The Stability of Nonlinear Differential Systems with Random Parameters

Abstract

The paper deals with nonlinear differential systems with random parameters in a general form. A new method for construction of the Lyapunov functions is proposed and is used to obtain sufficient conditions for 𝐿₂-stability of the trivial solution of the considered systems.

1. Introduction

1.1. The Aim of the Contribution

The method of Lyapunov functions is one of the most effective methods for investigation of self-regulating systems. It is important for determining the fact of stability or instability of given systems among other purposes. A successfully constructed Lyapunov function for given nonlinear self-regulating systems makes it possible to solve all the complex problems important in practical applications such as estimation of changes of a self-regulated variable, estimation of transient processes, estimation of integral criteria of the quality of self-regulation, or estimation of what is called guaranteed domain of stability.

In [1] it is explained why not every positive definite function can serve as a Lyapunov function for a system of differential equations. As experience shows, the most suitable Lyapunov functions have physical meaning. The Lyapunov function method is an effective method for the investigation of stability of linear or nonlinear differential systems that are explicitly independent of time (see, e.g., [1–9]). But there are no universal methods for constructing appropriate Lyapunov functions because, as well-known, in nonlinear differential systems, each case considered requires an individual method for constructing a Lyapunov function.

However, the method of Lyapunov functions is often difficult to apply to the investigation of some kinds of stability of nonstationary differential systems because the concept of Lyapunov stability can make the Lyapunov functions inconvenient to use. This problem was solved by a new definition of what is called 𝐿₂-stability of the trivial solution of the nonstationary differential (or difference) systems [10, 11], which is compatible with the method of Lyapunov functions.

In this paper, we deal with much more complicated investigation of the Lyapunov stability of differential systems with random parameters. We define a concept of 𝐿₂-stability of the trivial solution of the differential systems with semi-Markov coefficients and give an analogy between the 𝐿₂-stability and the stability obtained by Lyapunov functions. A new method of constructing Lyapunov functions is proposed for the study of stability of systems, and Lyapunov functions are derived for systems of differential equations with coefficients depending on a semi-Markov process. Sufficient conditions of stability are given, and it is proved that the condition of 𝐿₂-stability implies the existence of Lyapunov functions. In addition to this, the case of the coefficients of the considered systems depending on Markov process is analyzed.

1.2. Systems Considered

In this part, a new concept of semi-Markov function is proposed. It will be used later for the construction of Lyapunov functions.

Consider nonlinear 𝑛-dimensional differential system 𝑑𝑋(𝑡)((𝑡)(𝑡()𝑑𝑡=𝐹𝑡,𝑋,𝜉)),𝐹𝑡,0,𝜉=0, (1.1) on the probability space (Ω≡{𝜔},𝔗,𝐏,𝐅≡{𝐅_𝐭∶𝑡≥0}). A vector-function 𝑋=𝑋(𝑡), 𝑡≥0, is called a solution of (1.1) if 𝑋(𝑡) is a random vector-function from the set of random vector-functions defined on Ω, there exists mathematical expectation of {𝑋²(𝑡)}, and (1.1) is satisfied for 𝑡≥0. The derivative is understood in the meaning of differentiability of a random process [12].

A space of solutions 𝑋 can be interpreted as a phase space of states of a random environment. Measurable subsets of a random environment form a collection of its states. As a phase space of states serves a complete metric separable space (as a rule the Euclidean space or a finite space equipped with 𝜎-algebra of all subsets of 𝑋). Under assumptions of our problem (and in similar problems as well), solutions are defined in the meaning of a strong solution of the Cauchy problem [13].

Together with (1.1), we consider the initial condition 𝑋(0)=𝜑(𝜔),𝜑∶Ω→ℝ^𝑛. (1.2) In fact, any solution 𝑋(𝑡) of (1.1) depends on the random variable 𝜔, that is, 𝑋(𝑡)≡𝑋(𝑡,𝜔).

