Example 6.5. Consider the following IVP 2
4 1 0 0 0 0 0 3 5
•x˙1(t)
˙ x2(t)
‚
= 2 4
0 0
0 0
0 α(t) 3 5
•x1(t) x2(t)
‚ +
2 4
0 0 0 1 0 0 3 5
•x1(t−τ) x2(t−τ)
‚ +
2 4
cos(t)
−cos(t−τ)
−a(t)cos(t) 3
5, (6.14a)
for t∈[0, 10τ]. The delayτis chosen to be1and the initial function is φ(t)=
•si n(t) cos(t)
‚
, for t∈[−τ, 0]. (6.14b)
The functionαis given by
α(t)=
(0 for t∈[0, 2τ], 1 for t∈(2τ, 10τ].
We see that on the interval[0, 2τ]the DDAE(6.14a)is noncausal, and therefore the solver RADAR5 [60] fails to handle the IVP(6.14). The BVP method, however, successfully solves the IVP(6.14). To compute the numerical solution to the IVP(6.14), we implemented the three stage Gauß- Lobatto collocation method, as in [80], to the strangeness-free DDAE (6.8), which is pointwise computed automatically. The numerical solution and the ab-solute error are presented in Figure6.2. The stepsize is h=0.01.
0 2 4 6 8 10
−1
−0.5 0 0.5
1 x1
x2
0 2 4 6 8 10
10−16 10−14 10−12 10−10
error x1 error x2
Figure 6.2:Numerical solution and absolute error of the IVP (6.14) with constant stepsize h=0.01.
Remark 6.6. Since the functionG in the DAE (6.8a) depends linearly onφ, the suffi-cient smoothness condition so that the claim ii) of Theorem6.4holds, isφ∈Cµ+1, and for the super-convergence in claim iii), the functionφshould be inC2s+µ.
6.3. Examples 94 the number of stagess is three. First we consider their performance for retarded and neutral DDAEs.
Example 6.7. Consider the DDAE
•0 t 0 0
‚ •x˙1(t)
˙ x2(t)
‚
=
•1 0 0 1
‚ •x1(t) x2(t)
‚ +
•0 1 0 0
‚ •x1(t−τ) x2(t−τ)
‚ +
•−et+1
−t
‚
, (6.15) in the time intervalI=[0, 10τ], with the delayτ=1. We choose an initial functionφto be
•et t
‚
for t∈[−τ, 0]. Taking the derivative of the second equation of (6.15)and scaling it with t , we deduce that
0=tx˙2(t)−t.
Adding this equation to the first equation of (6.15)and eliminating tx˙2(t)on both sides, we see that
0=x1(t)+x2(t−τ)−et+1−t.
Therefore, the strangeness-free formulation of the DDAE(6.15)is
•0 0 0 0
‚
˙ x(t)=
•1 0 0 1
‚ x(t)+
•0 1 0 0
‚
x(t−τ)+
•−et+1−t
−t
‚ , which implies that(6.15)is of neutral type.
The relative errors with respect to the exact solution x(t)=
•et t
‚
for the two methods are presented in Figure6.3. There, by GMOS (resp. BVP) we denote the relative errors of the solution obtained by the generalized method of steps (resp., the BVP method). The constant stepsize is h=0.01.
In the next example we consider a causal DDAE of advanced type.
Example 6.8. Consider the DDAE
•0 1 0 0
‚
˙ x(t)=
•1 0 0 1
‚ x(t)+
•1 0 0 1
‚
x(t−τ)+
•1−et−et−1 1−2t
‚
, (6.16)
in the time interval I= [0, 10τ], with the delay τ =1. From (6.16), one obtains the strangeness-free DDAE
•0 0 0 0
‚
˙ x(t)=
•1 0 0 1
‚ x(t)+
•1 0 0 1
‚
x(t−τ)+
•0 1 0 0
‚
˙
x(t−τ)+
•−1−et−et−1 1−2t
‚ , which implies that (6.16) is of advanced type. The relative errors with respect to the exact solution x(t)=
•et t
‚
are also presented in Figure6.3. There, by GMOS (resp. BVP) we denote the relative errors of the solution obtained by the generalized method of steps (resp., the BVP method). The constant stepsize is h=0.01.
