On the decay rate of solutions of non-autonomous differential systems

Electronic Journal of Differential Equations, Vol. 2001(2001), No. 05, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ejde.math.unt.edu (login: ftp) ON THE DECAY RATE OF SOLUTIONS OF NON-AUTONOMOUS DIFFERENTIAL SYSTEMS TOM´ AS CARABALLO Abstract. Some results on the asymptotic behaviour of solutions of differential equations concerning general decay rate are proved. We prove general criteria on the exponential, polynomial, and more general decay properties of solutions by using suitable Lyapunov’s functions. We also present a detailed analysis of the perturbed linear and nonlinear differential systems. The theory is illustrated with several examples. 1. Introduction The asymptotic behaviour of systems described by differential equations is a very important topic as the vast literature on this field shows. To study the stability of a nonlinear system one can, on the one hand, analyze its linear approximation (see Brauer and Nohel [1], Yoshizawa [9] among others); on the other hand, one can use another method which relies in the technique discovered by Lyapunov (see Yoshizawa [9]). This is called the direct method (or Lyapunov’s Second Method) because it can be applied directly to the differential equation without any knowledge of its solutions, provided one is clever enough to construct the suitable auxiliary functions (called Lyapunov’s functions). But, a major limitation of this procedure is that there are no general methods to construct such auxiliary functions, much more in the nonautonomous case in which we are most interested. In this respect, there exist some interesting results due to Yoshizawa (see [9]- [10]) and LaSalle (see [7]-[8]), among others, which ensure asymptotic approach of trajectories to some closed attracting sets for the differential system (see also Kloeden [6] for another approach). However, apart from the usual exponential stability results obtained by the first approximation technique, in general, almost nothing is said about how fast is the convergence of solutions in dealing with the Lyapunov Second Method. Motivated by this fact, we shall first establish a sufficient condition for the exponential decay of solutions which allows the derivative of the Lyapunov function along the trajectories of the system to be bounded by a definite negative function plus an additional nonnegative function with exponential decay. Mathematics Subject Classification. 34D05, 34D10, 34D20. Key words. Asymptotic behaviour, exponential and polynomial stability, rate of decay. c 2001 Southwest Texas State University. Submitted December 7, 2000. Published January 3, 2001. supported by DGICYT Project PB98-1134. 1 2TOM ´ AS CARABALLO EJDE–2001/05 Another interesting problem arises when one is not able to prove exponential stability but knows that the null solution is asymptotically stable. In this case, an interesting question concerns the possibility of deciding the decay rate of solutions (to zero or to other solution). As far as we know, most stability results related to the Lyapunov method are devoted to provide results that ensure stability, asymptotic stability, etc. but, in general, do not give any further information about the decay rate of solutions (see Haraux [4, pp. 45-47] for a study of the energy decay of a particular second order equation). We shall partially cover this gap by providing some conditions which permit us to estimate the decay rates related to certain general functions (e.g. polynomials, logarithmics, etc.), by introducing a generalization of