In this chapter the problem of exponential stability of the zero state equilibrium of a class of jump linear differential equations on a real ordered Banach space is investigated. These equations are defined by a special class of jump operator valued functions. The results presented here could be viewed, in some sense, as an extension to the non-continuous case of the results developed in this chapter from Drăgan et al. in mathematical methods in robust control of linear stochastic systems (2013) [1]. The differential equations addressed in Drăgan et al. in mathematical methods in robust control of linear stochastic systems (2013) [1] (and named differential equations with positive evolution) are defined by a special class of strongly continuous operator valued functions. These differential equations are natural extensions to the time-varying case of the linear differential equations with constant coefficients on an ordered Banach space defined by a linear and bounded operator with positive semigroup. In the present chapter, discontinuities of the operator valued functions are allowed at some time instances called jump time instances. The jump linear differential equations with positive evolution studied in this chapter contain as special cases Lyapunov-type jump differential equations arising in a natural way in connection to the problem of exponential stability in the mean-square sense of a large class of jump stochastic linear differential equations. Properties of forward/backward jump operator valued functions on a Banach space are emphasized in Sect. 2.1. Section 2.2 is devoted to the characterization of the exponential stability and asymptotic stability of forward jump linear differential equations. In the main result of this section, the stability of the forward jump linear equations is equivalently formulated as the stability of a corresponding class of discrete-time linear equations (Theorem 2.1). The periodic case as well as the time-invariant case are also treated in this section. Sections 2.3 and 2.4 are devoted to the case of jump linear differential equations on a Hilbert space. The fundamental results obtained in these sections are then exploited in the last part of this chapter in order to obtain criteria for the exponential stability of a class of generalized Lyapunov linear differential equations with jumps on a Hilbert space as well as for the existence of some bounded solutions on the whole real semiaxis of the corresponding non-homogeneous generalized Lyapunov differential equations with jumps.

Stability of Jump Linear Differential Equations

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发表日期：

2025年

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Vasile Drăgan,
Samir Aberkane,
Ioan Lucian Popa

摘要

In this chapter the problem of exponential stability of the zero state equilibrium of a class of jump linear differential equations on a real ordered Banach space is investigated. These equations are defined by a special class of jump operator valued functions. The results presented here could be viewed, in some sense, as an extension to the non-continuous case of the results developed in this chapter from Drăgan et al. in mathematical methods in robust control of linear stochastic systems (2013) [1]. The differential equations addressed in Drăgan et al. in mathematical methods in robust control of linear stochastic systems (2013) [1] (and named differential equations with positive evolution) are defined by a special class of strongly continuous operator valued functions. These differential equations are natural extensions to the time-varying case of the linear differential equations with constant coefficients on an ordered Banach space defined by a linear and bounded operator with positive semigroup. In the present chapter, discontinuities of the operator valued functions are allowed at some time instances called jump time instances. The jump linear differential equations with positive evolution studied in this chapter contain as special cases Lyapunov-type jump differential equations arising in a natural way in connection to the problem of exponential stability in the mean-square sense of a large class of jump stochastic linear differential equations. Properties of forward/backward jump operator valued functions on a Banach space are emphasized in Sect. 2.1. Section 2.2 is devoted to the characterization of the exponential stability and asymptotic stability of forward jump linear differential equations. In the main result of this section, the stability of the forward jump linear equations is equivalently formulated as the stability of a corresponding class of discrete-time linear equations (Theorem 2.1). The periodic case as well as the time-invariant case are also treated in this section. Sections 2.3 and 2.4 are devoted to the case of jump linear differential equations on a Hilbert space. The fundamental results obtained in these sections are then exploited in the last part of this chapter in order to obtain criteria for the exponential stability of a class of generalized Lyapunov linear differential equations with jumps on a Hilbert space as well as for the existence of some bounded solutions on the whole real semiaxis of the corresponding non-homogeneous generalized Lyapunov differential equations with jumps.