In Figure6.3, we see that if a DDAE is of retarded or neutral type, then the general-ized method of steps seems to be more efficient than the BVP method. Otherwise, if a DDAE is of advanced type then the BVP method seems to be a better candidate. The
0 2 4 6 8 10 10−18
10−16 10−14 10−12 10−10
GMOSBVP
0 2 4 6 8 10
10−16 10−14 10−12 10−10 10−8
GMOSBVP
Figure 6.3:Relative error of the DDAE (6.15) (left) and of the DDAE (6.16) (right) with con-stant stepsizeh=0.01.
reason is that the generalized method of steps solves the discretized system on consec-utive intervals and hence, due to the advancedness of a DDAE, the error on each inter-val [(i−1)τ,iτ] is rapidly amplified when one proceeds to the next interval [iτ, (i+1)τ].
Furthermore, the BVP method can also handle advanced DDAEs, whose the strangeness-free formulations are of high order. This is demonstrated in the following example.
Example 6.9. Consider the DDAE 2
6 6 6 4
0 0 1 0 0 0 0 1 0 0 0 0 3 7 7 7 5 2 4
x˙1(t)
˙ x2(t)
˙ x3(t)
3 5=
2 6 6 6 4
0 1 0
0 0 1
0 0 0
H(t−τ) 0 0 3 7 7 7 5 2 4
x1(t) x2(t) x3(t) 3 5+
2 6 6 6 4
0 0 0 1 0 0 0 0 0 0 0 0 3 7 7 7 5 2 4
x1(t−τ) x2(t−τ) x3(t−τ) 3 5+
2 6 6 6 4
−t
−1−et−1 1
−H(t−τ)et 3 7 7 7 5 , (6.17) on the time intervalI=[0,`τ)with`=6,τ=1. Here H is the Heaviside function given by
H(t)=
(0, if t60, 1, if t>0.
By directly verifying Hypothesis5.22, one sees that the shift index of (6.17)is given by κ=
(1, if t6τ, 0, if t>τ.
By performing as in Example5.8, we deduce that the regular, strangeness-free formula-tion of the DDAE(6.17)takes the form
− 2 4
0 0 0 0 0 0 1 0 0 3 5 2 4
¨ x1(t)
¨ x2(t)
¨ x3(t)
3 5=
2 4
0 1 0 0 0 1 0 0 0 3 5
2 4
x1(t) x2(t) x3(t) 3 5+
2 4
0 0 0 1 0 0 0 0 0 3 5
2 4
x1(t−τ) x2(t−τ) x3(t−τ) 3 5
+ 2 4
1 0 0 0 0 0 0 0 0 3 5 2 4
˙ x1(t)
˙ x2(t)
˙ x3(t)
3 5+
2 4
−t−et−1
−1−et−1
−et 3
5, for t∈(0,τ),
(6.18a)
6.3. Examples 96 and
2 4 0 0 0 3 5=
2 4
0 1 0 0 0 1 1 0 0 3 5
2 4
x1(t) x2(t) x3(t) 3 5+
2 4
0 0 0 1 0 0 0 0 0 3 5
2 4
x1(t−τ) x2(t−τ) x3(t−τ) 3 5
+ 2 4
1 0 0 0 0 0 0 0 0 3 5 2 4
˙ x1(t)
˙ x2(t)
˙ x3(t)
3 5+
2 4
−t−et−1
−1−et−1
−et 3
5, for t∈(τ,`τ),
(6.18b)
Unfortunately, the generalized method of steps fails to determine the strangeness-free formulation(6.18), because it is not a first order DDAE. Furthermore, it is complicated to implement Runge-Kutta methods or BDF methods for(6.18)and the error in the ap-proximation of x(t˙ −τ)is also an important issue. For higher dimensional DDAEs, if the considered system is causal and of advanced type, then typically the strangeness-free formulation can contain high order derivatives of x(t)and x(t−τ), which makes it im-possible to use the generalized method of steps. The BVP method, however, can work fine in this case, if the strangeness indexµis sufficiently small. For example, the BVP method successfully handles the corresponding IVP for the DDAE (6.17), since the strangeness-index of the DAE (6.8a) is always two. The numerical solution and the relative error are presented in Figure6.4. Here we use again the three stage Gauß-Lobatto collocation method, as in [80], to the strangeness-free DDAE(6.9), which is pointwise computed au-tomatically. The constant stepsize is h=0.01.
0 2 4 6 8 10
−1 0 1 2 3 4
5 x1
x2 x3
0 2 4 6 8 10
10−20 10−15 10−10 10−5
error x1 error x2 error x3
Figure 6.4:The solution and the absolute error of the DDAE (6.17) with constant stepsize h=0.01.
