A DIRECT LYAPUNOV METHOD FOR DELAY DIFFERENTIAL EQUATIONS

This presentation is devoted to well-known tools of stability theory: Lyapunov functions and functionals. The development of delay equation theory through time based on some results of Russian mathematicians is discussed. In particular, there are two different approaches: one could measure the distance between points in a finite dimension space or between short curve segments (points in a functional space). Some definitions, theorems and examples are presented. AMS Subject Classification: 34K20

Lyapunov theorems for the system (1) without any changes.The method of Lyapunov functions was considered to be unpromising because no inverses of the theorems had been proven at this time.One of Elzgoltz's theorems is: Theorem 1.If there exists a definitely positive function, whose trajectory derivative with respect to the system (1) is not positive, then the zero solution of the system (1) is stable according to Lyapunov.
Example 1.For a given delay equation dx(t) dt = −ax(t)x 2 (t − h), a, h > 0 we could use the Lyapunov function V (x) = x 2 , and the corresponding trajectory derivative is always negative: dV dt = −ax 2 (t)x 2 (t − h) < 0. So, the zero solution is stable in accordance with the theorem.
To proceed with the discussion of this method, we will assume that the following conditions must be met: (i) the zero solution exists, (ii) the solution of the Cauchy problem of the system (1) must exist and be unique, (iii) this solution can be extended to positive infinity from some neighbourhood of zero.Theorem 1 is not practically important, as the question of the existence of such Lyapunov functions V (t, X) is still open.Furthermore, the trajectory derivative of function V (t, X) with respect to the system (1) will be a functional which is determined on the segments of the trajectory, thus complicating the check of the property that the derivative has a fixed sign.
In 1956 N. N. Krasovskii offered a method of studying the stability of solutions of the system (1) by using functionals.Instead of controlling the deviation of the vector X(t) from zero, he suggested separating the segment of the trajectory X t (θ), θ ∈ [0, h] and considering some functionals V (t, X(•)) as a generalised distance, i.e., to go from the space E n to the space of piecewisecontinuous vector-functions.All of the definitions of the second method of Lyapunov in this space can be transferred without any special changes.N. N. Krasovskii proved a number of theorems about stability and asymptotic stability of the zero solution of the system (1), and these theorems can be inversed [5,6,7,8].We will state one of these theorems without proof.
Theorem 2. For the trivial solution of the system (1) to be uniformly and asymptotically stable with respect to t 0 and ϕ t 0 (•) , it is necessary and sufficient for the functional V (ψ(•), t), to exist and have the following properties: 2. V (ψ(•), t), is continuous with respect to t and satisfies a Lipschitz condition with respect to ψ(•) in the domain (2); 3. V (ψ(•), t), is definitely positive and uniformly continuous at zero in the domain (2); 4. the upper trajectory derivative from the right with respect to system (1) is a definitely negative functional in the domain (2).
B. S. Razumikhin in 1956 [15] continued the study of the use of Lyapunov functions to determine the stability of the system (1) because the problem of the construction of such Krasovskii functionals was still, apparently, far from a constructive solution of many technical problems.However, the Lyapunov functions had been constructed in specific areas of research.We will talk about Razumikhin's results in detail.Let's impose some restrictions on the functionals F i (t, X(•)) that will guarantee the existence, uniqueness and continuability of solutions of the Cauchy problem for the system (1) in the domain (2), namely F i (t, ϕ(•)) ≡ 0 at ϕ(•) ≡ 0, and (h) , L > 0 First we will consider the question of stability of the zero solution for the system (1).
Theorem 3. The zero solution of the system (1) will be stable in the Lyapunov sense, if there exists such a function V (t, X), that 1. V (t, X) is determined and continuously differentiable in the domain 2. V (t, X) is positively determined in (3); 3. the trajectory derivative of V (t, X) with respect to the system (1) is a functional, which is determined, generally speaking, on any continuous curves Y t (•) of (2), and which has the property along any continuous curves that satisfy the condition Theorem 4. If the conditions of Theorem 3 are satisfied and V (t, X) is uniformly continuous, then the zero solution of the system (1) will be uniformly stable with respect t 0 according to Lyapunov.
Let's state now a theorem about asymptotic stability.
Theorem 5.The zero solution of the system (1) will be asymptotically stable, if there exists a function V (t, X) in ( 3) and having the properties: 1. V (t, X) has bounded and continuous partial derivatives in (3); 2. V (t, X) is definitely positive and uniformly continuous at zero; 3. the trajectory derivative of V (t, X) with respect to the system (1) is a functional which is determined, generally speaking, on any continuous curves y t (•) of ( 2) and which has a property along any continuous curves satisfying the condition (5).Here W (r) > 0 at r = 0, W (0) = 0.
