- Open Access
Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term
© Domoshnitsky; licensee Springer. 2014
- Received: 23 May 2014
- Accepted: 9 September 2014
- Published: 24 September 2014
Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation can be oscillating and asymptotically unstable, the delay equation , where , can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper refute this delusion.
- exponential stability
- maximum principles
- Cauchy function
- positivity of solutions
- positivity of the Cauchy function
- differential inequalities
for the difference .
and are measurable essentially bounded functions (below we write this as ) ().
We understand the solution of (1.7) in the traditional sense like in the book , i.e. as a function with absolutely continuous derivative on every finite interval such that , satisfying (1.7).
where and is an ω-periodic symmetric matrix function, were obtained.
where , q and λ are constants, was proposed in  and developed in , where the equations with a combination of delayed and advanced arguments are considered. A study of advanced equations (see, for example, (1.7), where ) can be found in the paper  in which results on boundedness, stability, and asymptotic representations of solutions are obtained.
with constant delay τ and coefficients and and without damping term were obtained in the paper . These results were based on Pontryagin’s technique for analysis of the roots of quasi-polynomials  and could not be used in the case of (1.8) with variable coefficients and/or delays. Note also that the assumption as regards the absence of the delay in the first term does not allow us to use this result of  in the stabilization in the model of the motion described in (1.1) by the delay control (1.2). Let us try to imagine situations in which variable delays and coefficients arising in the delayed feedback control may be important: (1) the case of control for missiles, where the delay depends on their distance from the controller and is variable; (2) spending of fuel implies the change of the mass of the missiles, which leads to variable coefficients in the delay system. As a conclusion, it should be stressed that there are no results for the exponential stability of the second order equation (1.8) without damping term in the case of variable coefficients and/or delays.
It will be demonstrated that the control of the form (1.2), where , can stabilize system (1.1). We obtain results as regards the exponential stability of (1.8), which are based on the maximum principles for the second order delay differential equation (1.8). Denoting , we can formulate the maximum inequalities principle  in the form: it follows from the inequalities , , that for all . Using the solutions’ representation (see (2.5) below), we can see that this principle is reduced to positivity of the Cauchy (fundamental) function of (1.8) and positivity of its nontrivial solution with nonnegative initial conditions. Nonoscillation of solutions in many cases leads to this property of the solution of second order equation (1.8). The positivity of the Cauchy function will open a way to the analysis of the asymptotic stability of nonlinear delay differential equations on the basis of the known schemes of upper and lower functions. The maximum boundedness principle  can be described in the form: there exists a constant Ω such that the solution of the initial value problem , , satisfies the inequality for . A generalization of the Perron theorem (see, for example, Theorem 3.5 in ) claims the exponential stability in this case.
The paper is organized as follows. After formulation of stabilization problem in Section 1, we discuss formulas of solution representations in Section 2. Simple results on stability of delay differential equations in the case , which will be proven on the basis of general assertions obtained in Section 5, are formulated in Section 3. Auxiliary results used in the proof of our main assertions can be found in Section 4. Results on the positivity of the Cauchy function , nonoscillation and exponential stability are obtained in Sections 5 and 6. Discussion and open problems are presented in Section 7.
This definition of the homogeneous equation allows researchers to build an analog of the classical general theory of ordinary differential equations (ODEs) for functional differential equations using the key notions of the classical ODEs theory, and it is based on the fact that the space of the solutions of the second order equation (2.3), (2.4) becomes two-dimensional. Note that (2.3), (2.2) can be written as (2.1), (2.4), where , for , and for . Thus (2.1), (2.4) is a nonhomogeneous one according to the approach of the book .
and the initial conditions , . Note that we will use this definition of the Cauchy function for the construction of below. The behavior of the fundamental system , of solutions of (2.3), (2.4) determines the existence and uniqueness of solutions of the boundary value problems for this equation. Positivity of the Cauchy function of (2.1), (2.4) is a basis of various theorems about differential inequalities (under corresponding conditions, the solution of an inequality is greater than the solution of an equation). These theorems help essentially in the study of stability for delay differential equations and their natural generalization - functional differential equations (FDEs) .
Let us formulate several definitions concerning stability.
where N and α do not depend on .
It is well known that for (2.1) with bounded delays these two definitions are equivalent .
can be oscillating and asymptotically unstable, the delay equation (2.3), (2.4) under corresponding conditions on the coefficients and delays is nonoscillating and exponentially stable. The basic idea of our approach is to avoid the condition on the nonnegativity of the coefficients for all , and to allow terms with positive and terms with negative coefficients to compensate each other.
the Cauchy function of (3.2) is nonnegative for ;
- (2)if there exists such positive ε such that(3.7)
- (3)if there exists , with , then(3.9)
Then the assertions of Theorem 3.1 are true.
for the difference . The stabilization has been reduced to the stability of this equation. A possible algorithm to construct this stabilizing control is clear now: first we choose the delay θ close to τ such that condition (3.11) is fulfilled, then we choose close to such that condition (3.10) is fulfilled.