Solvability Analysis of General Nonlinear DDAEs
This chapter is devoted to the solvability analysis of initial value problems for general nonlinear DDAEs of the form
F(t,x(t), ˙x(t),x(t−τ))=0, (7.1a) for allt∈I=(0,tf), together with an initial function
x(t)=φ(t), for allt∈[−τ, 0]. (7.1b) The extensions of the results in Chapters5,6for general nonlinear, noncausal DDAEs are desired. However, this is still an open question and therefore, in this chapter we only present some important results in prior investigations about the numerical anal-ysis of nonlinear DDAEs.
For notational convenience, we consider only real-valued problems where bothF and xare real vector-valued functions. To obtain the results for complex-valued problems, we can analyze the real and imaginary part of the equation and the unknown sepa-rately. Furthermore, let us restrict our consideration to square systems, i. e.,m=n.
Together with (7.1a), we are also interested in the associated DAE
F(t,x(t), ˙x(t))=0. (7.2)
In order to discuss the theoretical and numerical solution of the IVP (7.1), within this chapter we assume that the IVP (7.1) has a unique solution. Similar to the linear case, we see that the piecewise differentiable solution concept is more suitable for IVPs of DDAEs. Therefore, we recall this concept in the following definition.
Definition 7.1. i) A functionx:Iτ→Rnis called apiecewise differentiable solutionof (7.1a), if it is continuous, piecewise continuously differentiable and satisfies (7.1a) al-most everywhere. Throughout this chapter whenever we speak of a solution, we mean a piecewise differentiable solution.
ii) An initial functionφis calledconsistentif the IVP (7.1) has at least one solution.
iii) The DDAE (7.1a) is called solvableif it has at least one solution. It is called regu-97
98 lar if in addition, the solution to the IVP (7.1) is unique, provided a consistent initial function.
There are very few references about the solvability analysis of nonlinear DDAEs and among them we want to mention the following papers [5,60,68,86] for solution pro-cedures, and [10] for some difficulties that may occur for the numerical integration of DDAEs. Except [5], where the type of a DDAE is examined and the index reduction procedure is performed whenever it is necessary, the other references directly apply numerical methods to a given DDAE, without any regularization procedure.
Precisely, the authors showed in [5] that a retarded, Hessenberg DDAE of differentia-tion index at most three, takes the following form
˙
x(t) = f(x(t),x(t−τ),y(t),y(t−τ)),
0 = g(x(t),x(t−τ),y(t)), (7.3)
(where∂∂yg is nonsingular) for differentiation index one,
˙
x(t) = f(x(t),x(t−τ),y(t)),
0 = g(x(t)), (7.4)
(where∂∂gx∂∂yf is nonsingular) for differentiation index two,
˙
y(t) = f(x(t),x(t−τ),y(t),y(t−τ),z(t)),
˙
x(t) = g(x(t),x(t−τ),y(t)),
0 = h(x(t)),
(7.5)
(where∂h∂x ∂g∂y∂f∂z is nonsingular) for differentiation index three.
Furthermore, in the cases that the functiong in (7.3) is allowed to depend ony(t−τ) and g in (7.4) is allowed to depend on x(t−τ) then the corresponding system is of neutral type. Then, in [5] the authors constructed an index reduction procedure to derive adelay-essential-underlying ODE(DEUODE) and investigated the convergence and order of numerical methods like BDF methods, projected implicit Runge-Kutta methods applied to index one and index two retarded and neutral Hessenberg DDAEs.
It is also shown that a DDAE is (numerically) stable, or well-conditioned if its DEUODE is (numerically) stable.
Without considering any regularization or any index reduction procedure, in the article [60] the authors applied the 3-stage Radau IIA method and implemented it in the solver RADAR5 to compute the solution of IVPs for quasi-linear DDAEs of the form
My˙(t)=f(t,y(t),y(α(t,y(t)))), (7.6a) with the initial condition
y(t0)=y0, y(t)=g(t), for t<t0, (7.6b) where the matrixM is constant but allowed to be singular. The state-dependent delay α(t,y(t)) satisfiesα(t,y(t))6t for allt ≥t0. Note that this includes small delays, i.e.,
t−α(t,y(t))¿t, and vanishing delays, i.e.,α(t,y(t))=tfor somet.
The Radau IIA method has successfully handled a variety of systems including retarded and neutral DDEs and also DDAEs where the associated non-delayed DAE is regular and strangeness-free. However, since no regularization procedure was considered for the DDAE (7.6a), this method may not work, or it may give wrong results for general high index, noncausal DDAEs.