The random process 𝜉(𝑡), 𝑡≥0, is a semi-Markov process with the states 𝜃₁,𝜃₂,…,𝜃_𝑛. (1.3) We assume 𝜉(𝑡₀)=0 where 𝑡₀=0, and moments of jumps 𝑡_𝑗, 𝑗=0,1,…,𝑛, 𝑡₀<𝑡₁<⋯<𝑡_𝑛 of the process 𝜉 are such that 𝜉(𝑡_𝑗)=lim_{𝑡→𝑡𝑗+0}𝜉(𝑡) and 𝜉(𝑡)=𝜃_𝑠, 𝑠∈{1,2,…,𝑛} if 𝑡_𝑗≤𝑡<𝑡_𝑗+1, 𝑗=0,1,…,𝑛−1.

The transition from state 𝜃_𝑙 to state 𝜃_𝑠 is characterized by the intensity 𝑞_𝑙𝑠(𝑡),  𝑙,𝑠=1,2,…,𝑛, and the semi-Markov process is defined by the intensity matrix 𝑄(𝑡)=𝑞_𝑙𝑠(𝑡)^𝑛_𝑙,𝑠=1, (1.4) whose elements satisfy the relationships 𝑞_𝑙𝑠(𝑡)≥0,^𝑛_𝑙=1^∞₀𝑞_𝑙𝑠(𝑡)𝑑𝑡=1. (1.5) Let mutually different functions 𝑤_𝑠(𝑡,𝑥), 𝑠=1,2,…,𝑛, be defined for 𝑡>0, 𝑥∈ℝ^𝑛.

Definition 1.1. The function 𝑤(𝑡,𝑥,𝜉(𝑡)) is called a semi-Markov function if the equalities 𝑤𝑡,𝑥,𝜉(𝑡)=𝜃_𝑠=𝑤_𝑠𝑡−𝑡_𝑗,𝑥,𝑠=1,2,…,𝑛 (1.6) hold for 𝑡_𝑗≤𝑡≤𝑡_𝑗+1.

It means that the semi-Markov function 𝑤(𝑡,𝑥,𝜉(𝑡)) is a functional of a random process 𝜉(𝑡). The value of 𝑤(𝑡,𝑥,𝜉(𝑡)) is determined by the values 𝑡,𝑥,𝜉(𝑡) at the time 𝑡 and also by the value of the jump of the process 𝜉(𝑡) at time 𝑡_𝑗, which precedes time 𝑡. In fact, the system (1.1) means 𝑛 different differential systems in the form 𝑑𝑋(𝑡)𝑑𝑡=𝐹_𝑠(𝑡,𝑋(𝑡)),𝑠=1,2,…,𝑛, (1.7) where 𝐹_𝑠(𝑡,𝑥)≡𝐹𝑡,𝑥,𝜃_𝑠. (1.8) We assume that there exists a unique solution of (1.7) for every point (𝑡,𝑥) such that 𝑡≥0, ‖𝑥‖<∞ (‖⋅‖ stands for Euclidean norm), continuable on [0,∞).

1.3. Auxiliaries

In the paper, in addition to what was mentioned above, the following notations and assumptions are introduced:(1) the functions 𝐹_𝑠(𝑡,𝑥),  𝑠=1,2,…,𝑛, are Lipschitz functions with the Lipschitz constants 𝜌_𝑠, that is, the inequalities 𝐹_𝑠(𝑡,𝑥)−𝐹_𝑠𝑡,𝑦≤𝜌_𝑠𝑥−𝑦,𝑠=1,2,…,𝑛 (1.9) hold.(2) If 𝑥=0, then 𝐹_𝑠(𝑡,0)≡0,𝑠=1,2,…,𝑛,𝑡≥0. (1.10)(3) The inequalities 𝑁_𝑠(𝑡,𝑥)≤𝜌_𝑠𝑒^{−𝛼𝑠𝑡},𝑠=1,…,𝑛,𝑡≥0 (1.11) are valid. Here 𝜌_𝑠, 𝑠=1,…,𝑛 are the Lipschitz constants, 𝛼_𝑠, 𝑠=1,…,𝑛 are positive constants, and 𝑁_𝑠(𝑡,𝑋(0)), 𝑠=1,…,𝑛 is the solution 𝑋(𝑡) of (1.7) in the Cauchy form, that is, 𝑋(𝑡)=𝑁_𝑠(𝑡,𝑋(0)),𝑠=1,2,…,𝑛. (1.12)(4) We introduce the Lyapunov functional 𝑉=^∞₀𝐸(𝑤(𝑡,𝑥,𝜉(𝑡)))𝑑𝑡, (1.13) where 𝐸(⋅) denotes mathematical expectation, and we assume that the integral is convergent.