In the following assertion we assume the smallness of the difference of the delays and do not assume the smallness of the delay θ.
with constant delays τ and θ and positive coefficients and .
the Cauchy function of (3.19) is nonnegative for ;
the solutions , of (3.19), satisfying initial conditions (2.6), are nonegative for ;
the Cauchy function of (3.19) satisfies the exponential estimate (2.11) and the integral estimate (3.8);
if there exists , then equality (3.9) is fulfilled.
which Cauchy function is . Let us demonstrate that condition (3.6) is essential for positivity of the Cauchy function in Theorem 3.1, assuming that the numbers b and θ are chosen such that all other conditions of Theorem 3.1 are fulfilled. If , then in the triangle , its Cauchy function changes its sign. The same example demonstrates that condition (3.22) is essential in Theorem 3.2 for the positivity of . Note that in the case , the solution satisfying conditions (2.6) changes its sign on the interval . It is clear that inequality (3.22) implies .
is nonoscillating and exponentially stable if , , .
Let us formulate several results on systems of functional differential equations with Volterra operators. We mean the Volterra operators according to the following Tikhonov definition. Assume that is a linear bounded operator, where and are the spaces of continuous and essentially bounded functions respectively.
Definition 4.1 We say that an operator is a Volterra one if for every two functions and and every ω, the equality for implies the equality for .
where is called the Cauchy matrix of system (4.1), , .
and for ;
- (2)there exists a vector function with absolutely continuous components and essentially bounded derivatives such that(4.4)
This lemma follows from Theorem 16.6 for the case (see [, p.409]).
In order to formulate the conditions of positivity of the Cauchy function of the scalar first order equation (4.3), let us introduce the function which describes the ‘size’ of the memory of the operator .
Definition 4.2 Let us define the function as the maximal possible value on for which the equality for for every two continuous functions and implies the equality for almost every .
have no zeros for and its Cauchy function for .
This lemma follows from Theorem 15.7 (see [, p.358]).
Denoting the norm of the operator , as a corollary of this assertion we get the following.
Lemma 4.3 If , then the Cauchy function of (4.3) is positive for .
then , .
- (1)there exists a function v with absolutely continuous and essentially bounded derivative and essentially bounded derivative such that(5.4)
- (2)there exists a bounded absolutely continuous function u with essentially bounded derivative such that(5.5)
the Cauchy function of (5.1) is nonnegative for , and solutions , of (5.2), satisfying initial conditions (2.6), are such that , for .
The proof of Theorem 5.1 is based on several auxiliary results.
Lemma 5.1 Assertions (1) and (2) of Theorem 5.1 are equivalent.
It is clear now that inequalities (5.5) imply that . The implication (2) ⇒ (1) has been proven.
(1) ⇒ (2). Denote , where the function from the assertion (1), and demonstrate that this function satisfies the assertion (2). Obviously the function is nonnegative. The function satisfies the equation and consequently it can be represented in the form (5.6). The derivatives of the function are defined by (5.7) and (5.8). From the inequality we get inequality (5.9) and consequently condition (5.5) is fulfilled.
The implication (1) ⇒ (2) has been proven. □
Under the conditions , , of Theorem 5.1, the operators , , and are positive. Note that, according to the condition of Theorem 5.1, the Cauchy function of the first order equation (5.3) is positive for . The vector function , where , , satisfies all conditions of assertion (2) of Lemma 4.1. According to Lemma 4.1, we get the positivity of the element and the nonnegativity of the element of the Cauchy matrix of system (5.11) for . Remark 4.1 completes now the proof of the implication (1) ⇒ (3).
Noting that the equivalence (1) ⇔ (2) was obtained in Lemma 5.1, we can conclude that Theorem 5.1 has been proven. □
Denote . For bounded for , we have .
Remark 5.2 There is a corresponding inconvenience in choosing the functions because of the equality for . Trying to avoid this inconvenience, we can make the following trick.
where the parameter β will be defined below in the formulation of Theorem 5.2. Define the norm of this operator as .
with the same β as in (5.18), which will be defined below.
It is clear from the definition of the Cauchy function by (2.7), (2.8) that the Cauchy functions of (5.1) and (5.19) coincide in .