The other important studies are [68,86] where the authors considered retarded DDAEs in the semi-explicit form
˙
x(t) = f(t,x(t),y(t),x(α(t,x(t)))),
0 = g(t,x(t),y(t),x(α(t,x(t)))), (7.7) (where∂∂yg is nonsingular) for differentiation index one, and
˙
x(t) = f(t,x(t),y(t),x(α(t,x(t)))),
0 = g(t,x(t)), (7.8)
(where∂∂gx∂∂yf is nonsingular) for differentiation index two.
Here the delayα(t,x(t)) is state-dependent and non-vanishing, i.e., there existδ>0 such thatα(t,x)6t−δ. There in [68], the author investigated the convergence of col-location methods for the system (7.7), (7.8). One important contribution of the studies [60,68] is the discontinuity detection in the state dependent delay case.
We further note that all the investigation presented above, similar to the other in-vestigation about the solvability analysis of IVPs for linear DDAEs [25,29,83,122,142], are restricted to systems that have the following features:
i) The associated non-delayed DAE is uniquely solvable, and hence the considered DDAE is causal;
ii) The considered DDAE is of either retarded or neutral type.
Furthermore, prior work on DDAEs (for both linear and nonlinear systems) usually make use of the method of steps for the solution procedure. Because of this reason, we now recall the method of steps for nonlinear, causal DDAEs.
Introducing the vector-valued functions for eachi∈N xi: [0,τ] → Rn
t 7→ x(t+(i−1)τ), (7.9a)
and
Fi: [0,τ]×Rn×Rn×Rn → Rn
(t,xi(t), ˙xi(t),xi−1(t)) 7→ F(t+(i−1)τ,xi(t), ˙xi(t),xi−1(t)), (7.9b) we rewrite the IVP (7.1) as a sequence of DAEs
Fi(t, ˙xi(t),xi(t),xi−1(t))=0, (7.10a) for all t ∈ (0,τ), and for all i = 1, 2, . . . ,`. The function x0 is prescribed by
100 x0(t) :=φ(t−τ). Here equation (7.10a) is a parameter dependent DAE in variablexi with a function parameterxi−1. The initial condition for the DAE (7.10a) is
xi(0)=xi−1(τ). (7.10b)
Assuming that the DDAE (7.1a) is causal, it implies that the IVP (7.10) is uniquely solv-able for any sufficiently smooth function parameterxi−1and any consistent initial vec-torxi(0). Having found the functionxi, we can proceed in the same way to obtainxi+1 and so on. Therefore, the solutionx(t) to the IVP (7.1) is successfully constructed step by step, provided that an initial functionφis sufficiently smooth and satisfies certain consistency conditions.
In analogy to the linear DDAE case, we see that there are two unmentioned issues in prior investigations that motivates our future research:
1. How to categorize general nonlinear, noncausal DDAEs by their types (retarded, neutral and advanced) and to solve IVPs for retarded and neutral systems.
2. How to classify advanced DDAEs in order to figure out which class of advanced DDAEs can be efficiently integrated.
In this chapter we present some conclusions and some possible open problems found during the work.
Conclusion
The combination of differential-algebraic equations arising from an automatic model-ing based approach and the appearance of time delays due to the physical properties or feedback control naturally leads to delay differential-algebraic equations (DDAEs).
Despite of their natural importance and broad range of applications, DDAEs are not well understood even for fundamental problems such as the solvability analysis. In this thesis we have mainly studied the analytical and numerical solution to initial value problems for linear DDAEs.
Important properties inherited from the theories of DAEs and of DDEs lead to crit-ical consequences for DDAEs such as a solution concept, the discontinuity propaga-tion or a system classificapropaga-tion. However, the combinapropaga-tion between these two sub-classes DAEs and DDEs has lead to many interesting properties. In Chapter3we have discussed some characteristics of DDAEs, which are the starting points for important consequences associated with the computational solution of the corresponding IVPs.
In summary, initial value problems for DDAEs present four different difficulties:
1. The hidden type of a system, i. e., retarded, neutral or advanced.
2. The causality or noncausality of a system.
3. The linearity or nonlinearity of a system.
4. The type of delays and the number of delays.
Different from prior studies, which mostly considered the last two difficulties, in this work we have considered the first two difficulties. Until now, there is no treatment for all these four difficulties.