Definition 1.2. The trivial solution of the differential systems (1.1) is said to be 𝐿₂-stable if, for any solution 𝑋(𝑡) with bounded initial values of the mathematical expectation 𝐸𝑋(0)𝑋^∗(0), (1.14) the integral 𝐽=^∞₀𝐸‖𝑋(𝑡)‖²𝑑𝑡 (1.15) converges.

Remark 1.3. It is easy to see that (1.15) converges if and only if the matrix integral ^∞₀𝐸𝑋(𝑡)𝑋^∗(𝑡)𝑑𝑡 (1.16) is convergent.

Lemma 1.4. Let the function 𝑤(𝑡,𝑥,𝜉) be bounded, that is, there exists a constant 𝛽 such that the inequalities 0≤𝑤(𝑥)≤𝑤(𝑡,𝑥,𝜉)≤𝛽‖𝑥‖²,for𝜉=𝜃_𝑠,𝑠=1,2,…,𝑛, (1.17) or the inequalities 0≤𝑤(𝑥)≤𝑤_𝑠(𝑡,𝑥)≤𝛽‖𝑥‖²,𝑠=1,…,𝑛 (1.18) hold where 𝑤(𝑥) is a positive definite and differentiable function satisfying the inequality 𝐸(𝑤(𝑋(𝑡)))≤𝐸(𝑤(𝑡,𝑋(𝑡),𝜉(𝑡)))≤𝛽𝐸‖𝑋(𝑡)‖². (1.19) Let, moreover, the Lyapunov functional (1.13) exist for the system (1.7) with an 𝐿₂-stable trivial solution.
Then the Lyapunov functional (1.13) can be expressed in the form 𝑉=_𝐸𝑛^𝑛_𝑠=1𝑣_𝑠(𝑥)𝑓_𝑠(0,𝑥)𝑑𝑥,𝑑𝑥≡𝑑𝑥₁⋯𝑑𝑥_𝑛 (1.20) if the particular Lyapunov functions 𝑣_𝑠(𝑥)=^∞₀𝐸𝑤(𝑡,𝑋(𝑡),𝜉(𝑡))∣𝑋(0)=𝑥,𝜉(0)=𝜃_𝑠𝑑𝑡,𝑠=1,…,𝑛 (1.21) are known.

Proof. The functions 𝑣_𝑠(𝑥), 𝑠=1,…,𝑛, will be defined using auxiliary functions 𝑢_𝑠(𝑡,𝑥)=𝐸𝑤(𝑡,𝑋(𝑡),𝜉(𝑡))∣𝑋(0)=𝑥,𝜉(0)=𝜃_𝑠,𝑠=1,…,𝑛. (1.22) The mathematical expectation in (1.22) can be calculated by the transition intensities 𝑞_𝑙𝑠(𝑡),  𝑙,𝑠=1,…,𝑛,  𝑡≥0Ψ_𝑠(𝑡)=^∞_𝑡𝑞_𝑠(𝜏)𝑑𝜏,𝑞_𝑠(𝑡)≡^𝑛_𝑙=1𝑞_𝑙𝑠(𝑡),𝑠=1,…,𝑛, (1.23) whence the system 𝑢_𝑠(𝑡,𝑥)=Ψ_𝑠(𝑡)𝑤_𝑠𝑡,𝑋_𝑠(𝑡,𝑥)+^𝑛_𝑙=1^𝑡₀𝑞_𝑙𝑠(𝜏)𝑢_𝑙𝑡−𝜏,𝑁_𝑠(𝜏,𝑥)𝑑𝜏,𝑠=1,…,𝑛 (1.24) is obtained. Integrating the system of equations (1.24) with respect to 𝑡, we get the system of functional equations 𝑣_𝑠(𝑥)=^∞₀Ψ_𝑠(𝑡)𝑤_𝑠𝑡,𝑁_𝑠()+𝑡,𝑥𝑑𝑡_{𝑛𝑙=1∞0𝑡0}𝑞_𝑙𝑠(𝜏)𝑣_𝑙𝑡−𝜏,𝑁_𝑠(𝜏,𝑥)𝑑𝜏𝑑𝑡,𝑠=1,…,𝑛, (1.25) for 𝑣_𝑠(𝑥)=^∞₀𝑢_𝑠(𝑡,𝑥)𝑑𝑡,𝑠=1,…,𝑛. (1.26) The system (1.25) thus obtained can be solved by successive approximations 𝑣⁰_𝑠(𝑥)≡0,𝑠=1,…,𝑛,𝑣_𝑠^(𝛼+1)(𝑥)=^∞₀Ψ_𝑠(𝑡)𝑤_𝑠𝑡,𝑁_𝑠()+𝑡,𝑥𝑑𝑡_{𝑛𝑙=1∞0}𝑞_𝑙𝑠(𝑡)𝑣_𝑙^(𝛼)𝑁_𝑠(𝑡,𝑥)𝑑𝑡,𝑠=1,…,𝑛,𝛼=0,1,2,…. (1.27)