B. S. Razumihkin showed that the set of continuous curves y(t) satisfying the inequality (5), on which the derivative V (t, x) is negative, can be reduced.Namely, we will designate S i (t, x) as the largest value of |F i (t, y(t + •))|, for all continuous curves satisfying (5).Then it is possible to assert that Theorems 3, 4 and 5 will remain valid, if instead of set of the curves y t (•) we will consider a set of continuously differentiatable curves z t (•) satisfying the inequalities: This process of reduction of the set of the curves can be continued in a similar way.Let K i (t, X) = sup|F i (t, z t (•))| at every z t (•) satisfying ( 5) and (6).It is obvious that K i (t, X) ≤ S i (t, X).The negativeness of U (t, X t (•)) will be checked now only on such continuously differentiatable curves ξ t (•) such that Simple examples show that this process makes the estimation of the area of stability in the space of parameters of the system more precise, but whether it gives the exact solution of the problem of stability hasn't been determined as yet.
Example 2. Let it be required to find an area in the space of parameters a and b, for which the zero solution of the equation will be asymptotically stable.Applying Theorem 5 with the function V (t, X) = x 2 2 , we have V = ax 2 + bxx h , where x h = x(t − h).From the inequality (5) we will have for and then V ≤ (a + |b|)x 2 , i.e., for a < −|b| the zero solution is asymptotically stable (Fig. 1).
Figure 2 shows the exact boundary of the area of stability in the plane (a, b). Figure 3 shows the area specified by the method described above.From Equation (7) we can obtain an estimate with the help of (8): h .We use this inequality to estimate V at negative values of b: It is obvious that the right part of the inequality will be negative for Getting rid of the parameter h by introducing new variables ξ = bh and η = ah , we obtain the area represented in Figure 3. Theorem 6.If the previous inequality is satisfied then equation ( 7) is asymptotically stable.
Together with Theorem 5, the following theorem of N. N. Krasovskii appeared [7].In this theorem the role of the condition (5) was given to the inequality where f (r) > r and f is strongly monotonic.The theorems with the conditions ( 5) and ( 9) were applied and generalised by various Russian and foreign mathematicians but preference was given to the condition (9).

Instability
Let's consider the criteria of instability of solutions.Let's say that the instability of the zero solution of the system (1) takes place, if negation of the definition of stability according to Lyapunov is satisfied.If the negation of the definition of the uniform stability is satisfied, we will say, that the instability takes place in a broad sense.In 1960 S. N. Shimanov applied the method of functionals of Lyapunov-Krasovski to studying the zero solution of the system (1) [16].He proved analogues of the theorems of Lyapunov and Chetaev in the following formulations.
Theorem 7. If the functional V (t, X t (•)) exists and is determined and bounded in the domain being uniformly continuous at zero and such that its trajectory derivative with respect to the system (1) ∆V ∆t is a positively determined functional, then the zero solution is unstable.
The analogue of the first theorem of Lyapunov about instability is similarly formulated.For studying the instability in a broad sense we will present a theorem Prasolov A. V. [10], which has an inverse.Theorem 9.If the zero solution of the system (1) is unstable in a broad sense then there is a functional V (t, X(t + •)) with the properties: 1. it is bounded in the domain (2); 2. for any ε > 0 there exists a time t * and a curve X * (•), such that t * > 0, ||X * (•)|| (h) < ε V (t * , X * (•)) > 0; 3. the trajectory derivative of V (t, X(t + •)) with respect to the system (1) can be represented as: where λ = const > 0, W ≥ 0.
The following theorems give the reasons for using Lyapunov functions for studying the instability [11].
Theorem 10.The zero solution of the system (1) will be unstable in a broad sense, if there exists a uniformly continuous function V (t, X) in the domain (3) which has the properties: 1. in an arbitrary small neighbourhood of zero there will be a such point X * and a time t * > h, such that V (t * , X * ) > 0; 2. the right lower trajectory derived number V (t, X) with respect to the system (1) is a functional U (t, X(t + •)), which is assigned, generally speaking, on any piecewise-continuous curves y(t + •) of the domain (2) and, furthermore, it is positively determined on the curves satisfying the condition (5).