Chapter4addresses linear time invariant DDAEs, where all the system coefficients are constant matrices. The DDAEs are studied by three approaches: first, analyzing the structure of matrix triples, second, a matrix polynomial approach, and third, an al-gebraic method. An important contribution of this chapter is to point out the link be-tween the existence and uniqueness of solutions (of IVPs for DDAEs) and the regularity of either the matrix pair (E,A) or the matrix triple (E,A,B), depending on whether the time interval is bounded or not. This result allows to study the solvability of a DDAE by investigating spectral properties of its matrix coefficients. Furthermore, the
ma-101
102 trix polynomial approach is applicable for a much broader class of time delay systems including both underdetermined and overdetermined systems. Finally, another ap-proach in the theory of DAEs, namely an algebraic method, is examined to study gen-eral noncausal DDAEs of retarded and neutral types.
Chapter5studies the solvability analysis and reformulation of general linear time variable coefficients DDAEs, with a special focus on the numerical solution. IVPs for DDAEs are studied by two different approaches, which aim at different types of DDAEs.
While the first approach, the generalized method of steps, can successfully handle re-tarded and neutral DDAEs, the second one, the BVP method, is advantageous for study-ing advanced DDAEs.
The first approach, by introducing the shift index concept in order to estimate the noncausality of a DDAE, has extended the results in the theory of DAEs [75] to de-duce a strangeness-free formulation (and consequently, the underlying DDE), and also to obtain the necessary and sufficient conditions for a consistent initial function and a consistent inhomogeneity. To compute the solution of IVPs for DDAEs, numerical methods for DDEs are applied to the strangeness-free formulation, which is pointwise computed automatically. Furthermore, the nonadvancedness of a considered DDAE is checked by a verifiable condition.
Different from the first approach, the second approach removes the delay and studies IVPs for DDAEs as BVPs for DAEs. Thus, one does not need to worry about the delay and the type of the system. The interesting feature of this approach is that it is not be-ing limited to retarded and neutral DDAEs but clearly shows which advanced DDAEs (and consequently, advanced DDEs) can be efficiently integrated. This concept has not yet been mentioned in the literature, even for advanced DDEs.
Based on the two methods presented in Chapter 5, we have developed in Chap-ter6two strategies for the numerical integration of general linear DDAEs. The first strategy is based on the observation that the generalized method of steps is applica-ble for retarded and neutral DDAEs. As discussed in Chapter5, the strangeness-free formulation is pointwise computed in parallel with verifying the type of the consid-ered system to prevent the advanced situation. Following the method of steps, the Radau collocation method with Lagrange interpolation for the history function is then applied to this strangeness-free formulation. On the other hand, the second strategy makes use of well-known integration methods for the corresponding boundary value problems of the reformulated DAEs. The Gauß-Lobatto collocation method is applied to the strangeness-free formulation of the reformulated DAEs. It turns out that, for different type of DDAEs, one should choose different integration strategies. In more details, the generalized method of steps has shown better performance for retarded or neutral DDAEs, while the BVP method is more suitable for advanced DDAEs. The BVP method has shown the possibility to treat not only causal, advanced DDAEs but even noncausal, advanced DDAEs, whose the strangeness-free formulation is of high order and has complicated structure.
Outlook
There are many possible open research problems related to the work in this thesis. In the following, we list some related open problems and research directions, which may be of interest in the future.
Regularization and numerical solution to general nonlinear DDAEs. Linear DDAEs are treated within the scope of this thesis. However, most problems in real life appli-cations are nonlinear and therefore, a systematic study for nonlinear DDAEs is indis-pensable. Nevertheless, only two special classes of causal systems, namely Hessenberg systems ([5]) and semi-explicit, index one systems ([10], [60]) have been studied. The cases of advanced DDAEs and general noncausal DDAEs are still entirely open. We expect that the two approaches presented in Chapter 5can be extended to general nonlinear DDAEs.
Regularization of delay descriptor systems. DDAEs with control variable are often calleddelay descriptorsystems, which naturally occurs in many applications. Besides the two approaches presented in Chapter5, it would be interesting to extend the be-havior approach ([76]) to delay descriptor systems.
Fundamental control properties of delay descriptor systems. Once the regulariza-tion are well studied for delay descriptor systems, we wish to investigate fundamental control concepts such as stability, stabilization, controllability, observability, etc. Also of interest is the spectral analysis of Delay-DAEs and the computation of spectral ex-ponents and spectral intervals.
104
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