2. Main Results

2.1. The Case of a Semi-Markovian Random Process 𝜉(𝑡)

Theorem 2.1. Let the functions 𝐹_𝑠(𝑡,𝑥), 𝑠=1,2,…,𝑛, in the system (1.7) satisfy conditions (1.9), (1.11), let the semi-Markov process 𝜉(𝑡) be determined by the transition intensities 𝑞_𝑙𝑠(𝑡),𝑙,𝑠=1,…,𝑛, 𝑡≥0 satisfying (1.5), and let the functions 𝑤(𝑡,𝑥,𝜉(𝑡)) satisfy (1.17). Then the following statements are true. (1) The relationships 𝑣_𝑠^(𝛼)(𝑥)≤𝐶_𝑠^(𝛼)‖𝑥‖²,𝑠=1,…,𝑛,𝛼=0,1,…,_∞0Ψ_𝑠(𝑡)𝑒^{−2𝛼𝑠𝑡}𝑑𝑡<∞,^∞₀𝑞_𝑠(𝑡)𝑒^{−2𝛼𝑠𝑡}𝑑𝑡<∞,𝑠=1,…,𝑛, (2.1)(2.2) imply that, for the system (1.7), the particular Lyapunov functions can be established in the form 𝑣_𝑠(𝑥)=Ψ_𝑠(𝑡)𝑤_𝑠𝑡,𝑁_𝑠(𝑡,𝑥)+^𝑛_𝑙=1^𝑡₀𝑞_𝑙𝑠(𝜏)𝑣_𝑙𝑁_𝑠(𝜏,𝑥)𝑑𝜏,𝑠=1,…,𝑛, (2.3)(2) If the spectral radius of the matrix Γ=(𝛾_𝑙𝑠)^𝑛_𝑙,𝑠=1 is less than one, then the particular Lyapunov functions 𝑣_𝑠(𝑥), 𝑠=1,…,𝑛, can be found by the method of successive approximations (1.27). (3) Under assumption (1.11), the method of successive approximations (1.27) converges and the inequalities 𝑣_𝑠^(𝛼+1)≥𝑣_𝑠^(𝛼)(𝑥),𝑣_𝑠^(𝛼)(𝑥)≤𝑣_𝑠(𝑥),𝑠=1,…,𝑛 (2.4) hold. Then the sequence of functions 𝑣_𝑠^(𝛼)(𝑥), 𝑠=1,…,𝑛,𝛼=0,1,2,…, is monotone increasing and bounded from above by the functions 𝑣_𝑠(𝑥),𝑠=1,…,𝑛.