the concept of Lyapunov exponents. Another interesting fact is that, although our main interest will concern sub-exponential decay of solutions, our treatment also includes the case of super-exponential decay. This paper is organized as follows. In Section 2, we prove a sufficient condition ensuring exponential decay of solutions, and another one concerning asymptotic polynomial behaviour. Next, we introduce in Section 3 the concepts of generalized Lyapunov exponent with respect to a positive general function and the general decay rate of solutions, give some criteria for the asymptotic decay of solutions, and illustrate the results by showing some examples. Section 4 is devoted to the analysis of perturbed systems. In fact, we analyze the perturbations of linear and nonlinear differential systems. Finally, we include some remarks and ideas concerning the possibility of extending the results to the infinite dimensional framework and the functional one. 2. Exponential and polynomial asymptotic behaviour Consider the following initial-value problem for a system of differential equations in Rn: d dtX(t)=f(t, X(t)),t>t 0 X(t 0 )=X 0∈R n , (2.1) where f:R×D→Rnis a continuous function, and D⊂Rnis an open set such that 0 ∈D. It is well known (see, e.g., Coddington and Levinson [2]) that, given t0∈Rand X0∈Rn, there exists at least a solution to this problem defined in an open maximal interval. As we are interested in the stability or asymptotic behaviour of solutions, we assume that every solution to (2.1) is defined for t≥t0. When we deal with the stability analysis, we will also assume that f(t, 0) = 0, so that we consider the stability of the zero solution. Otherwise, we will not assume this and, we will therefore analyze the asymptotic behaviour of such solutions. Associated to the differential system in (2.1), we consider the derivative of a function along the system, i.e., for a continuously differentiable function V(·,·): R×D→Rwe define the function ˙ V(·,·):R×D→Ras follows ˙ V(t, x)=∂V (t, x) ∂t + n X i=1 ∂V (t, x) ∂xi fi(t, x). Remark. Observe that if X(t) is a solution to (2.1), then it holds d dtV(t, X(t)) = ˙ V(t, X(t)). EJDE–2001/05 ON THE DECAY RATE OF SOLUTIONS 3 Now we state a result which, in particular, ensures exponential decay to zero of solutions to (2.1). It is worth mentioning that ˙ Vdoes not need to be definite negative. Theorem 2.1. Assume V:R×D→Ris a continuously differentiable function satisfying: ∃c1>0and p>0such that c1|x|p≤V(t, x),for all (t, x)∈R×D, ∃c2>0such that ˙ V(t, x)≤−c 2 V(t, x)+λ(t),for all (t, x)∈R×D, where λ(·)is a nonnegative continuous function such that there exist M≥0,γ >0 satisfying λ(t)≤Me−γt,for all t∈R+. Then, there exists ε>0such that for any solution X(t)to (2.1) defined for t≥ t0≥0, there exists a constant C=C(X0)(which may depend on X0) such that |X(t)|≤C(X 0 )e −ε(t−t 0 )/p,for all t≥t0. Proof. Let us fix a positive number εsatisfying 0 <ε<min{c2,γ}, and estimate the following derivative for X(t), a solution to (2.1) defined for t≥t0, d dteεtV(t, X(t))=εeεtV(t, X(t)) + eεt ˙ V(t, X(t)) ≤eεt (εV (t, X(t)) −c2V(t, X(t)) + λ(t)) ≤eεtλ(t), and thus eεtV(t, X(t)) ≤eεt0V(t0,X 0)+Zt t 0 e εsλ(s)ds ≤eεt0V(t0,X 0)+Me (ε−γ)t 0 γ−ε ≤e εt0V(t0,X 0)+ M γ−ε. Therefore |X(t)|p≤1 c1V(t0,X 0)+ M γ−εe −ε(t−t 0) ,for all t≥t0, and the proof is complete. Example 1. Let us exhibit a simple example to illustrate this result. Consider the differential equation dX dt=−4X+e −t X1/3 ,(2.2) and take the usual auxiliary function V(x)=1 2x 2 .Then ˙ V(x)=dV(x) dx·−4x+e −t x 1/3=−4x 2+e −t x 4/3,(2.3) which is not definite