmodeled dynamics (De la Sen et al 2010 ; De la Sen 1986 ; Marchenko 2015a , b ), and ; . This implies that: provided that G n is nonsingular; .…”

“… for any initial condition , v n = Q n x n ; quantifies the unmodeled dynamics (De la Sen et al 2010 ; De la Sen 1986 ; Marchenko 2015a , b ), and ; . This implies that: provided that G n is nonsingular; .…”

“…Fixed point theory is also relevant to the stability properties of some iterative schemes or that of dynamic systems (De la Sen and Karapinar 2015 ; De la Sen and Ibeas 2014 , 2015 ; De la Sen et al 2015 ), as an alternative mathematical tool to other classical technique like, for instance, the analysis of Lyapunov stability or hyperstability. See, for instance Se la Sen et al ( 2010 , 2015 ), De la Sen ( 1986 ), Marchenko ( 2015a , b ). It can be pointed out that recent work in fuzzy metric spaces and probabilistic metric spaces can be found in Grabiec ( 1983 ), Cho et al ( 1998 ) and Rashid et al ( 2013a , b , c , 2015 ), respectively.…”

On Optimal Fuzzy Best Proximity Coincidence Points of Proximal Contractions Involving Cyclic Mappings in Non-Archimedean Fuzzy Metric Spaces

“…Fixed Point Theory is also relevant to the stability properties of some iterative schemes of that of dynamic systems [29][30][31][32][33], as an alternative tool to other classical techniques like Lyapunov stability. (See, for instance, [33][34][35][36][37].) There are also abundant studies on all such topics in classical metric spaces and Banach spaces, either in the fuzzy formalism or not necessarily under the fuzzy formalism, including a lot of research on contractive and non-expansive mappings, self-mappings and, in particular, cyclic proximal mappings.…”