Proof. Applying estimation (2.1) and assumption (2.2) to the successive approximations (1.27), we get 𝐶_𝑠^(𝛼+1)≤𝛽𝜌²_𝑠^∞₀Ψ_𝑠(𝑡)𝑒^{−2𝛼𝑠𝑡}𝑑𝑡+𝜌²_𝑠^𝑛_𝑙=1^∞₀𝑞_𝑙𝑠(𝑡)𝑒^{−2𝛼𝑠𝑡}𝑑𝑡𝐶_𝑙^(𝛼),𝑠=1,…,𝑛,𝛼=0,1,2,…. (2.5) It is sufficient to assume the existence of a bounded solution of the system of inequalities (2.1) whence the existence follows of a positive solution of the system of linear algebraic equations (2.3). Moreover, assumption (2.2) guarantees the convergence of the improper integrals in the system (1.27) and so, for the existence of a positive solution of the system (2.3), it is sufficient that the spectral radius 𝜌(Γ) of the matrix Γ=𝛾_𝑙𝑠^𝑛_𝑙,𝑠=1 (2.6) is less than one. For this, it is sufficient that ^𝑛_𝑙=1𝛾_𝑙𝑠≡𝜌²_𝑠^∞₀𝑞_𝑠(𝑡)𝑡^{−2𝛼𝑠𝑡}𝑑𝑡<1,𝑠=1,…,𝑛. (2.7) The convergence of the sequence 𝑣_𝑠^(𝛼)(𝑥),𝑠=1,…,𝑛, 𝛼=0,1,2,…, can be determined by the system 𝑣_𝑠^(𝛼+1)(𝑥)−𝑣_𝑠^(𝛼)(𝑥)=^𝑛_𝑙=1^∞₀𝑞_𝑙𝑠(𝑡)𝑣_𝑙^(𝛼)𝑁_𝑠(𝑡,𝑥)−𝑣_𝑙^(𝛼−1)𝑁_𝑠(𝑡,𝑥)𝑑𝑡,𝑠=1,…,𝑛,𝛼=1,2,3,…. (2.8) If there exist the inequalities 𝑣_𝑠^(𝛼)(𝑥)−𝑣_𝑠^(𝛼−1)(𝑥)≤^𝑛_𝑙=1^∞₀𝑞_𝑙𝑠(𝑡)𝜌²_𝑠𝑒^{−2𝛼𝑠𝑡}𝑑𝑡𝑑_𝑠^(𝛼)‖𝑥‖²,𝑠=1,…,𝑛, (2.9) where 𝑑_𝑠^(𝛼+1)=^𝑛_𝑙=1𝛾_𝑠𝑙𝑑_𝑙^(𝛼),𝑠=1,…,𝑛, (2.10) hold, then estimation (2.4) is true for all 𝛼=2,3,…, 𝑑_𝑠⁽¹⁾=𝐶_𝑠, 𝑠=1,…,𝑛.
Under assumptions (2.8), it follows lim_𝛼→+∞𝑑_𝑠^(𝛼)=0,𝑠=1,…,𝑛, (2.11) which implies a uniform convergence of the sequence 𝑣_𝑠^(𝛼)(𝑥), 𝑠=1,…,𝑛, 𝛼=0,1,2,….

Corollary 2.2. If the trivial solution of the differential systems (1.1) is 𝐿₂-stable, then there exist particular Lyapunov functions 𝑣_𝑠(𝑥), 𝑠=1,…,𝑛 that satisfy (2.3).

Corollary 2.3. Let the function 𝑤(𝑡,𝑥,𝜉(𝑡)) satisfy the inequality: 𝛽₁‖𝑥‖²≤𝑤(𝑡,𝑥,𝜉(𝑡))≤𝛽‖𝑥‖²,𝛽₁>0. (2.12) If there exist the Lyapunov functions 𝑣_𝑠(𝑥), 𝑠=1,…,𝑛 for the system (1.21), then the trivial solution of the differential systems (1.1) is 𝐿₂-stable.

Corollary 2.4. Let the semi-Markov process 𝜉(𝑡) in the system (1.1) have jumps at the times 𝑡_𝑗, 𝑗=0,1,2,…, 𝑡₀=0, in the transition from state 𝜃_𝑠 to state 𝜃_𝑙, and let the jumps satisfy the equation 𝑋𝑡_𝑗=Φ_𝑙𝑠𝑋𝑡_𝑗−0,Φ_𝑙𝑠(0)=0,𝑗=1,2,…, (2.13) where Φ_𝑙𝑠(𝑥) are any continuous Lipschitz vector functions. Then the system (2.3) has the form 𝑣_𝑠(𝑥)=^∞₀Ψ_𝑠(𝑡)𝑤_𝑠𝑡,𝑁_𝑠()+𝑡,𝑥𝑑𝑡_{𝑛𝑙=1∞0}𝑞_𝑙𝑠(𝑡)𝑣_𝑠Φ_𝑙𝑠𝑁_𝑠(𝑡,𝑥)𝑑𝑡,𝑠=1,…,𝑛, (2.14) and its solution can be found by the method of successive approximations.