negative. However, it follows by Young’s inequality (ab ≤ lap p+1 qlq/p bqwith 1 p+1 q= 1) for suitable l>0, p=3/2andq=3, ˙ V(x)=−4x 2+e −t x 4/3≤(−4+2 3l)x 2+1 3l 2e −3t , 4TOM ´ AS CARABALLO EJDE–2001/05 and, for l=3/2wehave−4+2 3l=−3, and therefore ˙ V(x)≤−3x 2+λ(t), where λ(t)= 4 27 e−3t. Now, the theorem ensures that solutions decrease towards zero with exponential decay. Remark. The exponential decay of λis essential to guarantee the same decay of solutions. Indeed, consider the following one dimensional equation dX dt=−X+1 1+t. It is clear that the null solution to the autonomous equation ˙ X=−Xis exponentially stable. Moreover, every solution to this equation converges exponentially to zero (i.e. the global attractor for this equation is the set {0}). However, as far as we consider the perturbed nonautonomous version, the solutions do not converge to zero, in general, with the same rate. To see this, notice that the solution to the problem dX dt=−X+1 1+t X(t 0 )=X 0 , is given by X(t)=X(t;t 0 ,X 0)=e −(t−t 0 ) X 0+Zt t 0 e −(t−s) (1 + s)−1ds . One can easily check that lim t→+∞ log |X(t)| t=0, so that we do not have exponential decay to zero. However, as a consequence of the theory we shall develop, we will be able to ensure that the solutions decay to zero with polynomial rate (see Example 3 below). This fact motivates our interest in analyzing the decay rate of solutions, that is, if we cannot prove exponential convergence of solutions and know that those are asymptotically stable, is it possible to ensure at least polynomial decrease?. The typical example related to nonexponential convergence of solutions to an equilibrium is given by the following simple ordinary differential equation (see Haraux [4, pp. 45-46]): ˙ X(t)=−X(t)|X(t)| p−1,t≥0,p>1. The solution starting in X0at time t=0isgivenby X(t)= sgn(X0) n(p−1) t+|X0|1−po1/(p−1) , so that |X(t)|behaves as {1/[(p−1) t]}1/(p−1) as time tgoes to ∞, and therefore it decreases polynomially to the equilibrium. Owing to this fact, in the following result we provide a sufficient condition guaranteeing polynomial convergence of solutions and, in the next Section, we will state a more general result concerning more general decay rates. EJDE–2001/05 ON THE DECAY RATE OF SOLUTIONS 5 Theorem 2.2. Assume that there exists a continuously differentiable function V: R×D→Rsatisfying ∃c1>0and p>0such that c1|x|p≤V(t, x),for all (t, x)∈R×D, ∃q>1such that ˙ V(t, x)≤−α(t)[V(t, x)]q,for all (t, x)∈R×D, where α(·)is a nonnegative continuous function such that lim inf t→∞ 1 tZt t0 α(s)ds ≥ν>0 (2.4) Then, there exists δ>0such that for any solution X(t)to (2.1) defined for t≥t0, there exists a constant C=C(X0)(which may depend on X0) such that |X(t)|≤C(X 0 )t −δ ,for all t≥t0. Proof. Let us consider X(t), a solution to (2.1) defined for t≥t0.Then d dt [V(t, X(t))] = ˙ V(t, X(t)) ≤−α(t)[V(t, X(t))]q. Denoting u(t)=V(t, X(t)), we have that this function satisfies the following differential inequality ˙u(t)≤−α(t)[u(t)]q, and, therefore its positive solutions satisfy ˙u(t) [u(t)]q≤−α(t). By a direct integration we easily obtain u(t)≤u(t0)1−q+(q−1) Zt t0 α(s)ds−1/(q−1) . Taking into account assumption (2.4), and given ε>0, we can ensure for t0large enough that Zt t0 α(s)ds ≥(ν−ε)t, for all t≥t0, and, consequently, u(t)≤C0(X0)t−1/(q−1),for all t≥t0. Noticing now the expression of u(t), it is clear that the result holds by setting δ=1/p(q−1) and a suitable C(X0). Example 2. We consider the following two dimensional system in order to apply the previous result. ˙y1=y2−y1|y1| ˙y2=−y1−y2|y2|. 