, where in (5.18) defining the operator B;
- (b)the inequality(5.20)
the Cauchy function of (5.19) is nonnegative for ;
- (2)if in addition there exists a positive ε such that(5.21)
Proof of Theorem 5.2 Let us start with the proof of the assertion (1). We set for in the assertion (1) of the following reformulation of the assertion (1) ⇒ (3) of Theorem 5.1 for (5.19).
- (1)there exists a function v with absolutely continuous and essentially bounded derivative and essentially bounded derivative such that
- (2)the Cauchy function of (5.19) is nonnegative for , and the solutions , of the homogeneous equation
satisfying the initial conditions (2.6), are such that , for .
and, after carrying the exponent out of the brackets, we get inequality (5.20). Thus inequalities (5.24) and (5.25) are results of a choice of the corresponding function in assertion (1) of Lemma 5.2.
where the operator B is defined by (5.18). The condition (a) implies, according to Lemma 4.3, the positivity of the Cauchy function of the first order equation (5.23).
Now we see that all conditions of Lemma 5.2 and assertion (1) are fulfilled. According to Lemma 5.2, we obtain assertion (1) of Theorem 5.2.
is bounded for every measurable bounded function f.
It follows now from the boundedness of the integral (5.37) for every bounded f that is bounded for .
satisfying conditions (5.27). Now it follows from the boundedness of the integral and solutions , that the integral is bounded, and, using the positivity of , we can claim that the integral is bounded for every measurable bounded , . We see that the component of the solution vector to system (5.41) is bounded for every measurable bounded and . The positivity of the Cauchy function of the first order equation (5.25) together with condition (5.21) imply the boundedness of the component of solution vector to system (5.41). Now a generalization of the Perron theorem (see, for example, Theorem 3.5 in ) claims that all entries of the Cauchy matrix satisfy the exponential estimate (2.11). Each of the solutions of (5.1) coincides with a corresponding solution of (5.19) with the measurable essentially bounded right-hand side such that for . Thus for every bounded right-hand side, the solution of (5.1) is bounded, and the exponential estimate (2.11) of the Cauchy function of (5.1), its derivative , and the solutions of the homogeneous equation satisfy the exponential estimate (2.11). □
The proof of Corollary 5.1 can be obtained from equality (5.31) and the fact of the exponential estimate of all elements of the Cauchy matrix of system (5.41).
Here and one of solutions is a constant and does not tend to zero when .
, , and are measurable essentially bounded functions for , .
Theorem 5.3 Let all assumptions of Theorem 5.2 be fulfilled.
if for , then the Cauchy function of (5.47) is positive for ;
- (2)if there exist positive and ε such that(5.50)
If , then and consequently . Reference to (2.5) completes the proof of the assertion (1).
In order to prove the assertion (2), we have to demonstrate that the solution x is bounded for every bounded right-hand side f and then the extension of the Bohl-Perron implies the exponential estimate (2.11) of the Cauchy matrix (see Theorem 3.5 in the book ). According to Theorem 5.2, the Cauchy function of (5.1) satisfies the exponential estimate (2.11). This implies the boundedness of ψ for every bounded f. If the norm of the operator is less than one, then the bounded operator exists. We have only to prove that .
This, according to our observations above, completes the proof of exponential estimate (2.11) of the Cauchy function of (5.47).
This completes the proof of the assertion (2) of Theorem 5.3. □
The inequality in the condition (a) of Theorem 5.2 is fulfilled if inequality (3.6) is satisfied.
Now Theorem 3.1 follows from Theorem 5.2 and Corollary 5.1. □
Remark 6.1 We have not yet used the assertion (2) of Theorem 5.1 to obtain sufficient conditions for the positivity of the Cauchy function of (3.2). Let us build the function , proving the following assertion.
For positivity of the function for , we have to assume that , which follows from inequality (3.22).
where is defined by (6.10), (6.12).
we have built the positive absolutely continuous function satisfying the condition (2) of Theorem 5.1. Now reference to Theorem 5.1 completes the proof of the assertions (1) and (2) of Theorem 3.2. To prove assertions (3) and (4) we actually repeat the proof of the assertion (2) of Theorem 5.2 (in the case ).
The proof of Theorem 3.2 have been completed. □
- 1.To obtain results on the exponential stability of the equation(7.1)
- 2.To obtain results as regards stabilization of the equation , where , to the trajectory by the control of the form(7.2)
To obtain results on oscillation/nonoscillation, existence of solutions tending to zero or tending to infinity for the second order equation (2.3) without the assumption of the nonnegativity of the coefficients.
- 4.To obtain results on stabilization of system (1.1) to the trajectory by the control of the form(7.4)
To obtain results as regards the exponential stability of systems of functional differential equations of different orders.
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