2.2. The Case of a Markovian Random Process 𝜉(𝑡)

Next result relates to the case of the semi-Markov process 𝜉(𝑡) being transformed into a Markov process described by the system of ordinary differential equations: 𝑑𝑃(𝑡)𝑎𝑑𝑡=𝐴𝑃,𝐴=_𝑙𝑠^𝑛_𝑙,𝑠=1, (2.15) under the influence of which the considered system 𝑑𝑋(𝑡)(𝑋(𝑡)(𝑡((𝑡𝑑𝑡=𝐹,𝜉)),𝐹0,𝜉))≡0, (2.16) takes the form 𝑑𝑋(𝑡)𝑑𝑡=𝐹_𝑠(𝑋(𝑡)),𝑠=1,…,𝑛. (2.17) We also assume that, if 𝑡_𝑗≤𝑡<𝑡_𝑗+1, 𝜉(𝑡)=𝜃_𝑠, then 𝑤(𝑡,𝑥,𝜉(𝑡))=𝑤_𝑠(𝑥),𝑤_𝑠(0)=0,𝑠=1,…,𝑛. (2.18) Then the system of equations (2.14) has the form 𝑣_𝑠(𝑥)=^∞₀𝑒^{𝑎𝑠𝑠𝑡}𝑤_𝑠𝑁_𝑠(𝑡,𝑥)𝑑𝑡+^𝑛_{𝑙=1𝑙≠𝑠}^∞₀𝑎_𝑙𝑠𝑒^{𝑎𝑠𝑠𝑡}𝑣_𝑙𝑁_𝑠(𝑡,𝑥)𝑑𝑡,𝑠=1,…,𝑛. (2.19)

Theorem 2.5. Let the nonlinear differential system (1.1), depending on the Markov process 𝜉(𝑡), be described by (2.15). Then the particular Lyapunov functions 𝑣_𝑠(𝑥) satisfy the linear differential system: 𝐷𝑣_𝑠(𝑥)𝐹𝐷𝑥^𝑠(𝑥)+𝑤_𝑠(𝑥)+^𝑛_𝑙=1𝑎_𝑙𝑠𝑣_𝑙(𝑥)=0,𝑠=1,…,𝑛. (2.20)

Proof. Let us write the solution of the system (2.17) in the Cauchy form: 𝑋(𝑡)=𝑁_𝑠(𝑡−𝜏,𝑋(𝜏)),𝑠=1,…,𝑛. (2.21) Differentiating (2.21) with respect to 𝜏, we get −𝜕𝑁_𝑠(𝑡−𝜏,𝑋(𝜏))+𝜕𝑡𝐷𝑁_𝑠(𝑡−𝜏,𝑋(𝜏))(𝜏)𝐹𝐷𝑋^𝑠(𝑋(𝜏))=0, (2.22) which, for 𝜏=0, 𝑋(0)=𝑥, takes the form: 𝐷𝑁_𝑠(𝑡,𝑥)𝐹𝐷𝑥^𝑠(𝑥)≡𝜕𝑁_𝑠(𝑡,𝑥)𝜕𝑥,𝑠=1,…,𝑛. (2.23) Then 𝐷𝑁_𝑠(𝑡,𝑥)𝐹𝐷𝑥^𝑠+𝑎_𝑠𝑠𝑣_𝑠(𝑥)=−^∞₀𝑎_𝑠𝑠𝑒^{𝑎𝑠𝑠𝑡}𝑤_𝑠𝑁_𝑠(𝑡,𝑥)+^𝑛_{𝑙=1𝑙≠𝑠}𝑎_𝑙𝑠𝑒^{𝑎𝑠𝑠𝑡}𝑣_𝑙𝑁_𝑠(+𝑡,𝑥)𝑑𝑡_∞0𝑒^{𝑎𝑠𝑠𝑡}𝜕𝑤𝜕𝑡^𝑠𝑁_𝑠(𝑡,𝑥)+^𝑛_{𝑙=1𝑙≠𝑠}𝑎_𝑙𝑠𝑣_𝑙𝑁_𝑠()=𝑡,𝑥𝑑𝑡_∞0𝜕𝑒𝜕𝑡^{𝑎𝑠𝑠𝑡}𝑤_𝑠𝑁_𝑠()+𝑡,𝑥_{𝑛𝑙=1𝑙≠𝑠}𝑎_𝑙𝑠𝑣_𝑙𝑁_𝑠(𝑡,𝑥)𝑑𝑡=−^𝑛_{𝑙=1𝑙≠𝑠}𝑎_𝑙𝑠𝑣_𝑙(𝑥), (2.24) which implies (2.20) if lim_𝑡→+∞𝑒^{𝛼𝑠𝑠𝑡}𝑤_𝑠𝑁_𝑠(𝑡,𝑥)+^𝑛_{𝑙=1𝑙≠𝑠}𝑎_𝑙𝑠𝑣_𝑙𝑁_𝑠(𝑡,𝑥)=0,𝑠=1,…,𝑛 (2.25) and the functions 𝑤_𝑠(𝑥), 𝑣_𝑠(𝑥), 𝑠=1,…,𝑛 are differentiable.