6TOM ´ AS CARABALLO EJDE–2001/05 It is easy to check that the unique stationary solution is the zero solution. Let us take V(t, y1,y 2)=1 2(y 2 1+y 2 2 ). Then ˙ V(t, y1,y 2)=−y 2 1|y 1 |−y 2 2|y 2 | =−|y 1 | 3+|y 2 | 3  ≤−c|y 1 | 2+|y 2 | 2 3/2 =−c[V(t, y1,y 2)]3/2, where c>0 is a suitable constant (notice that we have used the inequality a+b 2p≤ ap 2+bp 2,a, b > 0, p>1). Therefore, every solution to the system decays to zero with at least decay rate t−1. 3. General decay rate of solutions Firstly, we will introduce the concept of generalized Lyapunov exponent with respect to a positive function λ(·) which will enable us to establish a precise definition of stability or asymptotic behaviour with general decay function λ(·). Definition 3.1. Let the positive function λ(t)↑+∞be defined for all sufficiently large t>0, say t≥T>0. Let X(t) be a solution to (2.1). The number lim sup t→∞ log |X(t)| log λ(t) is called the generalized Lyapunov exponent of X(t) with respect to λ(t). The solution X(t) is said to decay to zero with decay function λ(t) of order at least γ>0, if its generalized Lyapunov exponent is less than or equal to −γ, i.e., lim sup t→∞ log |X(t)| log λ(t)≤−γ. If, in addition f(t, 0) = 0 for all t∈R, the zero solution is said to be globally asymptotically stable with decay function λ(t) of order at least γ>0, if every solution to (2.1) defined in the future decays to zero with decay function λ(t)of order at least γ>0. Remark. Clearly, replacing in the above definition the decay function λ(t)bye t leads to the usual Lyapunov exponents concept and exponential decay rate. Also, we point out that this definition includes both the case of sub-exponential decay functions (polynomials, logarithms) and the situation of super-exponential decay (e.g. λ(t)=exp{exp t}). Now, we can prove a sufficient condition ensuring almost sure stability of the solution of (2.1) with a general decay rate. Theorem 3.2. Let ϕ1(t),ϕ2(t)be two continuous functions with ϕ1nonnegative. Assume there exist a continuously differentiable function V:R+×D→R,and constants p>0,m≥0,ν≥0,θ∈Rsuch that (a): |x|pλ(t)m≤V(t, x),(t, x)∈R+×D. (b): ˙ V(t, x)≤ϕ1(t)+ϕ 2(t)V(t, x),(t, x)∈R+×D. EJDE–2001/05 ON THE DECAY RATE OF SOLUTIONS 7 (c): ∃T>0large enough such that for t0≥T, lim sup t→∞ log Rt t0ϕ1(s)expn−Rs t 0ϕ 2(r)drods log λ(t)≤ν, lim sup t→∞ Rt t0ϕ2(s)ds log λ(t)≤θ Then, if X(t)is a solution to (2.1) defined in the future (i.e. for t≥t0),then lim sup t→∞ log |X(t)| log λ(t)≤− m−(θ+ν) p. In particular, if m>θ+νand f(t, 0) = 0, the null solution is globally asymptotically stable with decay function λ(t)of order at least (m−(θ+ν)) /p. Proof. Given (t0,X 0)∈(T,+∞)×D,andX(t) a solution to the problem (2.1) defined in the future, let us compute d dtV(t, X(t)) = ˙ V(t, X(t)) ≤ϕ1(t)+ϕ 2(t)V(t, X(t)), which implies d dtexp −Zt t0 ϕ2(s)dsV(t, X(t))≤ϕ1(t)exp−Zt t 0 ϕ 2(s)ds, whence V(t, X(t)) ≤V(t0,X 0)+Zt t 0 ϕ 1 (s)exp−Zs t 0 ϕ 2(r)drdsexp Zt t0 ϕ2(s)ds. Given ε>0, there exists t1(ε) such that for all t≥max{t1(ε),t 0}we have Zt t0 ϕ1(s)exp−Zs t 0 ϕ 2(r)drds ≤λ(t)ν+ε,Zt t0 ϕ2(s)ds ≤log λ(t)(θ+ε). Consequently, it follows that log V(X(t),t)≤log((V(t0,X 0)) + λ(t)ν+ε)+(θ+ε)logλ(t) for all t≥min{t1(ε),t 0}, which immediately implies that lim sup t→∞ log V(X(t),t) log λ(t)≤ν+ε+θ+ε. As this holds for every ε>0, then lim sup t→∞ log V(X(t),t) log λ(t)≤ν+θ, and, therefore lim sup t→∞ log |X(t)| log λ(t)≤− m−(θ+ν) p, which completes the proof. Remarks. a) Observe that, if ϕ2(t)≥0, the result follows by replacing condition (c) by lim supt→∞ log Rt t0ϕ1(s)ds log λ(t)≤ν, lim supt→∞ Rt t0ϕ2(s)ds log λ(t)≤θ. 8TOM ´ AS CARABALLO EJDE–2001/05 