# zbMATH — the first resource for mathematics

Discontinuous solutions of semilinear differential-algebraic equations. I: Distribution solutions. (English) Zbl 0865.34003
Continuing previous work, the authors are studying the existence of classical and generalized solutions of semilinear implicit differential equations of the form: $$A(t) x'=G(t,x)$$, where $$A(t)$$, $$t\in J\subset\mathbb{R}$$ are $$n\times n$$ singular $$C^\infty$$-matrices of constant rank $$r<n$$.
Assuming that the system is “geometrically nonsingular of index 1” in a rather restrictive sense, the authors prove the existence and uniqueness of classical (smooth) solutions through some initial points as well as the existence of certain one-sided solutions and also the existence of generalized solutions in the sense of distributions through certain impasse points.

##### MSC:
 34A09 Implicit ordinary differential equations, differential-algebraic equations 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
Full Text:
##### References:
 [1] Rabier, P.J.; Rheinboldt, W.C., On impasse points of quasilinear differential-algebraic equations, J. math. analysis applic., 181, 429-454, (1994) · Zbl 0803.34004 [2] Rabier, P.J.; Rheinboldt, W.C., On the computation of impasse points of quasilinear differential-algebraic equations, Math. comp., 62, 133-154, (1994) · Zbl 0793.65050 [3] Chua, L.O.; Deng, A.-C., Impasse points. part I: numerical aspects, Int. J. circ. theor. applic., 17, 213-235, (1989) · Zbl 0676.94022 [4] Sastry, S.S.; Desoer, C.A., Jump behavior of circuits and systems, IEEE trans. circ. syst., 28, 1109-1124, (1981) · Zbl 0476.93036 [5] Takens, F., () [6] RABIER P. J. & RHEINBOLDT W. C., Classical and generalized solutions of time-dependent linear DAEs, Lin. Alg. Appl. (to appear). · Zbl 0792.65048 [7] RABIER P. J. & RHEINBOLDT W. C., Time-dependent linear DAEs with discontinuous inputs, Lin. Alg. Appl. (to appear). · Zbl 0864.65044 [8] Rabier, P.J.; Rheinboldt, W.C., A geometric treatment of implicit differential-algebraic equations, J. diff. eqns, 109, 110-146, (1994) · Zbl 0804.34004 [9] CAMPBELL S. L. & GRIEPENTROG E., Solvability of general differential-algebraic equations, SIAM J. Sci. Comp. (to appear). · Zbl 0821.34005 [10] Rabier, P.J., Implicit differential equations near a singular point, J. math. analysis applic., 44, 425-449, (1989) · Zbl 0683.34017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.