The introduction (Sect. 1) contains the review of natural scientific applications of boundary value problems (BVP) for differential-algebraic equations (DAEs), among them especially the references on optimal control problems, problems of electrical networks and multi-body systems. Also, here four substantive examples are presented.
This article deals with DAEs of the form \[ f((Dx)'(t), x(t),t)=0, \tag{1} \] most result remain valid for DAEs of standard form \[ \mathfrak{f}(x'(t), x(t),t)=0. \tag{2} \] The function \(f:\mathbb{R}^n\times \mathcal{D}_f\times \mathcal{I}_f \to \mathbb{R}^m\) (\(\mathcal{D}_f\times \mathcal{I}_f\subseteq \mathbb{R}^m \times \mathbb{R}\)) open is continuous and has continuous partial derivatives \(f_y\) and \(f_x\) with respect to the first two variables \(y \in \mathbb{R}^n\), \(x\in \mathcal{D}_f\), the partial Jacobian \(f_y(y,x,t)\) is everywhere singular, the matrix function \(D(t):\mathcal{I}_f\to \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)\) is at least continuous, often continuously differentiable, and \(D(t)\) has constant rank \(r\) on the given interval \(\mathcal{I}_f\); always, im D is supposed to be a \(\mathcal{C}^1\)-subspace varying in \(\mathbb{R}^n\). Two-point boundary conditions \(g(x(a),x(b))=0\) are considered, \(a,b \in \mathcal{I}_f\), where the function \(g:\mathcal{D}_f\times \mathcal{D}_f\) is continuous differentiable, the number \(l\leq m\) of boundary conditions depends on the structure of the DAEs.
Everywhere below classical solutions (1), (2) are lucking for, i.e., belonging to the interval \(\mathcal{I}\subseteq \mathcal{I}_f\) with values \(x(t)\in \mathcal{D}_f\), \(t\in \mathcal{I}\) and satisfying the DAE pointwise on \(\mathcal{I}\).
Then, on a compact interval \(\mathcal{I}\), the Banach functional spaces \(\mathcal{C}(\mathcal{I},\mathbb{R}^m)\) and \(\mathcal{C}^1_D(\mathcal{I},\mathbb{R}^m)\) are defining by the norms \(\|x\|_{\infty}:=\max_{t\in \mathcal{I}}|x(t)|\), \(x\in \mathcal{C}(\mathcal{I},\mathbb{R}^m)\) and \(\|x\|_{\mathcal{C}^1_D}:=\|x\|_{\infty}+\|(Dx)'\|_{\infty}\), \(x\in \mathcal{C}^1_D(\mathcal{I},\mathbb{R}^m)\).
Section 2 is devoted to the analytical theory of DAEs and contains the following subsections: Basic assumptions and terminology; The flow structure of regular linear DAEs; Accurately stated two-point boundary conditions; Conditioning constants and dichotomy; Nonlinear BVPs; Other boundary conditions; Further references, comments, and open questions. In Subsect. 2.1, three definitions of the considered BVP for DAEs are introduced.
Subsect. 2.2 is devoted to the flow structure of regular linear DAEs \(A(t)(Dx)'(t)+B(t)x(t)=q(t)\), \(t\in \mathcal{I}\). Solvability statements for the regular BVPs \( A(Dx)'+Bx=q\), \(G_ax(a)+G_bx(b)=\gamma\) with some restrictions on projectors are presented in Subsect. 2.3. Classical theory for ordinary BVPs is generalized on DAEs in theorems of Subsect. 2.4.
In Subsect. 2.5.1 (2.5.2), a series of equivalence conditions to the nonlinear BVP (1), (2) is established as well as their well-posedness in the natural and advanced setting. Here some results for index-1 and index-2 DAEs with the perturbed initial conditions are marked.
In Subsect. 2.6 for the nonlinear DAE (1), three types of boundary conditions are considered.
Subsect. 2.7 contains further references, comments and open problems.
Sect. 3 is devoted to collocation methods for the well-posed DAEs and contains Subsect 3.1 BVPs well-posed in the natural setting devoted to the BVPs \[ f((Dx)'(t),x(t),t)=0,\;\; g(x(a),x(b)=0, \] with basic assumptions from Subsect 2.1 and additionally in \(D(t)=\mathbb{R}^n\), \(t\in [a,b]=\mathcal{I}\), satisfying criteria for their local well-posedness in the natural setting, Subsect. 3.2 Partitioned equations (Partitioned component approximations); in Subsect. 3.3 separately for BVPs of index-2 DAEs; Subsect. 3.4 BVPs for singular index-1 DAEs in linear and nonlinear cases; Subsect. 3.5 Defect-based a posteriori error estimation for index-DAEs
Sect. 4 is devoted to the review of shooting numerical methods, at first for linear DAEs (computation of consistent initial values, single shooting, multiple shooting), then to nonlinear index-1 DAEs.
The Subsect. 5.1 of Sect. 5 Miscellaneous says about periodical solutions of DAEs. The works devoted to Lyapounov stability and Floquet exponents are marked, Subsect. 5.2 gives the presentation of the Abramov transfer method for index-1 and index-2 DAEs. Subsect. 5.3 says on Finite difference methods in DAEs that they turn out to be less efficient than collocation methods, the same is true for BVPs in DAEs. Two theorems from Subsect. 5.4 allow to formulate the BVP for (1), (2) as an operator equation. Therefore, to such DAEs, the Newton-Kantorovich method turns out to be applicable, that is done in the articles by R. März [Appl. Numer. Math. 18, No. 1–3, 267–292 (1995; Zbl 0840.65071)] and by T. Petry [Realisierung des Newton-Kantorovich-Verfahrens für nichtlineare Algebro-Differentialgleichungen mittels Abramov-Transfer. Berlin: Logos Verlag (1998; Zbl 0905.65090)]] and reflected in Subsect. 5.4. par The Appendix 6 contains two subsections: Basics concerning regular DAEs and List of symbols and abbreviations.
For the entire collection see [Zbl 1333.65004].