b) On the other hand, when m−(θ+ν)>0 it can be proved in the theorem that every solution to problem (2.1) is defined for all t≥t0, so that the limit makes sense for every solution. The next result is an improvement of theorem 2.2 to the more general case of considering a general decay function λ(t) instead of t. Theorem 3.3. Assume V:R×D→Ris a continuously differentiable function satisfying ∃c1>0and p>0such that c1|x|p≤V(t, x),for all (t, x)∈R×D, ∃q>1such that ˙ V(t, x)≤−α(t)[V(t, x)]q,for all (t, x)∈R×D, where α(·)is a nonnegative continuous function such that lim inf t→∞ log Rt t0α(s)ds log λ(t)≥ν>0 (3.1) Then, for any solution X(t)to (2.1) defined for t≥t0it holds lim sup t→∞ log |X(t)| log λ(t)≤− ν p(q−1). Proof. This follows the same lines as the proof of theorem 2.2, taking into account the new assumption (3.1). Now, we shall consider some examples in order to illustrate the results. Of course, as we are going to consider simple linear examples, the conclusions can be obtained by solving directly the equations, and the theory to be developed in the next Section can also be applied. However, our interest right now is to show the different situations which can appear in more complex systems. Example 3. Consider again the equation dX dt=−X+1 1+t. We know that every solution X(t) satisfies limt→+∞log |X(t)|/t = 0. But, taking V(t, x)=(1+t)x 2 , it is easy to check that ˙ V(t, x)=x 2+2x(1 + t)−x+1 1+t ≤x 2(−1−2t)+2x(1 + t)1/2 (1 + t)1/2 ≤x2(−1−2t)+x 2(1 + t)+ 1 1+t ≤1 1+t, so that setting ϕ1(t)= 1 1+tand ϕ2(t) = 0, we immediately obtain ν=θ=0in theorem 3.2, what implies that lim t→+∞ log |X(t)| log (1 + t)≤−1 2 . In other words, although the solutions do not approach zero exponentially, we can assure that their decay rate is at least t−1/2. EJDE–2001/05 ON THE DECAY RATE OF SOLUTIONS 9 Example 4. Now we include an example which does not contain any term causing exponential decay (as −Xin the previous one). Consider the following situation for p>1/2andq>0, dX dt=−p 1+tX+1 (1 + t)q. First, we take the function V(t, x)=(1+t) 2p x 2 , and evaluate ˙ V(t, x)=2p(1 + t)2p−1x2+2(1+t) 2px−p 1+tx+1 (1 + t)q ≤2(1 + t)2px (1 + t)q ≤2x(1 + t)p−1 2(1 + t)p+1 2 (1 + t)q ≤(1 + t)2p−1x2+(1+t) 2(p−q)+1. Now, observe that we can set ϕ1(t)=(1+t) 2(p−q)+1 and ϕ2(t)=(1+t) −1yielding lim t→+∞Rt 0ϕ2(s)ds log(1 + t)=1, and lim t→+∞ log Rt 0ϕ1(s)ds log(1 + t)=2(p−q)+2 if2(p−q)+2>0, 0 otherwise. Then, we can apply theorem 3.2 and obtain convergence to zero with decay rate at least (1 + t)−γin the following cases: If 2(p−q)+2 >0, i.e. if q<p+ 1 and, in addition, q>3/2, then γ=(−3+2q)/2. If 2(p−q)+2≤0, then γ=p−1/2. Example 5. Finally, we exhibit a situation with a more general decay rate. To this end, consider dX dt=−2X (1 + t)log(1+t)+1 (1 + t) [log (1 + t)]2. By using the Lyapunov function V(t, x)=x 2log (1 + t) (notice that we are considering λ(t)=log(1+t)), it holds ˙ V(t, x)= 1 1+tx 2+2xlog (t+1) −2x (1 + t)log(1+t)+1 (1 + t) [log (1 + t)]2! ≤−3x2 1+t+2x (1 + t)log(1+t) ≤−2x 2 1+t+1 (1 + t) [log (1 + t)]2, and we can set ϕ1(t)= 1 (1+t)[log(1+t)]2and ϕ2(t) = 0. Now, it is not difficult to check that (c) in theorem 3.2 is fulfilled with θ=ν= 0 and, consequently, γ=1/2. 