Corollary 2.6. If the solutions of the system (2.16) have the same jumps as the solution of the system (2.14) and converge to the jumps of the Markov process 𝜉(𝑡) such that Φ_𝑙𝑠(𝑥)≡𝐸, 𝑙=1,…,𝑛, then the system (2.19) takes the form 𝑣_𝑠(𝑥)=^∞₀𝑒^{𝑎𝑠𝑠𝑡}𝑤_𝑠𝑁_𝑠()+𝑡,𝑥𝑑𝑡_{𝑛𝑙=1𝑙≠𝑠∞0}𝑎_𝑙𝑠𝑒^{𝑎𝑠𝑠𝑡}𝑣_𝑙Φ_𝑙𝑠𝑁_𝑠(𝑡,𝑥)𝑑𝑡,𝑠=1,…,𝑛, (2.26) and the system (2.20) takes the form 𝐷𝑣_𝑠(𝑥)𝐷𝑥=𝐹_𝑠(𝑥)+^𝑛_𝑙=1𝑎_𝑙𝑠𝑣_𝑙Φ_𝑙𝑠(𝑥)=−𝑤_𝑠(𝑥),𝑠=1,…,𝑛. (2.27)

Example 2.7. Let us investigate the stability of solutions of two-dimensional system 𝑑𝑋(𝑡)=()𝑋(𝑡)(𝑋(𝑡)(𝑡𝑑𝑡𝜈−𝜆−𝛼+𝐺,𝜉)),𝛼+𝜆>𝜈,𝛼>0, (2.28) where 𝜉(𝑡) is a random Markov process having two states 𝜃₁, 𝜃₂ with probabilities 𝑝_𝑘=𝑃{𝜉(𝑡)=𝜃_𝑘}, 𝑘=1,2, that satisfy the equations 𝑑𝑝₁(𝑡)𝑑𝑡=−𝜆𝑝₁(𝑡)+𝜈𝑝₂(𝑡),𝑑𝑝₂(𝑡)𝑑𝑡=𝜆𝑝₁(𝑡)−𝜈𝑝₂(𝑡), (2.29) where 𝜆>0. The random matrix function 𝐺 is known: 𝐺₁(𝑥)=𝐺𝑥,𝜃₁=−𝛾₁𝑥₂−𝑥³₁𝛾₁𝑥₁−𝑥³₂,𝐺₂(𝑥)=𝐺𝑥,𝜃₂=𝛾₂𝑥₂−𝑥³₁−𝛾₂𝑥₁−𝑥³₂. (2.30) Taking the positive definite functions 𝑤₁(𝑥)=𝑤₂(𝑥)=𝑥²₁+𝑥²₂+1𝛼𝑥⁴₁+𝑥⁴₂,𝑥=𝑥₁,𝑥₂, (2.31) we can verify that the positive definite particular Lyapunov functions 𝜈₁(𝑥)=𝜈₂(𝑥)=1𝑥2𝛼²¹+𝑥²₂ (2.32) are the solutions to (2.20). Consequently, since the integral (1.13) 𝜈=^∞₀𝑥²₁+𝑥²₂+1𝛼𝑥⁴₁+𝑥⁴₂𝑑𝑡, (2.33) is convergent, the zero solution of the considered system is 𝐿₂-stable.

Acknowledgments

This paper was supported by Grant P201/11/0768 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00 216 30519, and by Grant no 1/0090/09 of the Grant Agency of Slovak Republic (VEGA).