By the positive definiteness of the functional we mean the existence of a positively defined function ω, such that The following theorem for the system (1) is proved in the same way as Theorem 10 with those natural modifications that distinguish the first theorem of Lyapunov about instability from the second one.
Theorem 11.The zero solution of the system (1) will be unstable in a broad sense, if there exists a function V (t, X) which is determined and uniformly continuous in the domain (3) and has the properties: 1. for any ε > 0 there exists a point X * and a time t * > t 0 , such that and 2. the right lower trajectory derived number V with respect to the system (1) can be represented as where λ > 0, and the functional W , determined on any piecewise-continuous curves y(t + •) from the domain (2), satisfy the inequality W (t, y(t + •)) ≥ 0 for all curves y(t + •) subject to the condition (5).
Example 3. We can obtain the following solution for the equation ẋ = ax + bx h by applying Theorem 10.Let V = x 2 2 .Then V = ax 2 + bxx h .The condition (5) will give inequality |x(t − h)| ≤ |x(t)|.Let's find a lower estimate V by using the last inequality: V ≥ (a − |b|)x 2 ; it means, that V will be positively determined, if a > |b|.
The determination of the property of instability of the zero solution of the system (1) by using functions is more convenient than by using functionals.The following theorem supports the existence of such functions [12,13].
Theorem 12.For the zero solution of the system (1) to be unstable in a broad sense, it is necessary and sufficient that there should be a function V (t, X) with the following properties: 1. V (t, X) is determined in the semi-cylinder (3); 2. it is bounded in (3) and at any fixed t > 0 lim ||X||→0 |V (t, X)| = 0; 3. for any ε > 0 there exists a pair {t * , X * }, such that 4. the trajectory derivative with respect to the system (1) has the form: where the functionals λ and W are determined along different piecewise-continuous curves y(t + •) of the domain (2) and λ is definitely limited from below by a positive number α, W ≥ 0, when the curves y(t + •) satisfy the inequality: Example 4. Let us consider the problem of single-axle stabilization of a solid body rotation with delayed control feedback moment.As it is known corresponding equations a form where Θ -an inertia tensor of the body, ω = {ω 1 , ω 2 , ω 3 } -angle velocity vector, M -external forces moment.The moment of forces was introduced by V.I.Zubov [17].Here a, b > 0, r = {r 1 , r 2 , r 3 } -unit vector being constant regarding a coordinate system fixed in the body, s = {s 1 , s 2 , s 3 } -an unit vector being constant in inertia space.To make the mathematical model ( 24), (25) complete one it needs to add a kinematical vector equation The moment of forces (25) brings to the body motions satisfying the limit relations: ω(t) → 0, s(t) → r, when t → ∞, if the initial data belong to some neighborhood of the relative equilibrium point Note the joint system (24), (25), (26) has the first integral ||s|| = const, and the problem sense gives us ||s|| ≡ 1.So further we shell talk about conditional stability only.Suppose the control moment of forces M is formed with time lag h > 0. This is more adequate description from engineering point of view.Thus, we have the system where where upper index "T" means transposition and α, β, γ are constants.It is possible to show that if the following inequality is hold: 2αΘ 1 − β 2 Θ 2 2 > 0, where Θ 1 and Θ 2 arethe least and the most Eigen values of the matrix Θ, then is true inequality where 0 < µ 1 ≤ µ 2 .It means the function V (ω, s, s h ) is a definitely positive in some neighborhood of the relative equilibrium point ω ≡ 0, s(t) ≡ r.Let us consider the derivative of the function V by virtue of the system (28): Estimating the terms of the third order with quadratic components we come to Using the Razumikhin conditions (inequalities) we have got the estimation This estimation is definitely negative if the following relations are true:  This example shows how effective the direct Lypunov method is: we considered a non-linear system of six order (in more details one could see A.V. Prasolov [14]).

Conclusion
We talked briefly about one small part of the delay differential equations theory, namely, the Lyapunov second method (or direct Lyapunov's method) of stability and instability study.This description certainly is not full either from theoretical or from practical points of view.A lot of names and interesting ideas are not mentioned here.More than this, the theory consists of numerical chapters, existence and uniqueness theorems, the problem of periodic solutions and many other staying outside of our attention. Fig.1Fig.2Fig.3
Every integral curve of system (28) is unique with continues initial functions {ω 0 (τ, s 0 (τ )} at τ ∈ [−h, 0].Let us research the relative equilibrium state ω ≡ 0, s(t) ≡ r with respect to stability by Lypunov function and Razumikhin condition.Choose the Lypunov function in form