16 TOM´ AS CARABALLO EJDE–2001/05 as above and using (4.9) |Y(t)|≤|Y(t)−X(t;t j 0 +i ,Y j 0+i)|+|X(t;t j 0+i,Y j 0+i)| ≤1 4λ(t j 0+i) −(δ−3ε)+1 4|Y j 0+i|λ(t−t j 0+i) −γ ≤1 4λ(t j 0+i) −(δ−3ε)+1 41 2(1 + |Yj0|)λ(i)−min{(δ−3ε),γ}λ(1)−γ ≤1 4λ(i+1) −min{(δ−3ε),γ} +1 41 2(1 + |Yj0|)λ(i)−min{(δ−3ε),γ}λ(1)−min{(δ−3ε),γ} ≤1 4λ(i+1) −min{(δ−3ε),γ}+1 41 2(1 + |Yj0|)λ(i+1) −min{(δ−3ε),γ} ≤1 2[1 + |Yj0|]λ(i+1) −min{(δ−3ε),γ}, and our claim is proved. Finally, (4.16) implies that, for t∈[tj0+i,t j 0+i+1] and for all i∈Nlarge enough, log |Y(t)| log λ(t)≤log 1 2(1 + |Yj0|) log λ(t)−min{(δ−3ε),γ}log λ(i) log λ(t), which allows us to ensure that lim sup t→∞ log |Y(t;t0,Y t 0)| log λ(t)≤−min{(δ−3ε),γ}, and since ε>0 is arbitrary, the proof is therefore complete. Remark. Notice that a more general result can also be proved by a suitable modification in the preceding proof. For instance, if gsatisfies |g(t, x)|≤β 1 (t)+β 2 (t)|x|,∀(t, x)∈Rn+1, instead of (4.12c) in the theorem, β1satisfies (4.12e), and for β2we assume that lim t→∞ Zt+1 t β2(s)ds=0, the assertion in the preceding theorem also holds. 5. Conclusions and final remarks We have developed a theory on general decay properties of solutions of differential systems by using the Lyapunov Second Method and some kind of first approximation results for perturbed systems. In particular, in order to prove our main results, we also have introduced the generalized Lyapunov exponents with respect to general positive functions which has permitted us to establish some criteria for general decay of solutions. However, a very interesting question is concerned with the possibility of determining how fast attract some closed set (e.g. attractors) the solutions of a differential system. Some results on this topic have previously been proved by Eden et al. [3] in the case of exponential attraction. But, to our knowledge, nothing is known about a weaker kind of attraction (e.g. polynomial) or a stronger one (super-exponential). EJDE–2001/05 ON THE DECAY RATE OF SOLUTIONS 17 On the other hand, our treatment could also be extended to the infinite-dimensional context, i.e. for partial differential equations, and some similar results could be proved for differential functional equations. We plan to investigate these in some subsequent works. Acknowledgments. I wish to express my sincere gratitude to the referee for the helpful and interesting comments and suggestions on this paper. I also want to thank Professors J. Real, J. A. Langa and M. J. Garrido for their helpful discussions and suggestions. References [1] F. Brauer and J.A. Nohel, The qualitative theory of ordinary differential equations, Dover, New York, (1989). [2] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw- Hill, New York, (1955). [3] A. Eden, C. Foias, B. Nikolaenko and R. Temam, Exponential attractors for dissipative evolution equations, Masson, Paris (1994). [4] A. Haraux, Syst`emes dynamiques dissipatifs et applications, Masson, Paris, (1991). [5] M.W. Hirsch and S. Smale, Ecuaciones Diferenciales, Sistemas Din´amicos y ´ Algebra Lineal, Alianza Editorial, Madrid (1983). [6] P.E. Kloeden, A Lyapunov function for pullback attractors of nonautonomous differential equations, Elect. J. Diff. Eqns. Conference 05 (2000), 91–102, http://ejde.math.swt.edu/conf-proc/05/toc.html [7] J.P. LaSalle, Stability theory of ordinary differential equations, J. Diff. Eqns. 4 (1968), 57-65. [8] J.P. LaSalle, Stability of nonautonomous systems, Nonlinear Anal. 1 (1976), 83-91. [9] T. Yoshizawa, Stability Theory by Liapunov’s Second Method, The Mathematical Society of Japan, Tokyo (1966). [10] T. Yoshizawa, Asymptotic behaviour of solutions in nonautonomous systems, in Trends in Theory and Practise of Nonlinear Differential Equations (Arlington, Texas, 1982), Lecture Notes in Pure and Appl. Math. 90, 553-562. Tom´ as Caraballo Departamento de Ecuaciones Diferenciales y An´ alisis Num´ erico Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain E-mail address:[email protected]